On Greedy Kaczmarz Method with Uniform Sampling for Consistent Tensor Linear Systems Based on T-Product: tGK method for consistent tensor linear systems

Abstract

Solving large system of tensor linear equations is a fundamental problem in mathematics. This paper proposes a sampling tensor greedy Kaczmarz method (tGK) to solve large-scale linear systems with a t-product structure by introducing an effective greedy criterion, which eliminates the entry with the largest residual in the submatrix system per iteration. Then a relaxed tensor greedy Kaczmarz method (tRGK (ω)) is obtained by introducing the relaxation parameter ω to tGK, which can effectively change the convergence rate. The linear convergence of the two methods is guaranteed when the tensor linear system is consistent. Several experiments show that the methods designed in this paper converge faster compared with tensor randomized Kaczmarz (tRK). Moreover, selecting appropriate parameters ω can improve the convergence rate of tGK.