Hyper-differential sensitivity analysis with respect to model discrepancy: Mathematics and computation

TL;DR

This paper develops a scalable, efficient framework for analyzing how model discrepancies affect PDE-constrained optimization solutions, combining sensitivity analysis, adjoint methods, and randomized SVD for practical computation.

Contribution

It introduces a novel hyper-differential sensitivity analysis method for PDECO problems with respect to model discrepancy, leveraging advanced linear algebra and computational techniques.

Findings

Framework efficiently quantifies sensitivity of PDECO solutions to model discrepancy.

Algorithm demonstrates computational efficiency on nonlinear PDECO problems.

Provides rich insights into the impact of model assumptions on optimization outcomes.

Abstract

Model discrepancy, defined as the difference between model predictions and reality, is ubiquitous in computational models for physical systems. It is common to derive partial differential equations (PDEs) from first principles physics, but make simplifying assumptions to produce tractable expressions for the governing equations or closure models. These PDEs are then used for analysis and design to achieve desirable performance. For instance, the end goal may be to solve a PDE-constrained optimization (PDECO) problem. This article considers the sensitivity of PDECO problems with respect to model discrepancy. We introduce a general representation of the discrepancy and apply post-optimality sensitivity analysis to derive an expression for the sensitivity of the optimal solution with respect to the discrepancy. An efficient algorithm is presented which combines the PDE discretization, post-optimality sensitivity operator, adjoint-based derivatives, and a randomized generalized singular value decomposition to enable scalable computation. Kronecker product structure in the underlying linear algebra and corresponding infrastructure in PDECO is exploited to yield a general purpose algorithm which is computationally efficient and portable across a range of applications. Known physics and problem specific characteristics of discrepancy are imposed through user specified weighting matrices. We demonstrate our proposed framework on two nonlinear PDECO problems to highlight its computational efficiency and rich insight.