strategies for solving large-scale optimization problems judith hill sandia national laboratories...
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Strategies for SolvingLarge-Scale Optimization Problems
Judith HillSandia National Laboratories
October 23, 2007
Modeling and High-Performance Computing Workshop
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration
under contract DE-AC04-94AL85000.
Overview
• Many engineering problems can be recast as an optimization question.
Water Distribution Systems:• Optimal sensor placement• Initial condition inversion problem
Identification of Airborne Contaminants• Initial condition inversion problem
Computational Biology• Material property inversion problem• Optimal control problem
Design Optimization• Boundary control problem
Optimization Formulation
• All of these problems are of the form
where the constraints are typically a partial differential equation (PDE).
PDE-Constrained Optimization
Example Problem
• Initial Condition Inversion under Convection-Diffusion Transport
Challenge: The state and design spaces are extremely large
Optimality Conditions
Implementation Challenges:• Large-scale coupled system
of equations• Adjoint is backwards in
time• Adjoints aren’t generally
available in legacy simulation codes
• Parallelizing this system of equations
• What happens for a non-linear case?
Requires a versatile large-scale PDE simulation tool with analysis capabilities
Nihilo-Sundance
• Nihilo-Sundance provides a suite of high-level, extensible, components to describe a PDE and its discretization with finite elements– Simple user-specification of PDE weak equations and
boundary conditions– Finite element method infrastructure– Access to linear operators – Analysis capabilities such as optimization algorithms– High-performance linear and nonlinear solvers and
preconditioners– Parallel capabilities under-the-hood
Nihilo allows for rapid creation of a 3-D, parallel simulation and analysis tool.
Forward Convection-Diffusion Problem
• Strong Form:
• Weak Form:
Eqn = Integral(interior, (u-uOld)/deltaT*psi + nu*(grad*u)*(grad*psi)
+ (v*(grad*u))*psi , new GaussianQuadrature(2)) ;
Adjoint for the Convection-Diffusion Problem
• Strong Form:
• Weak Form:
Eqn = Integral(interior, (lambdaOld-lambda)/deltaT*psi + nu*(grad*lambda)*(grad*psi) + (v*(grad*psi))*lambda
, new GaussianQuadrature(2)) + Integral(sensors, (u-uTarget)*psi , new GaussianQuadrature(2))
PDE-constrained optimization in Nihilo
• Nihilo Provides– Access to “black-box”
optimization algorithms
– Access to operators for intrusive optimization
– Finite element method infrastructure
– Parallel capabilities under-the-hood
• User Provides– Physics-specific information
• Forward Problem• Adjoint Problem• Sensitivity
– Problem-specific information
• User Chooses– Element type and order
– Quadrature scheme
– Linear/nonlinear solver
– Preconditioner
Complex Application: Biofilm Growth
• For a single-species, single nutrient biofilm, find the initial state of the biofilm:
Fully-Coupled, Non-linear System!
Simulation of biofilm growth
Experimental images courtesty S. Altman, Sandia
Summary
• Standard production codes are often difficult to manipulate for intrusive analyses
• Nihilo-Sundance represents a paradigm shift for looking at intrusive algorithms– The underlying symbolic engine allows for rapid creation of a
simulation tool.– Nihilo targets a modular design and implementation of
intrusive analysis algorithms, beyond that of optimization problems
• We demonstrated these capabilities on a complex problem, but could quickly move to a different application, reusing much of the infrastructure in place.
Acknowledgements
• Nihilo development team, including B. van Bloemen Waanders (Sandia) and K. Long (Texas Tech)
• For more information:
http://software.sandia.gov/sundance/
Questions
• Other Research Interests:– chemically reacting flows– aerosol modeling– parallel numerical algorithms– dynamic interface modeling– phase field and level set methods– inverse problems– uncertainty quantification
• Contact Information: