“smile”

by R. Krause, Instituteof Computatio

nal Science, USI, Lugano.

http://www.unige.ch/math/ascona2013/

The conference on domain decompo-sition methods for optimization withPDE constraints will be held in MonteVerita, Ascona, Switzerland fromSeptember 1 to 6, 2013. The focusof this conference is on new, innova-tive techniques for the parallel solu-tion of large scale constrained opti-mization problems where partial dif-ferential equations appear ans anessential part of the constraints. Suchproblems include structural optimiza-tion and optimal control problems aswell as inverse problems such as pa-rameter identification.

Sponsors

INVITED SPEAKERS

• Harbir Antil• Hans Georg Bock• Moritz Diehl• Eldad Haber• Laurence Halpern• Matthias Heinkenschloss• Ronald Hoppe• Axel Klawonn• Rolf Krause• Yvon Maday• Zden k Strakoš• Gerhard Wanner

Optimization and Model Reduction of Time DependentPDE-Constrained Optimization Problems

Harbir Antil

George Mason University

The optimal design of structures and systems described by partial differential equations(PDEs) often gives rise to large-scale optimization problems, in particular if the underlyingsystem of PDEs represents a multiscale, multiphysics problem. Therefore, reduced order mod-eling techniques such as balanced truncation model reduction (BTMR), proper orthogonaldecomposition, or reduced basis methods are used to significantly decrease the computationalcomplexity while maintaining the desired accuracy of the approximation. We are interestedin such shape optimization problems where the design issue is restricted to a relatively smallportion of the computational domain. In these cases, it appears to be natural to rely on afull order model only in that specific part of the domain and to use a reduced order modelelsewhere. A convenient methodology to realize this idea is a suitable combination of domaindecomposition and BTMR.

In the second part of the talk, we will demonstrate the existence of solution to the Stokesequations with slip boundary conditions in Lp spaces via domain decomposition, for a domainof class W

2−1/pp .

Joint work with M. Heinkenschloss, D.C. Sorensen (Rice University), R.H.W. Hoppe (Universityof Houston), R. H. Nochetto, P. Sodré (University of Maryland).

The Direct Multiple Shooting Method for

Optimization with Differential Equations

H. Georg BockUniversity of Heidelberg

Optimization problems with differential equations arise in many impor-tant and topical applications, in particular since differential equation (DE)models of real-life processes are penetrating all areas of research and de-velopment and have become the basis of strategic decisions. They includeparameter estimation, optimal design and optimal control as well as optimalexperimental design which plays a crucial role in model validation, e.g., formodel discrimination and calibration. The lecture will present the “directmultiple shooting method” which emerged in the late seventies as a blueprint of the “optimization boundary value problem” approach, or what isnow often called “simultaneous” or “all-at-once” approach to optimization,which retains the DE solution at least in discretized form as variables of theoptimization problem, which is then solved by non-feasible step methodssuch as Gauss-Newton, quasi-Newton or inexact Newton-type methods. Abrief historical review will explain the origin of the terms “direct” versus “in-direct” approach to DE optimization. A number of applications demonstratethe versatility and the advantages of the direct multiple shooting methodover the classical single shooting approach prevalent from the sixties untileven today, especially for DE dynamics with difficult intrinsic stability prop-erties such as chaotic systems, or strong non-linearities. Recent advancesand future challenges will be discussed, such as treatment of uncertainties,real-time optimization, non-stationary PDE as constraints, hybrid multi-scale models and non-linear mixed-integer optimal control.

Distributed multiple shooting for estimation and control

Moritz DiehlOptimization in Engineering Center OPTEC and Electrical Engineering Depart-ment ESAT, KU Leuven, Kasteelpark Arenberg 10, 3001 Leuven, Belgiummoritz.diehl@esat.kuleuven.be

Abstract:

This talk treats distributed estimation and control problems of large intercon-nected dynamic systems. These systems consist of subsystems described by ODE,DAE or instationary PDE, and we assume that the coupling equations betweensubsystems are of lower dimension than the internal subsystem states. Aim is tosolve a centralized optimization problem with a distributed algorithm, not onlyfor parallelization, but also to allow for localized data maintenance and decisionmaking. We describe the distributed multiple shooting method as one algorithmdesigned to achieve this task. The continuous time problem is reformulated intoa large structured nonlinear program by first dividing the domain of interest intospatial subsystems and temporal subintervals, on each of which we assume thereexists an efficient forward solver with sensitivity generation. Second, the spatialand temporal boundary conditions are finitely parameterized, e.g. using orthog-onal polynomials. The resulting nonlinear program is then solved by derivative-based, Newton-type algorithms such as sequential quadratic programming, or itsgeneralization, sequential convex programming (SCP). Fortunately, all necessaryderivatives can be easily computed in a distributed fashion. The remaining crucialstep is the solution of the convex subproblems, which contain all the interconnec-tions as linear sparse coupling constraints. We discuss a variety of approaches toaddress these large scale convex problems in a distributed fashion, most of whichare based on dual decomposition and gradient type methods in the space of multi-pliers. These methods only require local information exchange and are thus fullydistributed.

Big data and big models: Inverse Problems that are

Too Large to be Solved

Eldad Haber∗

Abstract

In recent years large data sets have been collected and analyzed, typ-ically, by using some machine learning algorithms. However, many typesof data demand a much more in-depth analysis that requires simulation,that is, solving partial differential equations, and optimization to estimateparameters.

In this talk we discuss examples for such data sets in earth science. Wedescribe the setting in which vast amounts of geophysical data is collectedfrom the air. We present the large scale modeling that is required to simulatesuch a data set and the inverse problems that arise from these types ofproblems. We show that by using traditional techniques these problemscannot be solved in reasonable time on reasonable hardware.

We then discuss a new set of algorithm that enable us to solve suchproblems. These algorithms are based on a concept we call domain of interestcomputation for the forward coupled with stochastic programming for theinverse. We show that by using this combination we are able to solve verylarge scale problems, using a rather modest hardware in reasonable time.

∗University of British Columbia, Vancouver, Canada

Optimized Schwarz waveform relaxation algorithms,theory and applications

Laurence HalpernLaboratoire d’Analyse, Geometrie et Applications (LAGA)

UMR 7539 CNRS, Universite Paris XIII

Avenue J-B Clement, 93430 Villetaneuse, France

Abstract

Schwarz waveform relaxation algorithms have been introduced in the last decadeto solve linear problems of various types in parallel by domain decomposition inspace. Each subdomain evolves with its own time grid in time windows, and theiteration between the subdomains takes place at the end of the time window. Opti-mized Schwarz waveform relaxation algorithms permit to accelerate the convergence,by choosing the type and the coefficients of the transmission conditions between thesubdomains. In this talk I will present an overview of the complete formulas foroptimized coefficients which have been obtained recently.

I will also show extensions to nonlinear time-dependent problems, involved inthe reactive transport process, where a small time scale is needed in a reactive zonelocalized in the neighbourhood of mobile interfaces between water and gas, whereaslarger time scales are involved in the mildly reactive zone (SHP-CO2 project withthe french ANR and the IFPEN) Within this project, we have introduced varioustypes of Schwarz waveform relaxation algorithms for non-linear systems. They arecoupled with Krylov acceleration and Newton algorithms for the non-linear part.

