Performance Bounds for PDE-Constrained Optimization under Uncertainty

TL;DR

This paper develops a theoretical framework to analyze the performance and optimality gaps of PDE-constrained optimization under uncertainty, accounting for various approximation inaccuracies and demonstrating applicability to elliptic PDE problems.

Contribution

It provides general estimates for optimality gaps in approximation-based PDE optimization under uncertainty, addressing multiple types of approximations and constraints.

Findings

Controls can achieve arbitrarily small optimality gaps as approximations improve.

The framework applies to problems with multiple expectation, risk, and reliability functions.

Demonstrated on an elliptic PDE with random coefficients and distributed control.

Abstract

Computational approaches to PDE-constrained optimization under uncertainty may involve finite-dimensional approximations of control and state spaces, sample average approximations of measures of risk and reliability, smooth approximations of nonsmooth functions, penalty approximations of constraints as well as many other kinds of inaccuracies. In this paper, we analyze the performance of controls obtained by an approximation-based algorithm and in the process develop estimates of optimality gaps for general optimization problems defined on metric spaces. Under mild assumptions, we establish that limiting controls have arbitrarily small optimality gaps provided that the inaccuracies in the various approximations vanish. We carry out the analysis for a broad class of problems with multiple expectation, risk, and reliability functions involving PDE solutions and appearing in objective as well as constraint expressions. In particular, we address problems with buffered failure probability constraints approximated via an augmented Lagrangian. We demonstrate the framework on an elliptic PDE with a random coefficient field and a distributed control function.