Obtaining Pseudo-inverse Solutions With MINRES

TL;DR

This paper introduces a simple refinement technique for MINRES that guarantees the minimum-norm solution for Hermitian and complex-symmetric systems, even with singular preconditioners, with minimal additional computational cost.

Contribution

A novel minimum-norm refinement method integrated into MINRES, extending its applicability to singular preconditioners and complex-symmetric systems.

Findings

MN refinement guarantees minimum-norm solutions

Effective with singular preconditioners

Supports complex-symmetric systems

Abstract

The celebrated minimum residual method (MINRES), proposed in the seminal paper of Paige and Saunders, has seen great success and widespread use in solving Hermitian (and complex-symmetric) linear systems. Unless the system is consistent, MINRES is not guaranteed to obtain the pseudo-inverse solution. We propose a novel and remarkably simple minimum-norm refinement (MN refinement) that seamlessly integrates with the final MINRES iteration, enabling us to obtain the minimum-norm solution with negligible additional computational cost. We extend our MN refinement to complex-symmetric systems, building on S.-C. Choi's extension of MINRES for solving these systems. Given the flexibility of MINRES to accommodate singular preconditioners, we further investigate the MN refinement in preconditioned settings that involve singular preconditioners. We also provide numerical experiments to support our analysis and showcase the effects of our MN refinement.