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Strategies for Solving Large-Scale Optimization Problems
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Strategies for SolvingLarge-Scale Optimization Problems
Judith HillSandia National Laboratories
October 23, 2007Modeling 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.
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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
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Optimization Formulation
• All of these problems are of the form
where the constraints are typically a partial differential equation (PDE).
PDE-Constrained Optimization
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Example Problem
• Initial Condition Inversion under Convection-Diffusion Transport
Challenge: The state and design spaces are extremely large
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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
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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.
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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)) ;
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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))
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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
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Complex Application: Biofilm Growth
• For a single-species, single nutrient biofilm, find the initial state of the biofilm:
Fully-Coupled, Non-linear System!
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Simulation of biofilm growth
Experimental images courtesty S. Altman, Sandia
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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.
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Acknowledgements
• Nihilo development team, including B. van Bloemen Waanders (Sandia) and K. Long (Texas Tech)
• For more information:
http://software.sandia.gov/sundance/
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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: