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 and Koshy have developed an acceleration technique for the Kaczmarz method for homogeneous linear systems that minimises the distance to the desired solution in the direction of a full Kaczmarz step. Matthew Tam has recently generalised this acceleration technique to inhomogeneous linear systems.

In this talk, I will develop this technique into an acceleration scheme that minimises 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.

optimization and control

Audience: researchers in the topic

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