Introduction Optimization via Adaptive Model Reduction Large-Scale, Constrained Optimization Conclusion Accelerating PDE-Constrained Optimization Problems using Adaptive Reduced-Order Models Matthew J. Zahr Advisor: Charbel Farhat Computational and Mathematical Engineering Stanford University Lawrence Berkeley National Laboratory, Berkeley, CA February 9, 2016 Zahr PDE-Constrained Optimization with Adaptive ROMs
Advisor Charbel FarhatComputational and Mathematical Engineering
Stanford University
Lawrence Berkeley National Laboratory Berkeley CAFebruary 9 2016
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Multiphysics Optimization Key Player in Next-Gen Problems
Current interest in computational physics reaches far beyond analysis of a singleconfiguration of a physical system into design (shape and topology1) controland uncertainty quantification
14
Better simulation is critical to engine design and calibration optimization
bull Three levels of simulation with different purposes
‒ Predictive combustion combustion optimization and methods development
‒ Full engine simulation engine system optimization and model-based controls
‒ Full vehicle simulation technology interactions component optimization and supervisory controls
bull Each scale requires different level of fidelity
‒ High fidelity combustion on vehicle scales computationally impossible (at the moment)
bull Exponential increase in parameter space translates to exponential increase in simulation space and computational requirements
TAKEAWAY Need for faster simulation faster optimization methods and reduced models for on-board controls
Engine System
EM Launcher Micro-Aerial Vehicle
1Emergence of additive manufacturing technologies has made topology optimizationincreasingly relevant particularly in DOE
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Topology Optimization and Additive Manufacturing2
Emergence of AM has made TO anincreasingly relevant topic
AM+TO lead to highly efficient designsthat could not be realized previously
Challenges smooth topologies requirevery fine meshes and modeling ofcomplex manufacturing process
2MIT Technology Review Top 10 Technological Breakthrough 2013
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
PDE-Constrained Optimization I
Goal Rapidly solve PDE-constrained optimization problem of the form
minimizeuisinRnu microisinRnmicro
J (u micro)
subject to r(u micro) = 0
where
r Rnu times Rnmicro rarr Rnu is the discretized partial differential equation
J Rnu times Rnmicro rarr R is the objective function
u isin Rnu is the PDE state vector
micro isin Rnmicro is the vector of parameters
red indicates a large-scale quantity O(mesh)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
micro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro) micro
u
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
dJdmicro (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Multiphysics Optimization Key Player in Next-Gen Problems
Current interest in computational physics reaches far beyond analysis of a singleconfiguration of a physical system into design (shape and topology1) controland uncertainty quantification
14
Better simulation is critical to engine design and calibration optimization
bull Three levels of simulation with different purposes
‒ Predictive combustion combustion optimization and methods development
‒ Full engine simulation engine system optimization and model-based controls
‒ Full vehicle simulation technology interactions component optimization and supervisory controls
bull Each scale requires different level of fidelity
‒ High fidelity combustion on vehicle scales computationally impossible (at the moment)
bull Exponential increase in parameter space translates to exponential increase in simulation space and computational requirements
TAKEAWAY Need for faster simulation faster optimization methods and reduced models for on-board controls
Engine System
EM Launcher Micro-Aerial Vehicle
1Emergence of additive manufacturing technologies has made topology optimizationincreasingly relevant particularly in DOE
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Topology Optimization and Additive Manufacturing2
Emergence of AM has made TO anincreasingly relevant topic
AM+TO lead to highly efficient designsthat could not be realized previously
Challenges smooth topologies requirevery fine meshes and modeling ofcomplex manufacturing process
2MIT Technology Review Top 10 Technological Breakthrough 2013
Zahr PDE-Constrained Optimization with Adaptive ROMs
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PDE-Constrained Optimization I
Goal Rapidly solve PDE-constrained optimization problem of the form
minimizeuisinRnu microisinRnmicro
J (u micro)
subject to r(u micro) = 0
where
r Rnu times Rnmicro rarr Rnu is the discretized partial differential equation
J Rnu times Rnmicro rarr R is the objective function
u isin Rnu is the PDE state vector
