Generalized hybrid iterative methods for large-scale Bayesian inverse problems

TL;DR

This paper introduces a generalized hybrid iterative method for efficiently solving large-scale Bayesian inverse problems, especially when covariance matrices are complex or only accessible via matrix-vector products.

Contribution

It develops a novel hybrid algorithm based on generalized Golub-Kahan bidiagonalization that handles large-scale problems without explicit covariance matrix computations.

Findings

Effective in large-scale problems with complex covariance structures

Avoids semi-convergence issues common in iterative methods

Automatically estimates regularization parameters

Abstract

We develop a generalized hybrid iterative approach for computing solutions to large-scale Bayesian inverse problems. We consider a hybrid algorithm based on the generalized Golub-Kahan bidiagonalization for computing Tikhonov regularized solutions to problems where explicit computation of the square root and inverse of the covariance kernel for the prior covariance matrix is not feasible. This is useful for large-scale problems where covariance kernels are defined on irregular grids or are only available via matrix-vector multiplication, e.g., those from the Mat\'{e}rn class. We show that iterates are equivalent to LSQR iterates applied to a directly regularized Tikhonov problem, after a transformation of variables, and we provide connections to a generalized singular value decomposition filtered solution. Our approach shares many benefits of standard hybrid methods such as avoiding semi-convergence and automatically estimating the regularization parameter. Numerical examples from image processing demonstrate the effectiveness of the described approaches.