# Randomized Krylov methods for inverse problems

## TL;DR

This paper introduces randomized Krylov subspace methods, including a randomized Golub-Kahan approach, for efficiently solving large-scale linear inverse problems with regularization, demonstrated through image deblurring and seismic tomography.

## Contribution

It develops new randomized iterative solvers for inverse problems, extending existing methods to rectangular matrices and enabling automatic regularization parameter selection.

## Key findings

- Effective in large-scale inverse problems
- Improves computational efficiency with randomized methods
- Demonstrated on image deblurring and seismic tomography

## Abstract

In this paper we develop randomized Krylov subspace methods for efficiently computing regularized solutions to large-scale linear inverse problems. Building on the recently developed randomized Gram-Schmidt process, where sketched inner products are used to estimate inner products of high-dimensional vectors, we propose a randomized Golub-Kahan approach that works for general rectangular matrices. We describe new iterative solvers based on the randomized Golub-Kahan approach and show how they can be used for solving inverse problems with rectangular matrices, thus extending the capabilities of the recently proposed randomized GMRES method. We also consider hybrid projection methods that combine iterative projection methods, based on both the randomized Arnoldi and randomized Golub-Kahan factorizations, with Tikhonov regularization, where regularization parameters can be selected automatically during the iterative process. Numerical results from image deblurring and seismic tomography show the potential benefits of these approaches.