micro isin Rnmicro is the vector of parameters
red indicates a large-scale quantity O(mesh)
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
micro
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro) micro
u
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
dJdmicro (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Topology Optimization and Additive Manufacturing2
Emergence of AM has made TO anincreasingly relevant topic
AM+TO lead to highly efficient designsthat could not be realized previously
Challenges smooth topologies requirevery fine meshes and modeling ofcomplex manufacturing process
2MIT Technology Review Top 10 Technological Breakthrough 2013
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
PDE-Constrained Optimization I
Goal Rapidly solve PDE-constrained optimization problem of the form
minimizeuisinRnu microisinRnmicro
J (u micro)
subject to r(u micro) = 0
where
r Rnu times Rnmicro rarr Rnu is the discretized partial differential equation
J Rnu times Rnmicro rarr R is the objective function
u isin Rnu is the PDE state vector
micro isin Rnmicro is the vector of parameters
red indicates a large-scale quantity O(mesh)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
micro
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro) micro
u
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
dJdmicro (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
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Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
PDE-Constrained Optimization I
Goal Rapidly solve PDE-constrained optimization problem of the form
minimizeuisinRnu microisinRnmicro
J (u micro)
subject to r(u micro) = 0
where
r Rnu times Rnmicro rarr Rnu is the discretized partial differential equation
J Rnu times Rnmicro rarr R is the objective function
u isin Rnu is the PDE state vector
micro isin Rnmicro is the vector of parameters
red indicates a large-scale quantity O(mesh)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
micro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro) micro
u
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
dJdmicro (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
micro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro) micro
u
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
dJdmicro (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
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Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
micro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro) micro
u
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
dJdmicro (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
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Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
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Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro) micro
u
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
dJdmicro (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro) micro
u
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
dJdmicro (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Nested Approach to PDE-Constrained Optimization
Virtually all expense emanates from primaldual PDE solvers
Primal PDE Dual PDE
Optimizer
J (u micro)
dJdmicro (u micro)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Projection-Based Model Reduction to Reduce PDE Size
Model Order Reduction (MOR) assumption state vector lies inlow-dimensional subspace
u asymp Φuurpartu
partmicroasymp Φu
parturpartmicro
where
Φu =[φ1u middot middot middot φku
u
]isin Rnutimesku is the reduced basis
ur isin Rku are the reduced coordinates of unu ku
Substitute assumption into High-Dimensional Model (HDM) r(u micro) = 0and project onto test subspace Ψu isin Rnutimesku
ΨuTr(Φuur micro) = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Connection to Finite Element Method Hierarchical Subspaces
S
Sh
Skh
S - infinite-dimensional trial space
Sh - (large) finite-dimensional trial space
Skh - (small) finite-dimensional trial space
Skh sub Sh sub S
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Few Global Data-Driven Basis Functions v Many Local Ones
Instead of using traditional localshape functions (eg FEM) useglobal shape functions
Instead of a-priori analyticalshape functions leverage data-richcomputing environment by usingdata-driven modes
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton0)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
]Use Proper Orthogonal Decomposition to generate reduced basis for eachindividually
ΦX = POD(X)
ΦY = POD(Y )
Concatenate to get reduced-order basis
Φu =[ΦX ΦY
]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of Ψu Minimum-Residual ROM
Least-Squares Petrov-Galerkin (LSPG)3 projection
Ψu =partr
partuΦu
Minimum-Residual Property
A ROM possesses the minimum-residual property if Ψur(Φuur micro) = 0 isequivalent to the optimality condition of (Θ 0)
