Convergence characteristics of the generalized residual cutting method

TL;DR

This paper analyzes the generalized residual cutting (GRC) method, proving its mathematical properties, its relation to existing Krylov subspace methods, and demonstrating its superior convergence and memory efficiency in large-scale problems.

Contribution

It formally proves GRC as a Krylov subspace method, relates it to the conjugate residual method, and demonstrates its advantages over GMRES in numerical experiments.

Findings

GRC is a Krylov subspace method.

GRC is equivalent to CR for symmetric matrices.

GRC shows more robust convergence and lower memory usage than GMRES.

Abstract

The residual cutting (RC) method has been proposed for efficiently solving linear equations obtained from elliptic partial differential equations. Based on the RC, we have introduced the generalized residual cutting (GRC) method, which can be applied to general sparse matrix problems. In this paper, we study the mathematics of the GRC algorithm and and prove it is a Krylov subspace method. Moreover, we show that it is deeply related to the conjugate residual (CR) method and that GRC becomes equivalent to CR for symmetric matrices. Also, in numerical experiments, GRC shows more robust convergence and needs less memory compared to GMRES, for significantly larger matrix sizes.