Randomized Flexible LSQR and LSMR with applications to inverse problems

TL;DR

This paper introduces randomized sketched variants of flexible LSQR and LSMR algorithms, enabling efficient large-scale inverse problem solutions while maintaining reconstruction quality through theoretical analysis and numerical validation.

Contribution

It presents novel randomized algorithms sFLSQR and sFLSMR, improving computational efficiency in large-scale inverse problems with theoretical guarantees.

Findings

sFLSQR and sFLSMR alleviate computational bottlenecks

They preserve reconstruction quality in inverse problems

Theoretical analysis supports their effectiveness

Abstract

LSQR and LSMR are iterative methods, based on the Golub-Kahan bidiagonalization algorithm, widely used for large-scale linear least squares problems. FLSQR and FLSMR are flexible variants of LSQR and LSMR, respectively, based on a flexible Golub-Kahan (Arnoldi-like) factorization algorithm, which naturally allow modifications of the solution approximation subspace and/or handling inexact matrix-vector multiplications with the (transpose of the) coefficient matrix, thereby enabling to enforce prior information into the computed solution. The goal of this paper is to introduce sFLSQR and sFLSMR, i.e., sketched variants of FLSQR and FLSMR, respectively, where randomization becomes particularly effective, as it allows to recover short recurrences for the solution approximation. In particular, this paper explores applications to large-scale inverse problems, showing the ability of the new randomized solvers to alleviate computational bottlenecks while preserving reconstruction quality. A theoretical analysis of sFLSQR and sFLSMR is provided, and their performance is validated through numerical experiments.