Optimization Based Domain Decomposition for

PDE Constrained Optimization Problems

Matthias Heinkenschloss

Department of Computational and Applied Mathematics - MS 134Rice University

Houston, Texas 77005, USA

I will present domain decomposition methods (DDMs) for optimizationproblems governed by partial differential equation (PDEs). The methodsapply domain decomposition to the optimization problem, rather than onlyto the governing PDEs. As a consequence, the resulting subdomain sub-problems are optimization problems. Numerical experiments indicate thatthe performance of these DDMs applied to linear-quadratic optimal con-trol problems is similar to their performance on single PDEs. However, thesubdomain optimization subproblems typically have to be solved iterativelyand inexact, ideally coarse, subproblem solves have to be considered. Forproblems with control or state constraints, or for nonlinear problems, suchas flow control problems governed by the Navier-Stokes equations, the op-timization based DDMs can be applied at different stages. I will presentdifferent strategies and compare them numerically.

Adaptive finite elements for optimally controlledelliptic variational inequalities of obstacle type

Ronald H.W. Hoppe1,2

1 Dept. of Math., Univ. of Houston, Houston, TX 77204-3008, U.S.A.2 Inst. of Math., Univ. of Augsburg, D-86159 Augsburg, Germany

Abstract

We are concerned with the numerical solution of distributed optimal controlproblems for second order elliptic variational inequalities by adaptive finiteelement methods. Both the continuous problem as well as its finite elementapproximations represent subclasses of Mathematical Programs with Equi-librium Constraints (MPECs) for which the optimality conditions are statedby means of stationarity concepts in function space and in a discrete, finitedimensional setting such as (ε-almost, almost) C- and S-stationarity. With re-gard to adaptive mesh refinement, in contrast to previous work which adoptsa goal oriented dual weighted approach, we consider standard residual-type aposteriori error estimators.The first main result states that for a sequence of discrete C-stationary pointsthere exists a subsequence converging to an almost C-stationary point, pro-vided the associated sequence of nested finite element spaces is limit dense inits continuous counterpart. As the second main result, we prove the reliabilityand efficiency of the residual-type a posteriori error estimators. Particularemphasis is put on the approximation of the reliability and efficiency relatedconsistency errors by heuristically motivated computable quantities and onthe approximation of the continuous active, strongly active, and inactive setsby their discrete counterparts.A detailed documentation of numerical results for two representative test ex-amples illustrates the performance of the adaptive approach.The results are based on joint work with A. Gaevskaya, M. Hintermuller, andC. Lobhard.

Nonlinear FETI-DP and BDDC methods

Axel Klawonn∗, Martin Lanser∗, and Oliver Rheinbach†

Abstract

New nonlinear FETI-DP (Dual-Primal Finite Element Tearing and Inter-connecting) and BDDC (Balancing Domain Decomposition by Constraints)domain decomposition methods are introduced. In all of these methods,in each iteration, local nonlinear problems are solved on the subdomains.The new approaches can significantly reduce communication and show asignificantly improved performance, especially for problems with localizednonlinearities, compared to a standard Newton-Krylov-FETI-DP or BDDCapproach. Moreover, the coarse space of the nonlinear FETI-DP methodscan be used to accelerate the Newton convergence. It is also found that thenew nonlinear FETI-DP and nonlinear BDDC methods are not as closelyrelated as in the linear context. Numerical results for the p-Laplace operatorare presented.

∗Mathematisches Institut, Universitat zu Koln, Weyertal 86-90, 50931 Koln, Germany.E-mail: {klawonn,mlanser}@math.uni-koeln.de

†Institut fur Numerische Mathematik und Optimierung, Fakultat fur Mathematikund Informatik, TU Bergakademie Freiberg, 09596 Freiberg, Germany. Email:oliver.rheinbach@math.tu-freiberg.de

Domain Decomposition and Multilevel Strategies

for Strongly Nonconvex Minimization Problems

Rolf Krause

Institute of Computational Science

Universita della Svizzera italiana

The design of multilevel or domain decomposition strategies for possiblyconstrained (non)convex minimization problems is far from trivial. In fact,when decomposing a minimization problem, different nonlinear subproblemson the resulting subdomains will arise, which have to be ”synchronzed”properly in order to achieve global convergence. A similar problem showsup in case the minimization is carried out employing a multilevel hierarchy,where different nonlinear functions on different levels interact. Here, wediscuss the design and the properties of domain decomposition and multilevelstrategies for the solution of non-convex — and possible constrained —minimization problems. Our approach is inherently nonlinear in the sensethat we decompose the original nonlinear problem into many small, butalso nonlinear, problems. In this way, strongly local nonlinearities or evenheterogeneous problems can be handled easily and consistently. Startingfrom ideas from Trust-Region methods, we show how global convergencecan be obtained for the case of a nonlinear domain decomposition as wellas for the case of a nonlinear multilevel method — or combinations thereof.These ideas also allow us for deriving a globally convergent variant of theASPIN method (G-ASPIN). We will illustrate our findings along examplesfrom computational mechanics in 3D.

Fast Domain Decomposition Algorithm for Continuum Solvation Models

by Eric Cancès, Filippo Lipparini, Yvon Maday, Benedetta Mennucci and Benjamin Stamm

Abstract : Continuum (or implicit) solvation models are nowadays largely used computational approaches to include solvent effects in the simulation of properties and processes of (supra)molecular systems in condensed phase. Their easiness of use combined with a favorable cost/effectiveness ratio has allowed their application in very different research fields, fromchemistry to biology and material sciences. Due to the very different typical ranges of accuracy and dimensions of these fields, continuum solvation models have been combined with both classical molecular mechanics (MM) approaches and quantum-mechanical (QM) descriptions e.g. to determine the shape of a molecule at ground state in a solvent resulting from a minimization procedure.

In this contribution we shall present, an efficient, parallel, linear scaling implementation of the conductor-like screening model (COSMO), based on a new domain decomposition (dd) algorithm recently proposed by Eric Cancès, Yvon Maday and Benjamin Stamm [1]. We shall present in detail the implementation together with its linear scaling properties, both in computational cost and memory requirements. We shall also provide numerical simulations that illustrate this behavior on linear and globular large-sized systems, for which the calculation of the energy and of the forces is achieved with timings compatible with the use of polarizable continuum solvation for molecular dynamics simulations; we refer to [1] and [2] for the full presentation of the ideas.

Eric Cancès, Yvon Maday, Benjamin Stamm : Domain decomposition for implicit solvation models. The Journal of Chemical Physics 2013, 139 (5), 054111

Filippo Lipparini, Benjamin Stamm, Eric Cancès, Yvon Maday, and Benedetta Mennucci : Fast Domain Decomposition Algorithm for Continuum Solvation Models: Energy and First Derivatives Journal of Chemical Theory and Computation 2013 9 (8), 3637-3648.

On preconditioned Krylov subspace methods in numerical PDEs

Zdenek StrakosFaculty of Mathematics and Physics, Charles University in Prague,Czech Republicstrakos@karlin.mff.cuni.cz

The algorithmic development in iterative solvers over the last few decadeshas focused more on the question on how to solve given problems then onthe the dual end equally important question on why things do or might notwork. Here the important point is considering the algebraic computationfully within the context of the underlying model and its discretization.

