Strategies for Solving Large-Scale Optimization Problems

Strategies for SolvingLarge-Scale Optimization Problems Judith Hill Sandia 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. Identification of Airborne Contaminants • Initial condition inversion problem Computational Biology • Material property inversion problem • Optimal control problem Water Distribution Systems: • Optimal sensor placement • Initial condition inversion 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

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? Optimality Conditions 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))

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 PDE-constrained optimization in Nihilo

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

Strategies for Solving Large-Scale Optimization Problems