Error estimates of Kaczmarz and randomized Kaczmarz methods

TL;DR

This paper extends error estimates for Kaczmarz and randomized Kaczmarz methods, providing more general convergence results based on Tanabe's theorem, applicable to inconsistent and ill-posed systems.

Contribution

It introduces generalized error estimates for these methods using Tanabe's convergence theorem, broadening understanding beyond previous bounds.

Findings

Error estimates based on x_k - P_{N(A)}x_0 - x^\u00a0daggera0 are established.

Results apply to inconsistent and ill-posed linear systems.

Provides theoretical convergence bounds for Kaczmarz methods.

Abstract

The Kaczmarz method is an iterative projection scheme for solving con-sistent system $Ax = b$. It is later extended to the inconsistent and ill-posed linear problems. But the classical Kaczmarz method is sensitive to the correlation of the adjacent equations. In order to reduce the impact of correlation on the convergence rate, the randomized Kaczmarz method and randomized block Kaczmarz method are proposed, respectively. In the current literature, the error estimate results of these methods are established based on the error $\|x_k-x_*\|_2$, where $x_*$ is the solution of linear system $Ax=b$. In this paper, we extend the present error estimates of the Kaczmarz and randomized Kaczmarz methods on the basis of the convergence theorem of Kunio Tanabe, and obtain some general results about the error $\|x_k-P_{N(A)}x_0-x^\dagger\|_2$.