Uncertainty quantification in large Bayesian linear inverse problems using Krylov subspace methods

TL;DR

This paper introduces Krylov subspace methods for efficient uncertainty quantification in large Bayesian linear inverse problems, providing theoretical guarantees and practical algorithms demonstrated in tomography applications.

Contribution

It develops new Krylov-based algorithms for approximating the posterior covariance and sampling, with theoretical error bounds and practical efficiency improvements.

Findings

Krylov methods accurately approximate posterior covariance matrices

Proposed algorithms efficiently generate posterior samples

Numerical tomography examples validate the methods' effectiveness

Abstract

For linear inverse problems with a large number of unknown parameters, uncertainty quantification remains a challenging task. In this work, we use Krylov subspace methods to approximate the posterior covariance matrix and describe efficient methods for exploring the posterior distribution. Assuming that Krylov methods (e.g., based on the generalized Golub-Kahan bidiagonalization) have been used to compute an estimate of the solution, we get an approximation of the posterior covariance matrix for `free.' We provide theoretical results that quantify the accuracy of the approximation and of the resulting posterior distribution. Then, we describe efficient methods that use the approximation to compute measures of uncertainty, including the Kullback-Liebler divergence. We present two methods that use preconditioned Lanczos methods to efficiently generate samples from the posterior distribution. Numerical examples from tomography demonstrate the effectiveness of the described approaches.