Hyper-differential sensitivity analysis for nonlinear Bayesian inverse problems

TL;DR

This paper extends hyper-differential sensitivity analysis to nonlinear Bayesian inverse problems governed by PDEs, enabling assessment of how uncertainties affect key posterior quantities like the MAP point and Bayes risk.

Contribution

It introduces a mathematical framework and computational method for HDSA in Bayesian PDE inverse problems, focusing on posterior-derived quantities.

Findings

Effective sensitivity analysis of the MAP point and Bayes risk demonstrated on a heat conduction PDE model.

The proposed approach quantifies the influence of model uncertainties on Bayesian inverse solutions.

Computational methods enable practical application of HDSA in complex PDE-based inverse problems.

Abstract

We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by PDEs with infinite-dimensional parameters. In previous works, HDSA has been used to assess the sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on a model inverse problem governed by a PDE for heat conduction.