Goal-oriented optimal approximations of Bayesian linear inverse problems

TL;DR

This paper develops optimal low-rank approximation methods for goal-oriented Bayesian inverse problems, enabling efficient computation of quantities of interest without full posterior calculations, and demonstrates their effectiveness in heat transfer applications.

Contribution

It introduces novel optimal low-rank approximations for the posterior covariance and mean of the QoI, tailored for large-scale inverse problems, with proven optimality under various metrics.

Findings

Optimal low-rank update of posterior covariance with respect to geodesic distance.

Optimal approximation of posterior mean as a low-rank linear function of data.

Successful application to a high-dimensional heat transfer inverse problem.

Abstract

We propose optimal dimensionality reduction techniques for the solution of goal-oriented linear-Gaussian inverse problems, where the quantity of interest (QoI) is a function of the inversion parameters. These approximations are suitable for large-scale applications. In particular, we study the approximation of the posterior covariance of the QoI as a low-rank negative update of its prior covariance, and prove optimality of this update with respect to the natural geodesic distance on the manifold of symmetric positive definite matrices. Assuming exact knowledge of the posterior mean of the QoI, the optimality results extend to optimality in distribution with respect to the Kullback-Leibler divergence and the Hellinger distance between the associated distributions. We also propose approximation of the posterior mean of the QoI as a low-rank linear function of the data, and prove optimality of this approximation with respect to a weighted Bayes risk. Both of these optimal approximations avoid the explicit computation of the full posterior distribution of the parameters and instead focus on directions that are well informed by the data and relevant to the QoI. These directions stem from a balance among all the components of the goal-oriented inverse problem: prior information, forward model, measurement noise, and ultimate goals. We illustrate the theory using a high-dimensional inverse problem in heat transfer.