Greedy Motzkin-Kaczmarz methods for solving linear systems

TL;DR

This paper introduces the greedy randomized Motzkin-Kaczmarz (GRMK) method for solving linear systems, which uses a greedy residual-based selection rule and demonstrates competitive convergence and performance, especially for sparse matrices.

Contribution

The paper proposes a new greedy randomized Motzkin-Kaczmarz method with convergence analysis and compares its performance to existing methods, including block variants.

Findings

GRMK performs similarly to GRK on dense matrices.

GRMK outperforms GRK in computing time for some sparse matrices.

Block versions of GRMK and GRK have comparable performance.

Abstract

The famous greedy randomized Kaczmarz (GRK) method uses the greedy selection rule on maximum distance to determine a subset of the indices of working rows. In this paper, with the greedy selection rule on maximum residual, we propose the greedy randomized Motzkin-Kaczmarz (GRMK) method for linear systems. The block version of the new method is also presented. We analyze the convergence of the two methods and provide the corresponding convergence factors. Extensive numerical experiments show that the GRMK method has almost the same performance as the GRK method for dense matrices and the former performs better in computing time for some sparse matrices, and the block versions of the GRMK and GRK methods always have almost the same performance.