minimizeurisinRku
||r(Φuur micro)||Θ
Implications
Recover exact solution when basis not truncated (consistent3)Monotonic improvement of solution as basis size increasesEnsures sensitivity information in Φ cannot degrade state approximation4
LSPG possesses minimum-residual property
3[Bui-Thanh et al 2008]4[Fahl 2001]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Traditional sensitivity analysis
parturpartmicro
=minus
Nsumj=1
rjΦuT partrjpartupartu
Φu +
(partr
partuΦu
)Tpartr
partuΦu
minus1
Nsumj=1
rjΦuT part2rjpartupartmicro
+
(partr
partuΦu
)Tpartr
partmicro
+ Guaranteed to give rise to exact derivatives of ROM quantities of interest
- Requires 2nd derivatives of r
- Φupartur
partmicro not guaranteed to be good approximate to full sensitivity partupartmicro
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Definition of parturpartmicro Minimum-Residual Reduced Sensitivities
Minimum-residual sensitivity analysis
parturpartmicro
= arg mina||Φuaminus
partu
partmicro||Θ = minus
[(partr
partuΦu
)Tpartr
partuΦu
]minus1(partr
partuΦu
)Tpartr
partmicro
+ Minimum-residual property - Φuparturpartmicro
is Θ-optimal solution topartu
partmicroin Φu
+ Does not require 2nd derivatives of r
-parturpartmicro6= partur
partmicro ie it is not the true ROM sensitivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM
ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Offline-Online Approach to Optimization
Offline
HDM
HDM
HDM
HDM
CompressROB Φ
ROM
Optimizer
Schematic
micro-space
HDM HDM middot middot middot HDM HDM ROB
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
RO
M
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Numerical Demonstration Offline-Online Breakdown
Optimal Solution (ROM) Optimal Solution (HDM)
HDM Solution ROB Construction Greedy Algorithm ROM Optimization
284times 103 s 548times 104 s 167times 105 s 30 s
126 2436 7437 001
HDM Optimization 197times 104 s
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M
middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M
middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M
middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M
middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M
middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M
middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM-Based Trust-Region Framework for Optimization
Compress
HDM
HDM
HDM
ROB Φ
ROM
Optimizer
Schematic
micro-space
HDM ROB
RO
M
RO
M
RO
M middot middot middot
RO
M
RO
M
HDM ROBR
OM
RO
M
RO
M middot middot middot
RO
M
RO
M middot middot middot
Breakdown of Computational Effort
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training
2 Step computation Approximately solve the reduced optimization problemwith non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training2 Step computation Approximately solve the reduced optimization problem
with non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training2 Step computation Approximately solve the reduced optimization problem
with non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training2 Step computation Approximately solve the reduced optimization problem
with non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Non-Quadratic Trust-Region Method with Adaptive Reduced-Order Models
1 Initialization Build Φu from sparse training2 Step computation Approximately solve the reduced optimization problem
with non-quadratic trust-region for a candidate microk
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Convergence to Critical Point of Unreduced Problem
Lim-Inf Convergence to Critical Point of Unreduced Optimization Problem
Let microk be a sequence of iterations produced by the Algorithm and suppose
J (u(microk)microk) = J (Φuur(microk)microk)
There exists ξ gt 0 such that
||nablaJ (u(microk)microk)minusnablaJ (Φuur(microk)microk)|| le ξ||nablaJ (Φuur(microk)microk)||
There exists ζ gt 0 such that for all micro isin micro | ||r(Φuur(micro)micro)|| le ∆k
|J (u(micro)micro)minus J (Φuur(micro)micro)| le ζ||r(Φuur(micro)micro)||
Thenlim infkrarrinfin
||nablaJ (u(microk) microk)|| = 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Assumptions of Convergence Theory Hold
If microk is a training point then
Minimum-residual formulation for theprimal reduced-order model implies
J (u(microk)microk) = J (Φuur(microk)microk)
Minimum-residual formulation for thereduced-order model sensitivity implies
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Assumptions of Convergence Theory Hold
If microk is a training point then
Minimum-residual formulation for theprimal reduced-order model implies
J (u(microk)microk) = J (Φuur(microk)microk)
Minimum-residual formulation for thereduced-order model sensitivity implies
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
ROM Solver Requires 4times Fewer HDM Queries
0 5 10 15 20 25 3010minus15
10minus11
10minus7
10minus3
101
Number of HDM queries
