On Solving Ill-Conditioned Linear Systems

TL;DR

This paper introduces a novel combination of spectral projection and multigrid methods to efficiently solve ill-conditioned linear systems, demonstrating significant iteration reduction in preliminary tests.

Contribution

It is the first to integrate spectral projection with multigrid methods for ill-conditioned systems, improving convergence of Krylov subspace methods.

Findings

Fewer iterations needed for ill-conditioned problems

Effective combination of spectral projection and multigrid methods

Preliminary results show promising efficiency gains

Abstract

This paper presents the first results to combine two theoretically sound methods (spectral projection and multigrid methods) together to attack ill-conditioned linear systems. Our preliminary results show that the proposed algorithm applied to a Krylov subspace method takes much fewer iterations for solving an ill-conditioned problem downloaded from a popular online sparse matrix collection.