A class of residual-based extended Kaczmarz methods for solving inconsistent linear systems

Abstract

In this paper, we propose four extended Kaczmarz methods: two partially randomized methods like probability proportional to residual or residual homogenizing, and two deterministic strategies with the maximal-residual control and the maximum-distance control. Without the full column rank and overdetermined assumptions on linear systems, we provide a thorough convergence analysis in terms of expectation, and derive upper bounds for the expected convergence rates of the new extended Kaczmarz methods. Numerical experiments on Gaussian models as well as 2D image reconstruction problems demonstrate that the new extended Kaczmarz methods can be much more effective than the existing ones. Especially, the improvements of two deterministic strategies are very prominent.