Ob
ject
ive
Fu
nct
ion
HDM-based optimizationROM-based optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
At the Cost of ROM Queries
0 20 40 60 80 100 120 140 16010minus18
10minus14
10minus10
10minus6
10minus2
Reduced optimization iterations
Ob
ject
ive
Fu
nct
ion
HDM sample0
20
40
60
RO
Msi
ze
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Next Shape Optimization of Full Aircraft (CRM)
ROMs are fast accurate and require limited resources
HDM solution (Drag = 142336kN) ROM solution (Drag = 142304kN)
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
At the Cost of ROM Queries
0 20 40 60 80 100 120 140 16010minus18
10minus14
10minus10
10minus6
10minus2
Reduced optimization iterations
Ob
ject
ive
Fu
nct
ion
HDM sample0
20
40
60
RO
Msi
ze
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Next Shape Optimization of Full Aircraft (CRM)
ROMs are fast accurate and require limited resources
HDM solution (Drag = 142336kN) ROM solution (Drag = 142304kN)
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Model Order ReductionNon-Quadratic Trust-Region SolverShape Optimization Airfoil Design
Next Shape Optimization of Full Aircraft (CRM)
ROMs are fast accurate and require limited resources
HDM solution (Drag = 142336kN) ROM solution (Drag = 142304kN)
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
PDE-Constrained Optimization II
Goal Rapidly solve PDE-constrained optimization problem of the form
minimizeuisinRnu microisinRnmicro
J (u micro)
subject to r(u micro) = 0
c(u micro) ge 0
where
r Rnu times Rnmicro rarr Rnu is the discretized partial differential equation
J Rnu times Rnmicro rarr R is the objective function
c Rnu times Rnmicro rarr Rnc are the side constraints
u isin Rnu is the PDE state vector
micro isin Rnmicro is the vector of parameters
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Problem Setup
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK6
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD7)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
6[Bonet and Wood 1997 Belytschko et al 2000]7[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
Restrict parameter to a low-dimensional subspace
micro asymp Φmicromicror
Φmicro =[φ1micro middot middot middot φ
kmicromicro
]isin Rnmicrotimeskmicro is the reduced basis
micror isin Rkmicro are the reduced coordinates of micronmicro kmicro
Substitute restriction into reduced-order model to obtain
ΦuTr(Φuur Φmicromicror) = 0
Related work[Maute and Ramm 1995 Lieberman et al 2010 Constantine et al 2014]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
micro-space Background mesh
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
micro-space Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Problem Setup
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK6
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD7)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
6[Bonet and Wood 1997 Belytschko et al 2000]7[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
Restrict parameter to a low-dimensional subspace
micro asymp Φmicromicror
Φmicro =[φ1micro middot middot middot φ
kmicromicro
]isin Rnmicrotimeskmicro is the reduced basis
micror isin Rkmicro are the reduced coordinates of micronmicro kmicro
Substitute restriction into reduced-order model to obtain
ΦuTr(Φuur Φmicromicror) = 0
Related work[Maute and Ramm 1995 Lieberman et al 2010 Constantine et al 2014]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
micro-space Background mesh
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
micro-space Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
Restrict parameter to a low-dimensional subspace
micro asymp Φmicromicror
Φmicro =[φ1micro middot middot middot φ
kmicromicro
]isin Rnmicrotimeskmicro is the reduced basis
micror isin Rkmicro are the reduced coordinates of micronmicro kmicro
Substitute restriction into reduced-order model to obtain
ΦuTr(Φuur Φmicromicror) = 0
Related work[Maute and Ramm 1995 Lieberman et al 2010 Constantine et al 2014]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
micro-space Background mesh
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
micro-space Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
micro-space Background mesh
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
micro-space Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Restrict Parameter Space to Low-Dimensional Subspace
micro-space Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
Selection of Φmicro amounts to arestriction of the parameter space
Adaptation of Φmicro should attemptto include the optimal solution inthe restricted parameter spaceie microlowast isin col(Φmicro)
Adaptation based on first-orderoptimality conditions of HDMoptimization problem
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Optimality Conditions to Adapt Reduced-Order Basis Φmicro
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Lagrangian Gradient Refinement Indicator
