Hyper-Differential Sensitivity Analysis of Uncertain Parameters in PDE-Constrained Optimization

TL;DR

This paper introduces hyper-differential sensitivity analysis for PDE-constrained optimization problems with uncertain parameters, providing a goal-oriented UQ tool that identifies influential uncertainties efficiently using low-rank structures and parallel computation.

Contribution

The paper develops a novel hyper-differential sensitivity analysis method specifically for PDE-constrained optimization, linking it to existing sensitivity and optimization theories, and demonstrating its efficiency with multi-physics examples.

Findings

Efficient identification of influential uncertain parameters in PDE-constrained optimization.

Use of low-rank structure and randomized solvers for computational efficiency.

Successful application to nonlinear steady state control and transient linear inversion examples.

Abstract

Many problems in engineering and sciences require the solution of large scale optimization constrained by partial differential equations (PDEs). Though PDE-constrained optimization is itself challenging, most applications pose additional complexity, namely, uncertain parameters in the PDEs. Uncertainty quantification (UQ) is necessary to characterize, prioritize, and study the influence of these uncertain parameters. Sensitivity analysis, a classical tool in UQ, is frequently used to study the sensitivity of a model to uncertain parameters. In this article, we introduce "hyper-differential sensitivity analysis" which considers the sensitivity of the solution of a PDE-constrained optimization problem to uncertain parameters. Our approach is a goal-oriented analysis which may be viewed as a tool to complement other UQ methods in the service of decision making and robust design. We formally define hyper-differential sensitivity indices and highlight their relationship to the existing optimization and sensitivity analysis literatures. Assuming the presence of low rank structure in the parameter space, computational efficiency is achieved by leveraging a generalized singular value decomposition in conjunction with a randomized solver which converts the computational bottleneck of the algorithm into an embarrassingly parallel loop. Two multi-physics examples, consisting of nonlinear steady state control and transient linear inversion, demonstrate efficient identification of the uncertain parameters which have the greatest influence on the optimal solution.