Hyper-differential sensitivity analysis in the context of Bayesian inference applied to ice-sheet problems

TL;DR

This paper introduces hyper-differential sensitivity analysis (HDSA) to evaluate how auxiliary parameters influence Bayesian inverse problems in ice-sheet modeling, providing a new interpretation of posterior correlations and demonstrating its application to Greenland ice sheet inversion.

Contribution

It presents a novel use of HDSA to analyze sensitivity of the MAP estimate to auxiliary parameters in Bayesian PDE-constrained inverse problems, with a new interpretation relating HDSA to posterior correlations.

Findings

HDSA effectively quantifies sensitivity of the MAP point to auxiliary parameters.

HDSA offers a new interpretation of posterior correlations in Bayesian inverse problems.

Application to Greenland ice sheet demonstrates practical utility.

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 which must be estimated. Although the Bayesian formulation is attractive for such problems, computational cost and high dimensionality frequently prohibit a thorough exploration of the parametric uncertainty. 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 approximate properties of the Bayesian posterior distribution. For instance, the maximum a posteriori probability (MAP) and the Laplace approximation of the posterior covariance can be computed. In this article, we propose using hyper-differential sensitivity analysis (HDSA) to assess the sensitivity of the MAP point to changes in the auxiliary parameters. We establish an interpretation of HDSA as correlations in the posterior distribution. Our proposed framework is demonstrated on the inversion of bedrock topography for the Greenland ice sheet with uncertainties arising from the basal friction coefficient and climate forcing (ice accumulation rate)