Convergence acceleration of Kaczmarz's method

TL;DR

This paper investigates the convergence of Kaczmarz's method for solving linear systems and proposes acceleration techniques based on sequence transformations, demonstrating their effectiveness through numerical examples.

Contribution

It introduces new acceleration procedures for Kaczmarz's method using sequence transformations, improving convergence speed.

Findings

Acceleration methods significantly improve convergence

Numerical examples confirm effectiveness of proposed procedures

Both direct and restart acceleration approaches are effective

Abstract

The method of alternation projections (MAP) is an iterative procedure for finding the projection of a point on the intersection of closed subspaces of an Hilbert space. The convergence of this method is usually slow, and several methods for its acceleration have already been proposed. In this work, we consider a special MAP, namely Kaczmarz' method for solving systems of linear equations. The convergence of this method is discussed. After giving its matrix formulation and its projection properties, we consider several procedures for accelerating its convergence. They are based on sequence transformations whose kernels contain sequences of the same form as the sequence of vectors generated by Kaczmarz' method. Acceleration can be achieved either directly, that is without modifying the sequence obtained by the method, or by restarting it from the vector obtained by acceleration. Numerical examples show the effectiveness of both procedures.