Techniques for Accelerating the Convergence of Restarted GMRES Based on the Projection

TL;DR

This paper introduces LGMRES and LLBGMRES, two enhanced restart methods for GMRES that utilize projection techniques to improve convergence speed, supported by theoretical analysis and numerical experiments.

Contribution

The paper proposes novel projection-based extensions, LGMRES and LLBGMRES, which significantly accelerate convergence of restarted GMRES methods.

Findings

LGMRES outperforms traditional GMRES(m) in convergence speed.

LLBGMRES further improves convergence with backtracking restart technology.

Theoretical analysis confirms better convergence properties of the new methods.

Abstract

In this paper, we study the restarted Krylov subspace method, which is typically represented by the GMRES(m) method. Our work mainly focused on the amount of change in the iterative solution of GMRES(m) at each restart. We propose an extension of the GMRES(m) method based on the idea of projection. The algorithm is named as LGMRES. In addition, LLBGMRE method is also obtained by adding backtracking restart technology to LGMRES. Theoretical analysis and numerical experiments show that LGMRES and LLBGMRES have better convergence than traditional restart GMRES(m) method.