In this contribution we focus on Krylov subspace methods. The key fea-ture is their adaptive behaviour given by their close link to approximatingcertain moment problems. As a consequence, we can often observe a sub-stantial acceleration of their convergence. The link to moments also bringsin some unexpected properties such as the possibility of detection of the un-known noise level in solving discrete ill-posed problems. On the other hand,the link to moments means a high nonlinearity of Krylov subspace methods,with each individual iteration depending in a very complex way on informa-tion contained in the data. This makes analysis of Krylov subspace methodschallenging and, at the same time, mathematically beautiful. In practice thedifficulties can show up, e.g., in the form of oscillatory spatial distributionof the algebraic error over the domain.

Efficient use of Krylov subspace methods presumes their understand-ing, including their numerical stability properties. The last is of particularimportance when short recurrences are required as an essential feature ofcomputations. A proper preconditioning is an inherent and fundamentalpart, with its motivation ideally based in the PDE context on the operatorconsiderations. In order to construct a reliable stopping criteria, the alge-braic error must be incorporated into locally efficient and fully computablea-posteriori error bounds.

In this contribution we will demonstrate on examples some issues men-tioned above.

On the discovery of the Lagrange multipliers

G. Wanner

Abstract. The talk explains how

• A thick book on mechanics (Varignon 1725),

• a letter by Johann Bernoulli to Varignon (1715),

• Euler’s Methodus (1744, on variational calculus),

• and d’Alembert’s Dynamique from 1743,

led to the famous Mecanique analytique (1788, 1811) by Lagrange, in which the

advantage of the methods of multipliers is demonstrated at many examples and, in

the second edition, finally receives its name.

In the second part of the talk we extend the ideas of Euler and Lagrange to the

problems of optimal control (Carateodory, Hestenes, Bellman, Pontryagin).

MINISYMPOSIA

• MS1: Multigrid and Iterative Strategies for Optimal Control Problems

• MS2: Optimal Control in Applications

• MS3: Shape Optimization and Optimal Control for PDE Constrained Optimization Problems

• MS4: Domain Decomposition Methods and Application to Control Problems

MS1: Multigrid and Iterative Strategies for Optimal Control Problems

John Pearson1, Stefan Takacs11 Mathematical Institute, 24–29 St. Giles’, Oxford, OX1 3LB

e-mail: john.pearson@worc.ox.ac.uk, takacs@maths.ox.ac.uk

ABSTRACT

In this minisymposium we focus on optimal control problems, which constitute an important class of

PDE-constrained optimization problems. There are many PDEs which can act as the constraints within

the problem, such as Stokes-type equations, PDEs with a time-dependent component, and many others

– consequently there is considerable potential for applications in applied sciences.

One of the major considerations in the field of optimal control problems is the development of fast and

effective methods for their numerical solution. A common approach is to develop efficient strategies for

solving the optimality system which characterizes the solution of the problem.

Upon discretization, this generally takes the form of a large and sparse (saddle point) system, the solu-

tion of which requires specialized methods tailored to the problem at hand. One technique for solving

these systems is to construct preconditioned iterative methods, incorporating techniques such as (alge-

braic or geometric) multigrid, multilevel and domain decomposition methods to approximate individual

blocks of the matrix. Alternatively, one may develop such routines to handle the entire matrix system.

A key advantage of both these approaches is that they provide the potential for the exhibition of mesh-

independent convergence.

There has been much recent progress in the construction of such methods, which have been applied

to many important problems. The aim of this minisymposium is to present state-of-the-art solution

strategies and their theoretical underpinning, as well as to highlight possible future directions for this

research such as parallelization of the methods and their application to problems arising in industry.

SPEAKERS

1. Andrew T. Barker (Max Planck Institute Magdeburg, Germany)

Title: Parallel preconditioning for all-at-once solution of time-dependent PDE constrained opti-

mization problems

2. Lorenz John (TU Graz, Austria)

Title: A multilevel preconditioner for an optimal Dirichlet boundary control problem

3. Monika Wolfmayr (Linz, Austria)

Title: A robust and optimal AMLI preconditioned MINRES solver for time-periodic parabolic

optimal control problems

4. Stefan Takacs (Oxford, UK)

Title: An all-at-once multigrid method applied to a Stokes control problem

Parallel preconditioning for all-at-once solution of time–dependent PDEconstrained optimization problems

Andrew T. Barker1 and Martin Stoll11 Max Planck Institute for Dynamics of Complex Technical Systems, D-39106 Magdeburg, Germany

e-mail: barker@mpi-magdeburg.mpg.de, stollm@mpi-magdeburg.mpg.de

ABSTRACT

We explore domain decomposition strategies for accelerating the convergence of all-at-once numerical

schemes for the solution of time-dependent PDE constrained optimization problems on parallel com-

puters. All-at-once schemes aim to solve for all time-steps at the same time, which has the important

advantages of preserving physical couplings in the solution, ensuring robustness with respect to the

regularization parameter, and accelerating convergence. However, this approach leads to very large lin-

ear systems, with resulting costs in computation and memory. We describe a parallel preconditioning

strategy for these systems that uses domain decomposition algorithms in the time domain and Schur

complement approximations for the resulting local saddle point systems. We describe the motivation

behind these algorithms and present numerical results showing their parallel performance. Finally, we

discuss possible practical applications of this approach.

A multilevel preconditioner for an optimal Dirichlet boundary controlproblem

Lorenz John1 and Olaf Steinbach1

1 Institute of Computational Mathematics, Graz University of Technology, Austria

e-mail: john@tugraz.at, o.steinbach@tugraz.at

ABSTRACT

We present a multilevel preconditioner for the discrete optimality system when considering finite el-

ement methods for optimal Dirichlet boundary control problems in appropriate energy spaces. While

for the interior degrees of freedom a standard multigrid method can be applied, a different approach

is required on the boundary. The construction of the preconditioner is based on a BPX type multilevel

method. We focus on the robustness with respect to the mesh size and the cost coefficient. Numerical

examples illustrate the obtained theoretical results.

An all-at-once multigrid method applied to a Stokes control problem

Stefan Takacs11 Mathematical Institute, 24–29 St. Giles’, Oxford, OX1 3LB

e-mail: takacs@maths.ox.ac.uk

ABSTRACT

In this talk we consider the Stokes control problem as model problem:

Minimise j(v, p, f) = 12‖v − vD‖2L2(Ω) +

α2 ‖f‖2L2(Ω)

subject to the Stokes equations

−Δv +∇p = f in Ω and ∇ · v = 0 in Ω and v = 0 on ∂Ω,

where vD is a given desired velocity field and is a given regularization parameter. The discretization of

the optimality system (KKT system) characterising the solution of such a PDE-constrained optimisation

problem leads to a large-scale sparse linear system. This system is symmetric but not positive definite.

