Multistep matrix splitting iteration preconditioning for singular linear systems

TL;DR

This paper introduces multistep matrix splitting iteration preconditioning techniques for Krylov subspace methods to efficiently solve large sparse singular linear systems, with theoretical analysis and numerical validation.

Contribution

It proposes new multistep splitting preconditioners, GSS and HSS, demonstrating improved robustness and efficiency over standard methods for singular systems.

Findings

GSS and HSS preconditioners outperform standard preconditioners in tests

Numerical experiments confirm robustness and efficiency

Theoretical justification supports practical application

Abstract

Multistep matrix splitting iterations serve as preconditioning for Krylov subspace methods for solving singular linear systems. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES) method. We present theoretical and practical justifications for using this approach. Numerical experiments show that the multistep generalized shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) iteration preconditioning are more robust and efficient compared to standard preconditioners for some test problems of large sparse singular linear systems.