On the Choice of Subspace for the Quasi-minimal Residual Method for Linear Inverse Problems

TL;DR

This paper investigates how choosing different subspaces affects the performance of Krylov subspace methods like GMRES and QMR in solving linear inverse problems, highlighting the advantages of range restricted variants.

Contribution

It analyzes the impact of subspace selection on solution quality and demonstrates the superior performance of range restricted QMR over standard methods.

Findings

Range restricted QMR outperforms standard QMR.

Range restricted GMRES can be more effective than conventional GMRES.

Range restricted QMR is less sensitive to noise due to its spectral properties.

Abstract

Inverse problems arise in various scientific and engineering applications, necessitating robust numerical methods for their solution. In this work, we consider the effectiveness of Krylov subspace iterative methods, including GMRES, QMR, and their range restricted variants for solving linear discrete ill-posed problems. We analyze the impact of subspace selection on solution quality. Our findings indicate that range restricted QMR can outperform standard QMR, and confirm the previously observed behavior that range restricted GMRES can be superior to conventional GMRES in terms of approximation efficacy. Notably, range restricted QMR demonstrates a key advantage over GMRES with respect to range restricted QMR's singular spectrum which can make the method less sensitive to errors that are naturally present making it particularly effective when the noise level in the problem is uncertain.