Generalized Gearhart-Koshy acceleration for the Kaczmarz method

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. When only a single Kaczmarz cycle is considered, the proposed affine search is more effective than the Gearhart-Koshy/Tam line-search, which in turn is more effective than the underlying Kaczmarz method. A numerical experiment from the context of computerized tomography suggests that the proposed affine search has the potential to outperform the the Gearhart-Koshy/Tam line-search and the underlying Kaczmarz method in terms of the computational cost that is needed to achieve a given error tolerance.