Greedy Randomized and Maximal Weighted Residual Kaczmarz Methods with Oblique Projection

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.

Faster Deterministic Pseudoinverse-Free Block Extension of Motzkin Method for Large Consistent Linear Systems

“…This is a pseudoinverse free method, which converges to the unique least-norm solution of the consistent linear system (1.1). Inspired by the works [16,31], we modify the greedy probability criterion to the almost-maximal residual principle and propose a new deterministic block Kaczmarz method which is also a deterministic pseudoinverse-free block extension of Motzkin (FBEM) method. In addition, we analyze its convergence and provide numerical results to show that FBEM has better convergence than FDBK and BEM in terms of computing time.…”

The randomized circumcentered-reflection iteration method for solving consistent linear equations

“…The Kaczmarz method with oblique projection were presented in [15], which could be regarded as a orthogonal projection based on two rows. Furthermore, its performance were enhanced by applying greedy strategies in [26]. Wu [27] extended the randomized two-subspace Kaczmarz method for inconsistent linear systems.…”