Strategies for Solving Large-Scale Optimization Problems

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.

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:

jhill@sandia.gov