Two-dimensional greedy randomized Kaczmarz methods for solving large-scale linear systems

TL;DR

This paper introduces new two-dimensional randomized Kaczmarz algorithms for efficiently solving large-scale linear systems, demonstrating improved convergence and computational performance over existing methods.

Contribution

The paper proposes novel two-dimensional randomized and semi-randomized Kaczmarz methods with simple random sampling, enhancing efficiency for large-scale problems.

Findings

Proven convergence to the least-norm solution.

Numerical results show faster computation times.

Superiority over existing methods in practical applications.

Abstract

In this paper, we consider a novel two-dimensional randomized Kaczmarz method and its improved version with simple random sampling, which chooses two active rows with probability proportional to the square of their cross-product-like constant, for solving large-scale linear systems. From the greedy selection strategy with grasping two larger entries of the residual vector at each iteration, we then devise a two-dimensional greedy randomized Kaczmarz method. To improve the above methods further, motivated by the semi-randomized Kaczmarz method and Chebyshev's law of large numbers, we propose a two-dimensional semi-randomized Kaczmarz method and its modified version with simple random sampling, which is particularly advantageous for big data problems. Theoretically, we prove that the proposed methods converge to the unique least-norm solution of the consistent linear systems. Numerical results on some practical applications illustrate the superiority of the proposed methods compared with some existing ones in terms of computing time.