Block regularization Kaczmarz method

Abstract

This article focuses on the modification of the iterative version of Kaczmarz block algorithm for solving the problem of regularization, which is a fairly effective method for large-scale problems. An important characteristic of iterative methods is the speed of convergence, which depends on the condition number of the original problem. The main drawback of many iterative methods is the large condition number, while methods based on normal equations have the condition number of the system equal to the square of the condition number of the original problem. At the present time to increase the speed of convergence of iterative methods different types of preconditioners are used reducing the condition number of the system. The disadvantages of this approach is manifested in high computational complexity and the lack of universal preconditioner, which could be applied to any iterative method. One of the most effective approaches for improving the convergence rate of the method is to use a block variant of the method used. In this regard, in this paper we propose a modification of the original block Kaczmarz method for the regularization of the problem, which will reduce the computational complexity, and thus increase the rate of convergence of the algorithm. The article provides a detailed derivation of the proposed modification of the method and the proof of the convergence of the proposed variant of the block Kaczmarz method.

Keywords

• regularization problem
• kaczmarz method
• regularized normal equations
• euler equations