Greedy Randomized and Maximal Weighted Residual Kaczmarz Methods with Oblique Projection

TL;DR

This paper introduces two new Kaczmarz-type algorithms with oblique projection that improve efficiency in solving large-scale linear systems, especially with correlated rows, by reducing iterations and runtime.

Contribution

It proposes the GRKO and MWRKO methods combining greedy randomized and maximal residual strategies with oblique projection, enhancing convergence and efficiency.

Findings

GRKO and MWRKO outperform existing methods in convergence speed.

Theoretical proofs confirm effectiveness of the new methods.

Numerical results demonstrate reduced iterations and runtime.

Abstract

For solving large-scale consistent linear system, we combine two efficient row index selection strategies with Kaczmarz-type method with oblique projection, and propose a greedy randomized Kaczmarz method with oblique projection (GRKO) and the maximal weighted residual Kaczmarz method with oblique projection (MWRKO) . Through those method, the number of iteration steps and running time can be reduced to a greater extent to find the least-norm solution, especially when the rows of matrix A are close to linear correlation. Theoretical proof and numerical results show that GRKO method and MWRKO method are more effective than greedy randomized Kaczmarz method and maximal weighted residual Kaczmarz method respectively.