Preconditioned steepest descent-like methods for symmetric indefinite systems

TL;DR

This paper explores PSD-like iterative methods for symmetric indefinite systems with SPD preconditioners, establishing their equivalence to restarted PMINRES and providing convergence analysis and practical insights.

Contribution

It demonstrates the equivalence of PSD-like schemes to restarted PMINRES and introduces simplified algorithms under spectral assumptions, bridging theory and practice.

Findings

PSD-like schemes are equivalent to restarted PMINRES with two-step restarts.

A convergence bound for the PSD-like method is derived.

Simplified PSD-like algorithms are proposed under spectral information.

Abstract

This paper addresses the question of what exactly is an analogue of the preconditioned steepest descent (PSD) algorithm in the case of a symmetric indefinite system with an SPD preconditioner. We show that a basic PSD-like scheme for an SPD-preconditioned symmetric indefinite system is mathematically equivalent to the restarted PMINRES, where restarts occur after every two steps. A convergence bound is derived. If certain information on the spectrum of the preconditioned system is available, we present a simpler PSD-like algorithm that performs only one-dimensional residual minimization. Our primary goal is to bridge the theoretical gap between optimal (PMINRES) and PSD-like methods for solving symmetric indefinite systems, as well as point out situations where the PSD-like schemes can be used in practice.