Therefore, standard iterative solvers are typically not the best choice. The KKT system is a linear system

for two blocks of variables: the primal variables (velocity field v, pressure distribution p and control f )

and the Lagrange multipliers introduced to incorporate the partial differential equation. Based on this

natural block-structure, we can verify that this system has a saddle point structure where the (1, 1)-block and the (2, 2)- block are positive semidefinite. Contrary to the case of elliptic optimal control

problems, the (1, 2)-block is not positive definite but a saddle point problem itself. We are interested in

fast iteration schemes with convergence rates bounded away from 1 by a constant which is independent

of the discretization parameter (the grid size) and of problem parameters, like in the regularization

parameter α in the model problem. To achieve this goal, we propose an all-at-once multigrid approach.

In the talk we will discuss the choice of an appropriate smoother and we will give convergence theory.

A robust and optimal AMLI preconditioned MINRES solver fortime-periodic parabolic optimal control problems

Monika Wolfmayr1, Ulrich Langer2, Johannes Kraus31 Doctoral Program ”Computational Mathematics”, Johannes Kepler University Linz, Austria

e-mail: monika.wolfmayr@numa.uni-linz.ac.at2 Institute of Computational Mathematics, Johannes Kepler University Linz, Austria

e-mail: ulanger@numa.uni-linz.ac.at3 Johann Radon Institute for Computational and Applied Mathematics, Linz, Austria

e-mail: johannes.kraus@oeaw.ac.at

ABSTRACT

In this talk, we will consider an optimal control problem with a parabolic time-periodic partial differ-

ential equation appearing in its PDE constraints. In order to solve the optimal control problem, we state

its optimality system and discretize it by the multiharmonic finite element method leading to a system

of linear algebraic equations that decouples into smaller systems, which can be solved totally in par-

allel. In [1], we construct preconditioners for these systems which yield robust and fast convergence

rates for the preconditioned minimal residual method with respect to all parameters. These block diago-

nal preconditioners are practically implemented by the algebraic multilevel iteration method presented

in [2]. The diagonal blocks of the preconditioners consist of a weighted sum of stiffness and mass

matrices. In [3], we discuss and prove the robustness and optimality of the AMLI method for solving

these reaction-diffusion type problems discretized by the finite element method. Moreover, the multi-

harmonic finite element analysis of time-periodic parabolic optimal control problems can also be found

in [4], where different variational settings are investigated and estimates of the complete discretization

error are derived.

References

[1] M. Kollmann, M. Kolmbauer, U. Langer, M. Wolfmayr, W. Zulehner: A Finite Element Solverfor a Multiharmonic Parabolic Optimal Control Problem, Comput. Math. Appl. 65(3), 2013,

469-486.

[2] J. Kraus: Additive Schur Complement Approximation and Application to Multilevel Precondition-ing, SIAM J. Sci. Comput., 34(6):A2872-A2895, 2012.

[3] J. Kraus, M. Wolfmayr: On the Robustness and Optimality of the Algebraic Multilevel Method forReaction-Diffusion Type Problems, submitted.

[4] U. Langer, M. Wolfmayr: Multiharmonic Finite Element Analysis of a Time-Periodic ParabolicOptimal Control Problem, submitted.

MS2: Optimal Control in Applications

Chamakuri Nagaiah1, Karl Kunisch1,2

1 Radon Institute of Computational and Applied Mathematics, Linz, Austria2 Institute of Mathematics and Scientific Computing

University of Graz, Graz, Austria

ABSTRACT

Optimization problems subject to constraints given by partial differential equations (PDEs) with addi-

tional constraints on the control and/or state variables belong to the most challenging problem classes in

natural sciences, engineering, and economics. Due to the complexities of the PDEs and the requirement

for rapid solution pose significant challenges for the computational scientists. A particularly challeng-

ing class of PDE-constrained optimization problems in several applications is characterized by the need

for real-time solution, i.e., in time scales that are sufficiently rapid to support simulation- based deci-

sion making. The main focus of this minisymposium is devoted to solve efficiently such large scale

application problems which includes the design of preconditioners for KKT systems arising within the

optimization, but also offer opportunities for storage management, subdomain model reduction and

parallelization techniques.

SPEAKERS

1. Martin Weiser (Zuse Institute Berlin, Germany)

Title: Data compression for parabolic optimal control problems with application to cardiac defib-

rillation

2. Armin Rund (Uni Graz, Austria)

Title: Time Optimal Control of the monodomain model in cardiac electrophysiology

3. Federico Negri (EPFL SB MATHICSE CMCS, Lausanne)

Title: Reduction strategies for PDE-constrained optimization with application in haemodynamics

4. Chamakuri Nagaiah (RICAM, Linz, Austria)

Title: A parallel PDE-constrained optimization: an application to cardiac electrophysiology

5. Lorenz John (TU Graz Austria)

Title: Optimal Dirichlet boundary control for arterial blood flow

6. Peng Chen (EPFL SB MATHICSE CMCS, Lausanne)

Title: Weighted reduced basis method for stochastic optimal control problems with PDE con-

straints

7. Lars Lubkoll (Zuse Institute Berlin, Germany)

Title: Optimal control in implant shape design

8. Henry Kasumba (Austrian Academy of Sciences, Linz, Austria)

Title: A bilevel shape optimization problem for the exterior Bernoulli free boundary value prob-

lem

WEIGHTED REDUCED BASIS METHOD FOR STOCHASTICOPTIMAL CONTROL PROBLEMS WITH PDE CONSTRAINTS

Peng Chen1, Alfio Quarteroni1,2 and Gianluigi Rozza3

1 Modelling and Scientific Computing, CMCS, Mathematics Institute of Computational Science and

Engineering, MATHICSE, Ecole Polytechnique Federale de Lausanne, EPFL, Station 8, CH-1015 Lausanne,

Switzerland. e-mail: peng.chen@epfl.ch, alfio.quarteroni@epfl.ch

2 Modellistica e Calcolo Scientifico, MOX, Dipartimento di Matematica F. Brioschi, Politecnico di Milano,

Piazza L. da Vinci 32, I-20133, Milano, Italy. e-mail: alfio.quarteroni@polimi.it

3 SISSA MathLab, International School for Advanced Studies, via Bonomea 265, 34136 Trieste, Italy.

e-mail: gianluigi.rozza@sissa.it

Abstract: Optimal control problems governed by partial differential equations (PDEs) are commonlyencountered in the optimal design, control and optimization of mathematical models for the underlyingphysical systems in many science and engineering fields. In practical applications, uncertainties mayinevitably arise from various sources, e.g. the PDE coefficients, initial and boundary conditions,external loadings and computational geometries, leading to stochastic optimal control problems thatrequire to solve the underlying PDEs at many realizations of the uncertainties. Whenever it is veryheavy to solve the PDEs or the dimension of the uncertainties is too high, which are frequently facedin practice, the solution demand can easily go beyond any available computational power. In orderto tackle these computational challenges, efficient combination of model reduction techniques andstochastic numerical methods have been recognized as one of the most promising approaches [1, 2, 3].