From Lagrange multiplier estimates only KKT condition not satisfiedautomatically
nablamicroL(micro λ) = 0
Use |nablamicroL(micro λ)| as indicator for refinement of discretization of micro-space
micro |nablamicroL(micro λ)|
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Constraints may lead to infeasible sub-problems
Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]
minimizeurisinRku microrisinRkmicro
J (Φuur Φmicromicror)
subject to c(Φuur Φmicromicror) ge 0
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Constraints may lead to infeasible sub-problems
Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]
minimizeurisinRku microrisinRkmicro
J (Φuur Φmicromicror)
subject to c(Φuur Φmicromicror) ge 0
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Constraints may lead to infeasible sub-problems
Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]
minimizeurisinRku microrisinRkmicro
J (Φuur Φmicromicror)
subject to c(Φuur Φmicromicror) ge 0
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Constraints may lead to infeasible sub-problems
Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]
minimizeurisinRku microrisinRkmicro
J (Φuur Φmicromicror)
subject to c(Φuur Φmicromicror) ge 0
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Constraints may lead to infeasible sub-problems
Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]
minimizeurisinRku microrisinRkmicro
J (Φuur Φmicromicror)
subject to c(Φuur Φmicromicror) ge 0
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Constraints may lead to infeasible sub-problems
Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]
minimizeurisinRku microrisinRkmicro
J (Φuur Φmicromicror)
subject to c(Φuur Φmicromicror) ge 0
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Constraints may lead to infeasible sub-problems
Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]
minimizeurisinRku microrisinRkmicro
J (Φuur Φmicromicror)
subject to c(Φuur Φmicromicror) ge 0
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Constraints may lead to infeasible sub-problems
Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]
minimizeurisinRku microrisinRkmicro
J (Φuur Φmicromicror)
subject to c(Φuur Φmicromicror) ge 0
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Constraints may lead to infeasible sub-problems
Non-Quadratic Trust-Region MOR [Zahr and Farhat 2014]
minimizeurisinRku microrisinRkmicro
J (Φuur Φmicromicror)
subject to c(Φuur Φmicromicror) ge 0
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Elastic constraints to circumvent infeasible subproblems
Constrained Non-Quadratic Trust-Region MOR (CNQTR-MOR)
minimizeurisinRku microrisinRkmicro tisinRnc
J (Φuur Φmicromicror)minus γtT1
subject to c(Φuur Φmicromicror) ge t
ΨuTr(Φuur Φmicromicror) = 0
||r(Φuur Φmicromicror)|| le ∆
t le 0
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Compliance Minimization 2D Cantilever
25
40
16000 8-node brick elements 77760 dofs
Total Lagrangian form finite strain StVK9
St Venant-Kirchhoff material
Sparse Cholesky linear solver (CHOLMOD10)
Newton-Raphson nonlinear solver
Minimum compliance optimization problem
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u micro) = 0
Gradient computations Adjoint method
Optimizer SNOPT [Gill et al 2002]
Maximum ROM size ku le 5
9[Bonet and Wood 1997 Belytschko et al 2000]10[Chen et al 2008]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Order of Magnitude Speedup to Suboptimal Solution
HDM CNQTR-MOR + Φmicro adaptivity
HDM Solution HDM Gradient HDM Optimization
7458s (450) 4018s (411) 8284s
HDMElapsed time = 19761s
HDM Solution HDM Gradient ROB Construction ROM Optimization
1049s (64) 88s (9) 727s (56) 39s (3676)
CNQTR-MOR + Φmicro adaptivityElapsed time = 2197s Speedup asymp 9x
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Better Solution after 64 HDM Evaluations
HDM CNQTR-MOR + Φmicro adaptivity
CNQTR-MOR + Φmicro adaptivity superior approximation to optimalsolution than HDM approach after fixed number of HDM solves (64)
Reasonable option to warm-start HDM topology optimization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 0 (1000)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
Macro-element Evolution
Iteration 1 (977)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Reduction of High-Dimensional Parameter SpaceElastic Nonlinear ConstraintsTopology Optimization 2D Cantilever
CNQTR-MOR + Φmicro adaptivity
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton1)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
An Adaptive Reduction Framework for Optimization under Uncertainty
Highly volatile systems tend to be plaguedby uncertainties which must be quantifiedfor meaningful problem formulation
Optimize moments of quantities of interestof stochastic partial differential equation
minimizeuisinRnu microisinRnmicro
intΞ
J (u micro ξ) dξ
