Controlling Numerical Uncertainty in PDE-Constratined Optimization

Denis Ridzal, John von Neumann Fellow, 2007-2008

Motivation / Importance / Goals

• Solution of optimization problems governed by large-scale PDEs is a critical enabling technology:  optimal design, optimal control, parameter estimation, optimization-based physics coupling.
• The solution process is plagued, among other things, by numerical uncertainty:
  (1)  loss of information involved in translating the mathematical model into its algebraic form
  (2)  inexactness in the solution of linear systems (the core component of the algebraic form)
• How do we control (1) and (2) to ensure accurate and efficient solution of above problems?

Technical Approach

• Analyze the impact of the PDE discretization on the solution of the optimization problem.
• Develop new optimization algorithms that efficiently manage linear solver inexactness. Research Highlights

Research Highlights

A. Compatibility of a spatial discretization with respect to a PDE need not imply its compatibility with respect to the optimization problem governed by that PDE! (novel numerical result - theoretical study in progress)

B. Developed and analyzed a new SQP algorithm that automatically adjusts solver tolerances to 1) ensure global convergence 2) match a prescribed local convergence rate

(Return to Applied Math program list)