Adopting this approach, we have developed and analyzed an efficient computational strategy forsolving stochastic optimal control problems with PDE constraints [4, 5, 6, 7]. We proposed a weightedalgorithm based on reduced basis method (RBM) [8] to solve uncertainty propagation problems [5] andapplied it to efficiently solve stochastic optimal control problems [6]. Based on constructive approxi-mation theory, we have proven that the weighted RBM results in an approximation error convergingexponentially fast as long as the solution is smooth in stochastic space, especially efficient for stochas-tic problems featuring various different probability distributions [5]. In analyzing the mathematicalwell-posedness of the stochastic optimal control problem, we first established a stochastic saddle pointformulation for the optimal control problem and demonstrated that there exists a global and uniquestochastic optimal solution for linear problems for the first time thanks to Brezzi’s theorem [6, 7]. Reg-ularity of the optimal solution in stochastic space was studied explicitly for the analysis of stochasticapproximation error. For a complete discretization of the associated stochastic optimality system, weused finite element method (FEM) with optimal preconditioning techniques for deterministic approxi-mation in physical space and applied the weighted RBM together with a stochastic collocation method(SCM) for stochastic approximation in probability space. A global error analysis of the FEM-wRBM-SCM strategy for solving the stochastic optimal control problems has been carried out in detail forlinear problems and ongoing for nonlinear cases. Numerical experiments ranging from small-scale,low-dimensional problems to large-scale and high dimensional problems (with dimension O(100)) havebeen performed to illustrate the computational efficiency and accuracy of our method.

Keywords: optimal control problems, stochastic partial differential equations, uncertainty quantifi-cation, model order reduction, weighted reduced basis method, stochastic collocation method

References

[1] C. Lieberman, K. Willcox, and O. Ghattas. Parameter and state model reduction for large-scalestatistical inverse problems. SIAM Journal on Scientific Computing, 32(5):2523–2542, 2010.

[2] Board on Mathematical Sciences and Their Applications. Assessing the Reliability of ComplexModels: Mathematical and Statistical Foundations of Verification, Validation, and UncertaintyQuantification. the National Academies Press, 2012.

[3] P. Chen, A. Quarteroni, and G. Rozza. Comparison of reduced basis method and collocationmethod for stochastic elliptic problems. submitted, 2012.

[4] P. Chen, A. Quarteroni, and G. Rozza. Stochastic optimal Robin boundary control problems ofadvection-dominated elliptic equations. to appear in SIAM Journal on Numerical Analysis, 2013.

[5] P. Chen, A. Quarteroni, and G. Rozza. A weighted reduced basis method for elliptic partialdifferential equations with random input data. to appear in SIAM Journal on Numerical Analysis,2013.

[6] P. Chen, and A. Quarteroni. Analysis of a weighted reduced basis method for stochastic optimalcontrol problems constrained by elliptic partial differential equations. submitted, 2013.

[7] P. Chen, A. Quarteroni, and G. Rozza. A global error estimate for stochastic optimal controlproblems constrained by Stokes equations. in preparation, 2013.

[8] A.T. Patera and G. Rozza. Reduced basis approximation and a posteriori error estimation forparametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu,2007.

Optimal Dirichlet boundary control forarterial blood flow

Domain Decomposition Methods for Optimization with PDE Constraints,Monte Verita, Ascona, Switzerland, Sep 1–6, 2013

Lorenz John1 Petra Pustejovska 2 Olaf Steinbach3

We consider an optimal Dirichlet boundary control problem for the Navier–Stokes equations. The control is considered in the energy space where therelated norm is realized by the so called Steklov–Poincare operator. We in-troduce a stabilized finite element method for the optimal control problem.Further we present some numerical results with emphasis on arterial bloodflow.

1john@tugraz.at Institute of Computational Mathematics, Graz University of Tech-nology, Austria

2pustejovska@tugraz.at Institute of Computational Mathematics, Graz Universityof Technology, Austria

3o.steinbach@tugraz.at Institute of Computational Mathematics, Graz Universityof Technology, Austria

A BILEVEL SHAPE OPTIMIZATION PROBLEM FOR THE EXTERIORBERNOULLI FREE BOUNDARY VALUE PROBLEM

H. KASUMBA, K KUNISCH, AND A. LAURAIN

ABSTRACT. A bilevel shape optimization problem with the exterior Bernoulli free bound-

ary problem as lower-level problem and the control of the free boundary as the upper-level

problem is considered. Using the shape of the inner boundary as the control, we aim at

reaching a specific shape for the free boundary. A rigorous sensitivity analysis of the

bilevel shape optimization in the infinite-dimensional setting is performed. The numeri-

cal realization using two different cost functionals is presented demonstrate the efficiency

of the approach.

Reduction strategies for PDE-constrained optimization with application inhaemodynamics.

Andrea Manzonib, Federico Negria, Alfio Quarteronia, Gianluigi Rozzab

a MATHICSE-CMCS, École Polytechnique Fédérale de Lausanne, Switzerlandb SISSA Mathlab, International School for Advanced Studies, Trieste, Italy

We present a reduced framework for the numerical solution of parametrized PDE-constrained op-timization problems. This framework is based on a suitable optimize-then-discretize-then-reduce ap-proach which takes advantage of the Reduced Basis method [2, 3] for the rapid and reliable solutionof parametrized PDEs.We mainly focus on control problems governed by advection-diffusion and Navier-Stokes equationsinvolving infinite-dimensional control functions, thus requiring a suitable reduction of the whole opti-mization problem [1], rather than of the sole state equation. We discuss the stability of the reducedbasis approximation and the convergence of a suitable Newton-SQP algorithm employed for the solu-tion of both the underlying finite element approximation in the Offline stage and the RB approximationin the Online stage. Finally, by employing the Brezzi-Rappaz-Raviart theory, we derive a rigorous aposteriori error estimate.We solve some problems arising from applications in haemodynamics, dealing with both data assimi-lation and optimal control of blood flows [4].

References

[1] Negri, F., Rozza, G., Manzoni, A. and Quarteroni, A. (2012) Reduced basis method for parametrizedelliptic optimal control problems. Submitted.

[2] Rozza, G., Huynh, D. B. P. and Manzoni, A. (2013) Reduced basis approximation and a posteriorierror estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants.Numerische Mathematik, 2013, in press.

[3] Rozza, G., Huynh, D. B. P. and Patera, A. T. (2008) Reduced basis approximation and a posteri-ori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch.Comput. Methods Eng. 15, 229–275.

[4] Rozza, G., Manzoni, A. and Negri, F. (2012) Reduction strategies for PDE-constrained optimizationproblems in haemodynamics. In Proceedings of the 6th European Congress on ComputationalMethods in Applied Sciences and Engineering (ECCOMAS 2012), Vienna, Austria.

