Title: Analysis and Control of Numerical Uncertainty in PDE-Constrained Optimization (John von Neumann Fellowship Lecture)

Speaker: Denis Ridzal, Sandia National Laboratories

Date/Time: Monday, November 26, 3-4pm (Mountain Time)

Location: CSRI/90-NM; 940/1103-CA

Brief Abstract: Optimal design, optimal control, and parameter estimation problems are ubiquitous in science and engineering. Their mathematical models often involve large-scale PDE-constrained optimization problems whose solution can be very challenging. A frequently ignored, yet critical challenge, is handling of the so-called numerical uncertainty, which can be defined as the combination of (1) the loss of information involved in translating the mathematical model into its algebraic form and (2) inexactness in the solution of linear systems, the core component of the algebraic form.

The first part of the talk focuses on a demonstration of numerical failure due to the choice of the algebraic representation. A comparative study of Galerkin and mixed Galerkin finite element discretizations used for the solution of semiconductor design problems is presented. A key result of the study is that compatibility of a spatial discretization with respect to a PDE may not be enough to ensure stable and accurate solution of the optimization problem governed by that PDE.

The second part of the talk addresses the control of inexactness in the solution of linear systems, in the context of sequential quadratic programming (SQP) algorithms for nonlinear constrained optimization. Each iteration within an SQP algorithm requires the solution of several linear systems involving the linearized constraints. For problems governed by PDEs, these systems are solved using iterative solvers, which are inherently inexact. In this case, the optimization algorithm must be responsible for dynamically managing the stopping tolerances for the linear solvers. We develop and analyze a novel matrix-free SQP algorithm that automatically adjusts solver tolerances in order to ensure global convergence, and at the same time provides a mechanism for matching a prescribed local convergence rate.