subject to r(u micro ξ) = 0 ξ isin Ξ
Combine adaptive model reductionframework with dimension-adaptive sparsegrids to enable stochastic optimization
14
Better simulation is critical to engine design and calibration optimization
bull Three levels of simulation with different purposes
‒ Predictive combustion combustion optimization and methods development
‒ Full engine simulation engine system optimization and model-based controls
‒ Full vehicle simulation technology interactions component optimization and supervisory controls
bull Each scale requires different level of fidelity
‒ High fidelity combustion on vehicle scales computationally impossible (at the moment)
bull Exponential increase in parameter space translates to exponential increase in simulation space and computational requirements
TAKEAWAY Need for faster simulation faster optimization methods and reduced models for on-board controls
Engine System
EM Launcher
Collaborators Drew Kouri (Sandia NM) Kevin Carlberg (Sandia CA)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
High-Order Methods for Optimization of Conservation Laws
Derived implemented fully discrete adjoint method for globally high-orderdiscretization of conservation laws on deforming domains
Incorporation of time-periodicity constraints
Energy = 94096e+00Thrust = 17660e-01
Energy = 49476e+00Thrust = 25000e+00
Energy = 46110e+00Thrust = 25000e+00
Initial Optimal ControlOptimal
ShapeControl
Collaborators Per-Olof Persson (UCB LBNL) Jon Wilkening (UCB LBNL)
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton2)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
High-Order Methods for Optimization of Conservation Laws
Derived implemented fully discrete adjoint method for globally high-orderdiscretization of conservation laws on deforming domains
Incorporation of time-periodicity constraints
Energy = 94096e+00Thrust = 17660e-01
Energy = 49476e+00Thrust = 25000e+00
Energy = 46110e+00Thrust = 25000e+00
Initial Optimal ControlOptimal
ShapeControl
Collaborators Per-Olof Persson (UCB LBNL) Jon Wilkening (UCB LBNL)
Zahr PDE-Constrained Optimization with Adaptive ROMs
var ocgs=hostgetOCGs(hostpageNum)for(var i=0iltocgslengthi++)if(ocgs[i]name==MediaPlayButton2)ocgs[i]state=false
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
proposed domain decomposition procedure will provide ldquooptimalrdquo global modes ndash at least in representing thepreviously available data ndash that will be used as the global shape functions in g3 Standard finite element ordiscontinuous Galerkin shape functions will comprise the local basis functions and adaptive mesh refinementwill be used to ensure as lean a discretization as possible The proposed mixture of local and global basisfunctions introduced at the continuous level leads to a trial space that supports discontinuous solutionsalong the interface g l Lagrange multipliers [12] penalty methods [13] and numerical fluxes [14] willbe considered to ensure a solution with appropriate continuity is computed Figure 6 provides a motivatingexample for this project using a simple heaving airfoil The vorticity snapshot in Figure 6a shows localpropagating vortices that will be dicult to capture with a single global basis A potential decompositionof into l (white mesh) and g = l is shown in Figure 6b
(a) Vorticity around heaving airfoil (b) Potential l g decomposition (c) Idealized sparsity structure
Figure 6 Left Vorticity around heaving NACA0012 airfoil Middle Possible feature-based decomposition of domain into l
ndash white triangulation ndash and g = l Right Schematic of sparsity of linear of equations arising from mixed local-globaldiscretization blue dot ( ) represents individual non-zero entry and red patches are dense rowscolumns
At this point a number of critical research tasks emerge The proposed method ndash and model reductionmethods in general ndash would benefit greatly from technology that enhances the utility of global modes ndashthereby minimizing the reliance on local shape functions which rapidly contribute to the dimensionality ofthe problem A severe drawback of global basis functions is they associate features with spatial locationsThis is a powerful result in that it enables a single vector to capture the most complex features yet limitedin that global vectors cannot ecrarrectively ldquomoverdquo features That is their ability to capture features in onelocation says nothing about their ability to capture the same feature in a dicrarrerent location Preliminaryattempts to propagatetransform features in basis functions based on the concept of Lax pairs [15] havebeen introduced I will consider an approach that implicitly transforms the global basis vectors by applyinga transformation to the underlying mesh ndash ecrarrectively modifying the association of features with fixed spatiallocations For example consider a global basis capable of representing a discontinuity at a given spatiallocation A transformation of the underlying spatial domain will modify the location (and possibly shape)of discontinuity that can be captured
