Inexact Generalized Golub-Kahan Methods for Large-Scale Bayesian Inverse Problems

TL;DR

This paper introduces an inexact generalized Golub-Kahan method tailored for large-scale Bayesian inverse problems, effectively handling inexact forward models and enabling automatic regularization parameter selection, with demonstrated success in tomography tasks.

Contribution

It develops a novel inexact Golub-Kahan decomposition and hybrid scheme for regularization parameter choice in large-scale Bayesian inverse problems.

Findings

Method effectively handles inexact forward models.

Numerical experiments show stability and accuracy.

Applicable to large-scale tomography reconstructions.

Abstract

Solving large-scale Bayesian inverse problems presents significant challenges, particularly when the exact (discretized) forward operator is unavailable. These challenges often arise in image processing tasks due to unknown defects in the forward process that may result in varying degrees of inexactness in the forward model. Moreover, for many large-scale problems, computing the square root or inverse of the prior covariance matrix is infeasible such as when the covariance kernel is defined on irregular grids or is accessible only through matrix-vector products. This paper introduces an efficient approach by developing an inexact generalized Golub-Kahan decomposition that can incorporate varying degrees of inexactness in the forward model to solve large-scale generalized Tikhonov regularized problems. Further, a hybrid iterative projection scheme is developed to automatically select Tikhonov regularization parameters. Numerical experiments on simulated tomography reconstructions demonstrate the stability and effectiveness of this novel hybrid approach.