A stabilized GMRES method for singular and severely ill-conditioned systems of linear equations

TL;DR

This paper introduces a stabilized GMRES method that uses Cholesky decomposition to improve the stability and accuracy of solutions for singular and severely ill-conditioned linear systems, outperforming traditional GMRES.

Contribution

The paper proposes a novel stabilized GMRES approach that addresses divergence issues by solving normal equations, enhancing robustness for ill-conditioned and inconsistent systems.

Findings

Method effectively stabilizes GMRES for ill-conditioned systems

Numerical results demonstrate improved accuracy and robustness

Applicable to both underdetermined and range-symmetric systems

Abstract

Consider using the right-preconditioned GMRES (AB-GMRES) for obtaining the minimum-norm solution of inconsistent underdetermined systems of linear equations. Morikuni (Ph.D. thesis, 2013) showed that for some inconsistent and ill-conditioned problems, the iterates may diverge. This is mainly because the Hessenberg matrix in the GMRES method becomes very ill-conditioned so that the backward substitution of the resulting triangular system becomes numerically unstable. We propose a stabilized GMRES based on solving the normal equations corresponding to the above triangular system using the standard Cholesky decomposition. This has the effect of shifting upwards the tiny singular values of the Hessenberg matrix which lead to an inaccurate solution. We analyze why the method works. Numerical experiments show that the proposed method is robust and efficient, not only for applying AB-GMRES to underdetermined systems, but also for applying GMRES to severely ill-conditioned range-symmetric systems of linear equations.