Randomized Krylov methods for inverse problems

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.