Flexible Modified LSMR Method for Least Squares Problems

TL;DR

This paper introduces a flexible modified LSMR method that reduces computational effort by solving only one linear system per iteration, enhancing efficiency for least squares problems.

Contribution

It proposes a novel flexible MLSMR method that integrates with flexible GMRES, reducing the number of linear systems solved per iteration.

Findings

FMLSMR demonstrates improved efficiency in numerical tests.

The method effectively handles preconditioning in least squares problems.

Numerical examples confirm the computational advantages of the proposed approach.

Abstract

LSMR is a widely recognized method for solving least squares problems via the double QR decomposition. Various preconditioning techniques have been explored to improve its efficiency. One issue that arises when implementing these preconditioning techniques is the need to solve two linear systems per iterative step. In this paper, to tackle this issue, among others, a modified LSMR method (MLSMR), in which only one linear system per iterative step needs to be solved instead of two, is introduced, and then it is integrated with the idea of flexible GMRES to yield a flexible MLSMR method (FMLSMR). Numerical examples are presented to demonstrate the efficiency of the proposed FMLSMR method.