Block sampling Kaczmarz–Motzkin methods for consistent linear systems

Abstract

The sampling Kaczmarz–Motzkin (SKM) method is a generalization of the randomized Kaczmarz method and the Motzkin method. It first samples some rows of coefficient matrix randomly to build a set and then makes use of the maximum violation criterion within this set to determine a constraint. Finally, it makes progress by enforcing this single constraint. In this paper, based on the SKM method and the block strategies, we present two block sampling Kaczmarz–Motzkin methods for consistent linear systems. Specifically, we also first sample a subset of rows of coefficient matrix and then determine an index in this set using the maximum violation criterion. Unlike the SKM method, in the block methods, we devise different greedy strategies to build index sets. Then, the new methods make progress by enforcing the corresponding multiple constraints simultaneously. Numerical experiments show that, for the same accuracy, our methods outperform the SKM method and the famous deterministic method, i.e., the CGLS method, in terms of the number of iterations and computing time.