A Parallel PDE-constrained Optimization: An

Application to Cardiac Electrophysiology

Chamakuri Nagaiah1, Karl Kunisch1,2, Gernot Plank3

1Radon Institute of Computational and Applied MathematicsLinz, Austria

2 Institute of Mathematics and Scientific ComputingUniversity of Graz, Graz, Austria

3 Institute of Biophysics, Medical University of Graz

Harrachgasse 21, Graz, A-8010 Austria

In this talk I will present domain decomposition techniques and their efficientimplementation for PDE constrained optimization of bidomain model in cardiacelectrophysiology. Anatomically realistic such multiscale models of torso embeddedwhole heart electrical activity are computationally expensive endeavor on its ownright and solving optimal control of such models in an optimal manner is a chal-lenging issue. The bidomain model consist of a system of elliptic partial differentialequations coupled with a non-linear parabolic equation of reaction-diffusion type,where the reaction term, modeling ionic transport is described by a set of ordinarydifferential equations. An extra elliptic equation for the solution of an extracellularpotential needs to be solved on the torso domain. The optimal control approach isbased on minimizing a properly chosen cost functional depending on the extracellu-lar current as input at the boundary of torso domain, which must be determined insuch a way that wavefronts of transmembrane voltage in cardiac tissue are smoothedin an optimal manner. In parallel computations, the domain decomposition of suchrealistic geometry consists of heart surrounded by torso is not a trivial task. First,we partition the heart domain into p-subdomains and similarly partition the torsodomain into p-subdomains and then we solve the PDEs as a decoupled system byexpense of one additional communication at each time step. In this talk, a par-allel finite element based algorithm is devised to solve an optimal control problemon such complex geometries and its efficiency is demonstrated not only for the di-rect problem but also for the optimal control problem. The computations realize amodel configuration corresponding to optimal boundary defibrillation of a reentryphenomenon by applying current density stimuli.

Time Optimal Control of the Monodomain Model

in Cardiac Electrophysiology

Karl Kunisch, Armin RundInstitute for Mathematics and Scientific Computing,

University of Graz

Abstract

The electrical behavior of the cardiac tissue is described by thebidomain equations, a set of semilinear reaction-diffusion equationscoupled with a set of ODE describing the cell level (e.g. of Hodgkin-Huxley type). The present work is concerned with time optimalcontrol of the monodomain model, which is a simplification of thebidomain model, with the focus on cardiac arrhythmias. The aim isto determine effective and short defibrillation shocks.

Specifically, we look at a sample part of heart tissue that exhibitsan undesired electrical behavior in the form of a reentry wave. Byplacing two electrode plates onto the heart tissue, the electrical be-havior can be influenced by the current applied to the electrodes.The task is to find short and low-energy pulses, that allow for aneffective termination of the reentry wave. The proper choice of thecost functional is crucial.

The time optimal control problem is reformulated as a bileveloptimization problem, where the terminal time is fixed in the lowerlevel problem. The lower level problem itself is solved via second-order methods, in particularly a trust-region Newton method. Thealgorithmic and numerical solution of the bilevel problem is de-scribed in detail. The numerical experiments demonstrate that de-fibrillation pulses designed by the time optimal control approachinfluence and terminate reentry phenomena effectively.

Data compression for parabolic optimal control problems

with applciation to cardiac defibrillation

M. Weiser, N. Chamakuri and S. Gotschel

Solutions of the monodomain equations describing the propagation of cardiacexcitation exhibit highly local features in their solutions, which has triggered theuse of adaptive mesh refinement and time step selection. This talk addresses opti-mal control problems in defibrillation and focuses on techniques for spatiotemporaladaptivity to achieve a reduction of computation time and storage requirement. Inparticular we present adaptive lossy compression of state trajectory data for theadjoint computation of reduced gradients, storing both FE coefficients and adap-tively refined meshes. A mixed a-priori/a-posteriori error estimator allows to choosethe quantization tolerance such as not to impede the convergence of a Newton-CGoptimization algorithm.

The decreased storage requirement allows to consider a longer time-horizon inthe optimizations, which results in improved stability of the optimal solutions. Theeffectivity of the algorithm is illustrated on several numerical examples.

MS3: Shape Optimization and Optimal Control for PDE ConstrainedOptimization Problems

Harbir Antil1, Mattias Heinkenschloss2, Ronald H. W. Hoppe31 Department of Mathematical Sciences, George Mason University at Fairfax (hantil@gmu.edu)

2 Department of Computational and Applied Mathematics, Rice University, Houston

(heinken@rice.edu)3 Department of Mathematics, University of Houston, Houston (rohop@math.uh.edu)

SPEAKERS

1. Carlos N. Rautenberg, University of Graz

Title: Approximation Methods for Elliptic and Evolution Quasi-variational Inequalities

2. Irwin Yousept, Department of Mathematics, University of Darmstadt

Title: Optimal control of quasilinear H(curl)-elliptic partial differential equations in magneto-

static field problems

3. Christopher Linsenmann, Department of Mathematics, University of Augsburg

Title: The finite element-based immersed boundary method: fully implicit time-stepping scheme

and optimal control of microfluidic devices

Approximation Methods for Elliptic and Evolution

Quasi-variational Inequalities

Carlos RautenbergUniversity of Graz

We consider quasi-variational inequalities (QVIs) where the constraintset mapping determines a bound on the norm of the gradient of the statevariable. The approximation of elliptic QVIs is based on a sequential mini-mization technique in combination of a penalization approach and a semis-mooth Newton iteration. The parabolic problem is handled by means ofthe time-step application of the elliptic solver and an appropriate stabilityresult for the convergent behavior of the method. An account of furtherapproximation techniques is given and methods for hyperbolic problems arediscussed. Numerical tests are also provided.

Optimal control of quasilinear H(curl)-elliptic partial differential equations inmagnetostatic field problems

Irwin YouseptDepartment of Mathematics, University of Darmstadt, Germany

We discuss the mathematical and numerical analysis for optimal control problems governedby quasilinear H(curl)-elliptic partial differential equations. The model problem involves 3Dbounded isotropic materials with magnetic permeability depending strongly on the magneticfield. Due to the physical and mathematical nature of the problem, it is necessary to includedivergence-free constraints on the state and the control. The divergence-free control constraint istreated as an explicit variational equality constraint, whereas a Lagrange multiplier is includedin the state equation to deal with the divergence-free state constraint. We present our recentmathematical and numerical results including the sensitivity analysis of the nonlinear control-to-state mapping, KKT optimality conditions based on the Helmholtz decomposition, regularityresults, and rigorous error estimates for the finite element approximations based on the curl-conforming Nedelec edge elements. Brief numerical results illustrating the theoretical findingsare presented.

The Finite Element-based Immersed Boundary Method:Fully Implicit Time-stepping Scheme and Optimal Control of

Microfluidic Devices

Christopher Linsenmann, Ronald H.W. Hoppe, and Thomas Franke

Institute of Mathematics/Institute of Physics, University of Augsburg,

Universitatsstr. 14/1, D-86159 Augsburg, Germany

We use the Finite Element-based Immersed Boundary Method (FE-IBM) forsimulating and optimizing the motion of elastic cells, such as red blood cells, im-mersed in an external viscous fluid. While the classical IBM uses Finite Differencesfor the spatial and temporal discretization, we employ the Taylor-Hood (P2/P1) Fi-nite Element for the spatial discretization of the governing time-dependent Stokesequations, periodic splines in a variational setting for the representation of theelastic structure, and for the temporal discretization a semi-implicit (Forward Eu-ler/Backward Euler, FwE/BwE) as well as a fully implicit (BwE/BwE) time-step-ping scheme.

For the fully implicit scheme in fully variational formulation, we present an adap-tive numerical solution technique that is based on a Newton continuation predictor-corrector method. It uses time-step sizes that are determined adaptively to meet theconvergence requirements of the correction step. This way, the fully implicit schemecan be realized numerically with acceptable computational effort. The feasibility ofthe approach is illustrated by a numerical example.

