Accelerated Kaczmarz methods via randomized sketch techniques for solving consistent linear systems

TL;DR

This paper introduces accelerated Kaczmarz methods enhanced with randomized sketch techniques, improving convergence speed and efficiency for solving consistent linear systems, supported by theoretical analysis and numerical experiments.

Contribution

It presents novel randomized sketch techniques to accelerate Kaczmarz methods, with convergence proofs and bounds on solution discrepancies, outperforming existing methods in speed.

Findings

Faster convergence compared to traditional methods

Theoretical bounds on solution discrepancy

Numerical experiments confirm improved running time

Abstract

Motivated by the randomized sketch to solve a variety of problems in scientific computation, we improve both the maximal weighted residual Kaczmarz method and the randomized block average Kaczmarz method using two new randomized sketch techniques. Besides, convergence analyses of the proposed methods are provided. Furthermore, we establish an upper bound for the discrepancy between the numerical solutions obtained via the proposed methods and those derived from the original approaches. Numerical experiments demonstrate that the new methods perform better than the existing ones in terms of the running time with the same accuracy.