Another diculty associated with the proposed approach is the integration of quantities arising in thevariational equation against the global basis functions In finite element methods the impact of ldquovariationalcrimesrdquo ndash associated with employing numerical quadrature in place of exact integration ndash are minimized byusing suciently small element sizes or high quadrature orders to ensure accurate integration The generalityof g precludes these approaches To mitigate issues related to variational crimes I propose the use of anintegration mesh ie a mesh of g that will be used only to integrate terms in the variational formulation
3This is equivalent to using proper orthogonal decomposition and the method of snapshots [11] using the previously availabledata in g
8
Methods to transform features in global basis functions - minimize relianceon local shape functions
Linear algebra for sparse operators with a few dense rows and columns
Elements of high-order methods (Mathematics Group) adaptive meshrefinement (Applied Numerical Algorithms Group and Center forComputational Science and Engineering) numerical linear algebra(Scalable Solvers Group)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Fewer Queries Second-Order Methods for Accelerated Convergence
Hessian information highly desired in optimization and UQ but expensive due toO(Nmicro) required linear system solves
SensitivityAdjoint Method for Computing Hessian
d2Jdmicrojdmicrok
=part2J
partmicrojpartmicrok+
part2Jpartmicrojpartu
partu
partmicrok+
partu
partmicroj
T part2Jpartupartmicrok
+partu
partmicroj
T part2Jpartupartu
partu
partmicrok
minus partJpartu
partr
partu
minus1 [ part2r
partmicrojpartmicrok+
part2r
partmicrojpartu
partu
partmicrok+
part2r
partmicrokpartu
partu
partmicroj+
part2r
partupartupartu
partmicrojotimes partu
partmicrok
]where
partu
partmicroj=
partr
partu
minus1 partr
partmicroj
Fast multiple right-hand side linear solver by building data-driven subspace
for image ofpartr
partu
minus1
partr
partu
minusT
MOR concepts in context of numerical linear algebra (Scalable SolversGroup)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Leveraging Inexactness For Acceleration of Many-Query Multiphysics Problems
Framework introduced for accelerating PDE-constrainedoptimization problems with side constraints andlarge-dimensional parameter space
Adaptive reduction of state and parameter spaces
Applied to aerodynamic design and topology optimization
Order of magnitude speedup speedup observedCompetitive warm-start method
Future work combine advantages of MORAMR fordrastic computational savings with in-situ trainingsecond-order methods for rapidly converging many-queryalgorithms new (multiphysics) applications
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Acknowledgement
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References I
Barbic J and James D (2007)
Time-critical distributed contact for 6-dof haptic rendering of adaptively sampled reduceddeformable models
In Proceedings of the 2007 ACM SIGGRAPHEurographics symposium on Computeranimation pages 171ndash180 Eurographics Association
Barrault M Maday Y Nguyen N C and Patera A T (2004)
An empirical interpolation method application to efficient reduced-basis discretization ofpartial differential equations
Comptes Rendus Mathematique 339(9)667ndash672
Belytschko T Liu W Moran B et al (2000)
Nonlinear finite elements for continua and structures volume 26
Wiley New York
Bonet J and Wood R (1997)
Nonlinear continuum mechanics for finite element analysis
Cambridge university press
Bui-Thanh T Willcox K and Ghattas O (2008)
Model reduction for large-scale systems with high-dimensional parametric input space
SIAM Journal on Scientific Computing 30(6)3270ndash3288
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References II
Carlberg K Bou-Mosleh C and Farhat C (2011)
Efficient non-linear model reduction via a least-squares petrovndashgalerkin projection andcompressive tensor approximations
International Journal for Numerical Methods in Engineering 86(2)155ndash181
Chapman T Collins P Avery P and Farhat C (2015)
Accelerated mesh sampling for model hyper reduction
International Journal for Numerical Methods in Engineering
Chaturantabut S and Sorensen D C (2010)
Nonlinear model reduction via discrete empirical interpolation
SIAM Journal on Scientific Computing 32(5)2737ndash2764
Chen Y Davis T A Hager W W and Rajamanickam S (2008)
Algorithm 887 Cholmod supernodal sparse cholesky factorization and updatedowndate
ACM Transactions on Mathematical Software (TOMS) 35(3)22
Constantine P G Dow E and Wang Q (2014)
Active subspace methods in theory and practice Applications to kriging surfaces
SIAM Journal on Scientific Computing 36(4)A1500ndashA1524
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References III