As a motivation for the subsequent optimal control problem in the second part ofthe talk, we consider a cell sorting technique developed by Franke et al. to separatedifferent cell types (e.g., red blood cells and cancer cells) from each other. A latelypatented microfluidic device relies on active sorting by surface acoustic waves. Theseare generated by a special electrode mounted on a piezoelectric material and canbe used to deflect cells on their way through a microchannel.

Having this application in mind, we study an optimal control problem featuringthe FE-IBM equations as state constraints plus box constraints for the control. Atracking-type functional with desired cell positions serves as objective functional.For the numerical solution a projected gradient method with Armijo line searchis used, where in each iteration the gradient of the reduced objective functionalis found by the adjoint approach. The computed time-dependent optimal controlrepresenting the electric power applied to the electrode mentioned above can beused to steer successive cells to desired target positions.The aim of this investigation is to improve the separation efficiency of active sortingdevices.

MS 4: Domain Decomposition Methods and Applications to ControlProblems

Martin J. Gander1 and Axel Klawonn2

1 Section de mathematiques, Universite de Geneve, Rue du Lievre 2–4, Geneva, Switzerland2 Mathematisches Institut, Universitat zu Koln, Weyertal 86-90, 50931 Koln, Germany

ABSTRACT

Algorithms for PDE constrained optimization problems are extremely demanding in computer

ressources: they require in general multiple solutions of partial differential equations during the iterative

process of the nonlinear optimization strategy. This minisymposium shows how domain decomposition

methods can be effectively used to parallelize large scale simulations within the optimization iteration,

and also points out new directions in domain decomposition.

SPEAKERS

1. Oliver Rheinbach (TU Freiberg)

Title: FETI-DP methods for Optimal Control Problems

2. Julien Salomon (Universite Paris-Dauphine)

Title: An intermediate states method for the time-parallelized solving of optimal control problems

3. Kevin Santugini (Institut Polytechnique de Bordeaux)

Title: A new discontinuous coarse space correction algorithm for Optimized Schwarz Methods

4. Lorenz John (TU Graz)

Title: Schur complement preconditioners for the biharmonic Dirichlet boundary value problem

and applications to boundary control problems

FETI-DP methods for Optimal Control Problems

Roland Herzog∗and Oliver Rheinbach†

We present a dual-primal FETI method applied to the optimal controlof two dimensional linear elasticity problems. The numerical results aresimilar to the ones presented by Heinkenschloss and Nguyen (2006) for bal-ancing Neumann-Neumann and seem to indicate numerical scalability withrespect to the number of subdomains and robustness with respect to thecost parameter α.

∗Fakultat fur Mathematik, TU Chemnitz, Reichenhainer Str. 41, 09126 Chemnitz†Fakultat fur Mathematik und Informatik, TU Bergakademie Freiberg, Akademiestr.

6, 09599 Freiberg

An intermediate states method for the time-parallelized

solving of optimal control problems

Julien Salomon

Universite Paris-Dauphinesalomon@ceremade.dauphine.fr

In this talk, we present a general approach to parallelize efficiently optimal con-trol solvers. This method, first introduced in 2006, is based on the introduction ofintermediate states that enables one to decompose the original optimality systeminto similar sub-systems. These ones can then be treated independently using stan-dard solvers. We present a recent improvement of the method that makes it fullyefficient and discuss the role of the solver used in parallel.

A new discontinuous coarse space correction algorithm

for Optimized Schwarz Methods

Kevin Santugini

Institut Polytechnique de BordeauxKevin.Santugini@math.u-bordeaux1.fr

For domain decomposition methods that produce discontinuous iterates suchas Optimized Schwarz methods, it is advantageous for the coarse space to be dis-continuous so as to be able to correct discontinuities during the coarse correctionstep. In this presentation, we introduce a new coarse space correction using thediscontinuous coarse spaces introduced in DD21. The basic idea is to choose thecoarse corrector that minimizes the jump of the optimized transmission conditions.A major difference is that the new algorithm is designed with a cell centered finitevolume discretization in mind, in contrary to the previous discontinuous coarse spacealgorithm that was designed for finite element discretizations.

Schur complement preconditioners for the biharmonic

Dirichlet boundary value problem and applications to

boundary control problems

Lorenz John and Olaf Steinbach

TU Graz, Austriajohn@tugraz.at, o.steinbach@tugraz.at

We propose and analyse preconditioners for the Schur complement matrix ofa mixed finite element discretisation of the biharmonic Dirichlet boundary valueproblem. Since the system matrix is spectrally equivalent to the piecewise defined

Sobolev space ˜H−1/2pw (Γ) we may use either an appropriate boundary element ap-

proximation of local single layer boundary integral operators to define an optimalpreconditioner, or we may consider a multilevel preconditioner where the resultingspectral condition number is only optimal up to logarithmic terms. The multilevelpreconditioner is then applied to the solution of unconstrained boundary controlproblems by using induced energy norms to describe the involved cost or regulari-sation. It turns out that the system matrices of the Dirichlet and Neumann controlproblems coincide which unifies the applied solution strategy. This approach canalso be applied to the particular case of a Neumann control problem for the Laplaceequation without any further modification. Numerical experiments confirm all thetheoretical results.

CONTRIBUTEDTALKS

Optimal control of elliptic boundary value problems in

polyhedral domains

Thomas Apel

Institut fur Mathematik und Bauinformatik,Universitat der Bundeswehr Munchen, Germany

thomas.apel@unibw.de

The presentation is concerned with the discretization of an optimal control prob-lem for an elliptic partial differential equation in a three-dimensional polyhedraldomain. Anisotropic, graded meshes basd on a domain splitting are used for deal-ing with the singular behaviour of the solution in the vicinity of the non-smoothparts of the boundary. The discretization error is analyzed by using a new quasi-interpolation operator. The results were achieved in joint work with Ariel Lombardi(Buenos Aires) and Max Winkler (Munchen).

The Interface Control Domain Decomposition Method for

the Stokes-Darcy Coupling

Marco Discacciati1, Paola Gervasio2, and Alfio Quarteroni3

1Laboratori de Calcul Numeric, Departament de Matematica Aplicada III, Universitat Politecnica de

Catalunya, C/Jordi Girona 1-3, Campus Nord, Edifici C2, E-08034 Barcelona, Spain.

marco.discacciati@upc.edu

2DICATAM, Universita di Brescia, via Branze 38, I-25123 Brescia, Italy. paola.gervasio@unibs.it3MOX, Politecnico di Milano, P.zza Leonardo da Vinci 32, I-20133 Milano, Italy, and MATHICSE,

Chair of Modelling and Scientific Computing, Ecole Polytechnique Federale de Lausanne, Station 8,

CH-1015 Lausanne, Switzerland. alfio.quarteroni@epfl.ch

AbstractFiltration of fluids through porous media occurs in many relevant applications: in industrialprocesses involving filtering devices; in the environment as concerns the percolation of waters ofhydrological basins through rocks and sand; in physiology when studying the filtration of bloodthrough arterial vessel walls.The modeling of such physical processes requires considering different systems of partial differentialequations in the domain of interest. Indeed, the motion of incompressible free fluids is typicallydescribed by the Navier-Stokes equations, while Darcy equations are adopted in the porous media.These equations must be suitably coupled to describe the filtration process that takes place acrossthe surface separating the fluid region from the porous media domain.A classic approach, based on a non-overlapping decomposition of the computational domain,introduces suitable coupling conditions across an interface separating the fluid region from theporous medium. Such conditions require continuity of normal velocities and normal stresses andinvolve the so-called Beavers-Joseph-Saffman condition [4, 1].In this talk, we present a different coupling method, called Interface Control Domain Decomposi-tion (ICDD) method [2], based on an overlapping decomposition of the domain which introduces atransition region between the two subproblems. We consider suitable control functions which playthe role of unknown boundary data on the interfaces, and we set up an optimal control problemcharacterizing a suitable cost functional to minimize the difference between physical quantities ofinterest (e.g., velocity, stresses, ...) with respect to a suitable interface norm [3].We show the effectiveness of the method on some numerical examples and we compare the resultswith those obtained with the non-overlapping coupling approach.

