Preconditioned steepest descent-like methods for symmetric indefinite systems
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.
Cite
@article{arxiv.1609.05407,
title = {Preconditioned steepest descent-like methods for symmetric indefinite systems},
author = {Eugene Vecharynski and Andrew Knyazev},
journal= {arXiv preprint arXiv:1609.05407},
year = {2017}
}