Fahl M (2001)
Trust-region methods for flow control based on reduced order modelling
PhD thesis Universitatsbibliothek
Gill P E Murray W and Saunders M A (2002)
Snopt An sqp algorithm for large-scale constrained optimization
SIAM journal on optimization 12(4)979ndash1006
Lawson C L and Hanson R J (1974)
Solving least squares problems volume 161
SIAM
Lieberman C Willcox K and Ghattas O (2010)
Parameter and state model reduction for large-scale statistical inverse problems
SIAM Journal on Scientific Computing 32(5)2523ndash2542
Maute K and Ramm E (1995)
Adaptive topology optimization
Structural optimization 10(2)100ndash112
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
References IV
Nguyen N and Peraire J (2008)
An efficient reduced-order modeling approach for non-linear parametrized partialdifferential equations
International journal for numerical methods in engineering 76(1)27ndash55
Nocedal J and Wright S (2006)
Numerical optimization series in operations research and financial engineering
Springer
Rewienski M J (2003)
A trajectory piecewise-linear approach to model order reduction of nonlinear dynamicalsystems
PhD thesis Citeseer
Zahr M J and Farhat C (2014)
Progressive construction of a parametric reduced-order model for pde-constrainedoptimization
International Journal for Numerical Methods in Engineering
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Relaxed Penalized Problem Setup
minimizeuisinRnu microisinRnmicro
fextTu
subject to V (micro) le 1
2V0
r(u microp) = 0
micro isin [0 1]kmicro
Effect of Penalization
Ke larr (microe)pKe
Ke eth element stiffness matrix
(a) Without penalization
(b) With penalization
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Binary Solutions
Implication for ROM
From parameter restriction microp = (Φmicromicror)p
Precomputation relies on separability of Φmicro and micror
Separability maintained if (Φmicromicror)p = Φmicromicro
pr
Sufficient condition columns of Φmicro have non-overlapping non-zeros
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Efficient Evaluation of Nonlinear Terms
Due to the mixing of high-dimensional and low-dimensional terms in theROM expression only limited speedups available
rr(ur micror) = ΦuT r(Φuur Φmicromicror) = 0
To enable pre-computation of all large-dimensional quantities intolow-dimensional ones leverage Taylor series expansion
[rr(ur micror)]i = D0im(micror)m + D1
ijm(ur times micror)jm + D2ijkm(ur times ur times micror)jkm
+ D3ijklm(ur times ur times ur times micror)jklm = 0
where
D3ijklm =
part3rtpartuppartuqpartus
(u φmmicro )(φiu times φju times φku times φlu)tpqs
Related work [Rewienski 2003 Barrault et al 2004Barbic and James 2007 Nguyen and Peraire 2008Chaturantabut and Sorensen 2010 Carlberg et al 2011]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Lagrange Multiplier Estimate
Lagrange Multiplier Constraint Pairs
λ λr τ τrc(u micro) ge 0 c(Φuur Φmicromicro) ge 0 Amicro ge b Armicror ge br
Goal Given ur micror τr ge 0 λr ge 0 estimate τ ge 0 λ ge 0 to compute
nablamicroL(Φmicromicror λ τ ) =partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λminusAT τ
Lagrange Multiplier Estimates
λ = λr
τ = arg minτge0
∣∣∣∣∣∣∣∣AT τ minus(partJpartmicro
(Φuur Φmicromicror)minuspartc
partmicro(Φuur Φmicromicror)
T λ
)∣∣∣∣∣∣∣∣Non-negative least squares [Lawson and Hanson 1974 Chapman et al 2015]
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Gradient Filtering Nodal Projection
Minimum length scale rmin
Gradient Filtering11
partJpartmicrok
=
sumjisinSk
HkjmicroipartJpartmicroi
microksumjisinSk
Hkj
Nodal Projection
microk =
sumjisinSk τ jHjksumjisinSk Hjk
(a) Without projectionfiltering
(b) With projection
11Hki = rmin minus dist(k i)
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Gradient of Lagrangian Updated Macroelements
Zahr PDE-Constrained Optimization with Adaptive ROMs
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs
Introduction
Optimization via Adaptive Model Reduction
Model Order Reduction
Non-Quadratic Trust-Region Solver
Shape Optimization Airfoil Design
Large-Scale Constrained Optimization
Reduction of High-Dimensional Parameter Space
Elastic Nonlinear Constraints
Topology Optimization 2D Cantilever
Conclusion
Ongoing Research Projects
Future Research
fdrm0
fdrm1
fdrm2
IntroductionOptimization via Adaptive Model Reduction
Large-Scale Constrained OptimizationConclusion
Ongoing Research ProjectsFuture Research
Standard Difficulty Checkerboarding
Implication for ROM
Nonlocality introduced through projectionfiltering
microe influences volume fraction of all elements within rmin of elementnode e
Clashes with requirement on Φmicro of columns with non-overlapping non-zeros
Handled heuristically by performing parameter basis adaptation to eliminateldquocheckerboardrdquo regions of parameter space uses concept of rmin
Next Helmholtz filtering
Zahr PDE-Constrained Optimization with Adaptive ROMs