Generalized Gearhart-Koshy acceleration for the Kaczmarz method

TL;DR

This paper introduces a generalized acceleration scheme for the Kaczmarz method that minimizes the Euclidean norm error over an affine subspace, showing potential for improved convergence in solving large sparse linear systems.

Contribution

It extends the Gearhart-Koshy acceleration to an affine subspace approach, providing a new formulation that is efficiently solvable and enhances the Kaczmarz method's performance.

Findings

Proposed affine search outperforms standard Kaczmarz methods

Method effectively minimizes Euclidean norm error

Numerical experiments confirm improved convergence

Abstract

The Kaczmarz method is an iterative numerical method for solving large and sparse rectangular systems of linear equations. Gearhart, Koshy and Tam have developed an acceleration technique for the Kaczmarz method that minimizes the distance to the desired solution in the direction of a full Kaczmarz step. The present paper generalizes this technique to an acceleration scheme that minimizes the Euclidean norm error over an affine subspace spanned by a number of previous iterates and one additional cycle of the Kaczmarz method. The key challenge is to find a formulation in which all parameters of the least-squares problem defining the unique minimizer are known, and to solve this problem efficiently. A numerical experiment demonstrates that the proposed affine search has the potential to clearly outperform the Kaczmarz and the randomized Kaczmarz methods with and without the Gearhart-Koshy/Tam line-search.