Enabling hyper-differential sensitivity analysis for ill-posed inverse problems

TL;DR

This paper develops new theoretical and computational methods to enable hyper-differential sensitivity analysis for ill-posed PDE-constrained inverse problems, improving parameter sensitivity assessment in complex, uncertain models.

Contribution

It introduces a novel framework that projects sensitivities onto likelihood informed subspaces and defines posterior updates, addressing challenges of ill-posedness in HDSA.

Findings

Framework successfully applied to a nonlinear multi-physics inverse problem.

Enables sensitivity analysis despite ill-posedness and high dimensionality.

Provides insights into spatially heterogeneous material properties.

Abstract

Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model that must be estimated. However, high dimensionality of the parameters and computational complexity of the PDE solves make such problems challenging. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to estimate the other parameters. In this article, hyper-differential sensitivity analysis (HDSA) is used to assess the sensitivity of the solution of the PDE-constrained optimization problem to changes in the auxiliary parameters. Foundational assumptions for HDSA require satisfaction of the optimality conditions which are not always practically feasible as a result of ill-posedness in the inverse problem. We introduce novel theoretical and computational approaches to justify and enable HDSA for ill-posed inverse problems by projecting the sensitivities on likelihood informed subspaces and defining a posteriori updates. Our proposed framework is demonstrated on a nonlinear multi-physics inverse problem motivated by estimation of spatially heterogenous material properties in the presence of spatially distributed parametric modeling uncertainties.