References

[1] M. Discacciati. Coupling free and porous-media flows: models and numerical approximation. InSimulation of Flow in Porous Media. De Gruyter, 2013. Accepted. To appear.

[2] M. Discacciati, P. Gervasio, and A. Quarteroni. The Interface Control Domain Decomposition (ICDD)method for elliptic problems. Technical Report 11, MOX - Politecnico di Milano, 2013.

[3] M. Discacciati, P. Gervasio, and A. Quarteroni. The Interface Control Domain Decomposition (ICDD)method for the Stokes-Darcy problem. Technical report, 2013. In preparation.

[4] W. Jager and A. Mikelic. On the interface boundary condition of Beavers, Joseph and Saffman. SIAM

J. Appl. Math., 60(4):1111–1127, 2000.

Interface Control Domain Decomposition (ICDD) Method

Marco Discacciati† , Paola Gervasio∗, Alfio Quarteroni‡

†Laboratori de Calcul Numeric (LaCaN), Escola Tecnica Superior d’Enginyers de Camins, Canals iPorts de Barcelona (ETSECCPB), Universitat Politecnica de Catalunya (UPC BarcelonaTech)

Campus Nord UPC - C2, E-08034 Barcelona, Spaine-mail: marco.discacciati@upc.edu, web page: http://sites.google.com/site/marcodiscacciati

∗DICATAM, Universita di Brescia, I-25123 Brescia, Italye-mail: gervasio@ing.unibs.it, web page: http://www.ing.unibs.it/gervasio

‡MOX, Politecnico di Milano, I-20133 Milano, ItalyMATHICSE, EPFL, CH-1015 Lausanne, Switzerland

e-mail: alfio.quarteroni@epfl.ch, web page: http://cmcs.epfl.ch/people/quarteroni

Keywords: domain decomposition, optimal control, heterogeneous problems.

ABSTRACT

We present the Interface Control Domain Decomposition (ICDD) method, a strategy introduced for thesolution of partial differential equations (PDEs) in computational domains partitioned into subdomainsthat overlap. After reformulating the original boundary value problem with the introduction of newadditional control variables, the unknown traces of the solution at internal subdomain interfaces, thedetermination of the latter is made possible by the requirement that the (a-priori) independent solutionsin each subdomain undergo a minimization of a suitable cost functional.

We illustrate the method on two kinds of boundary value problems, one homogeneous (an elliptic PDE),the other heterogeneous (a coupling between a second order advection-diffusion equation and a firstorder advection equation). The main advantage of applying this approach to heterogeneous problemsis to avoid sharp interfaces which would require an in depth knowledge of the local physical behavior(interface conditions) of the specific problem.

Finally we validate numerically our method through a family of numerical tests and investigate theconvergence properties of our iterative solution algorithm.

References

[1] M. Discacciati, P. Gervasio, and A. Quarteroni. The Interface Control Domain Decomposition(ICDD) Method for Elliptic Problems. tech. report, MOX, Politecnico di Milano, 2012. Submittedto SICON.

[2] M. Discacciati, P. Gervasio, and A. Quarteroni. Interface Control Domain Decomposition (ICDD)Methods for Coupled Diffusion and Advection-Diffusion Problems, tech. report, MOX, Politecnicodi Milano, 2013. Submitted to International Journal for Numerical Methods in Fluids.

Optimization Methods for Nonlinear Model Predictive

Control of Non-Stationary Partial Differential Equations

Ekaterina Kostina

University of Marburg, Germanykostina@mathematik.uni-marburg.de

Many spatio-temporal processes in the natural and life sciences, and engineer-ing are described by the mathematical model of non-stationary partial differentialequations (PDE). It would be of high practical relevance as well as a mathematicalchallenge to use such models for a process optimization subject to numerous im-portant inequality restrictions. However in the presence of disturbances and model-ing errors the real process will never follow the off-line computed optimal solution.Thus the challenge is to compute feedback controls that take these perturbationsinto account. We present a new optimization method for Nonlinear Model Predic-tive Control (NMPC). The NMPC principle is to solve a complete optimal controlproblem whenever new information about perturbations are available and to applythe first instant of the optimal control as a feedback law. However, the frequencyof perturbation information is orders of magnitude higher than even a single opti-mization iteration. Therefore we discuss innovtaive multi-level iterations strategyto make NMPC computations real-time feasible for PDE optimal control problems.Based on joint work with H.G. Bock, G. Kriwet and J.P.Schloeder.

Block Interface Preconditioners forOptimal Control of Elliptic PDE

Daniel Loghin

University of Birmingham, UK

The discretization of optimal control of elliptic partial differential equations

problems yields optimality conditions in the form of linear systems with a

block structure. Correspondingly, if the solution method is a non-overlapping

domain decomposition method, we need to solve interface problems which ex-

hibit a block structure. It is therefore natural to consider block precondition-

ers acting on the interface variables for the acceleration of Krylov methods

with substructuring preconditioners.

In this talk we describe a technique which employs a preconditioner block

structure based on the fractional Sobolev norms corresponding to the do-

mains of the boundary operators arising in the matrix interface problem,

some of which may include a dependence on the control regularization pa-

rameter. We illustrate our approach on standard elliptic control problems.

We present analysis which shows that the resulting iterative method con-

verges independently of the size of the problem. We include numerical results

which indicate that performance is also independent of the control regular-

ization parameter and exhibits only a mild dependence on the number of the

subdomains.

Dirichlet-Neumann Method for the Time-Dependent Problems

Bankim Chandra Mandal

University of Geneva

Abstract

We present a waveform relaxation version of the Dirichlet-Neumann method for the heat and wave equations.Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatialdomain decomposition, and the iteration involves subdomain solves with Dirichlet boundary conditions followedby subdomain solves with Neumann boundary conditions. However, each subdomain problem is now in spaceand time, and the interface conditions are also time-dependent. We call the algorithm the Dirichlet-NeumannWaveform Relaxation (DNWR) method. Using a Laplace transform argument, we show for the heat equationthat when we consider finite time intervals, the DNWR method converges superlinearly for a particular choice ofthe relaxation parameter. For the wave equation we prove convergence in finite number of steps for a particularparameter. We also present numerical experiments, comparing the DNWR method to the Schwarz WaveformRelaxation method with overlap.