Greedy randomized block Kaczmarz method for matrix equation AXB=C and its applications in color image restoration

2024/08/10 by Wenli Wang, Duo Liu, Wang, Wenli +4
Engineering · #Advanced Image Fusion Techniques #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2408.05444

openalex publication_date 2024/08/10 · openalex created_date 2024/09/23 · openalex updated_date 2026/07/28

Abstract

In view of the advantages of simplicity and effectiveness of the Kaczmarz method, which was originally employed to solve the large-scale system of linear equations Ax=b, we study the greedy randomized block Kaczmarz method (ME-GRBK) and its relaxation and deterministic versions to solve the matrix equation AXB=C, which is commonly encountered in the applications of engineering sciences. It is demonstrated that our algorithms converge to the unique least-norm solution of the matrix equation when it is consistent and their convergence rate is faster than that of the randomized block Kaczmarz method (ME-RBK). Moreover, the block Kaczmarz method (ME-BK) for solving the matrix equation AXB=C is investigated and it is found that the ME-BK method converges to the solution A⁺CB⁺+X⁰-A⁺AX⁰BB⁺ when it is consistent. The numerical tests verify the theoretical results and the methods presented in this paper are applied to the color image restoration problem to obtain satisfactory restored images.

Related

• On global randomized block Kaczmarz algorithm for solving large-scale matrix equations
• On the relaxed greedy randomized Kaczmarz methods with momentum acceleration for solving matrix equation AXB=C
• Block Matrix and Tensor Randomized Kaczmarz Methods for Linear Feasibility Problems
• Randomized Block Kaczmarz Method with Projection for Solving Least Squares
• Greedy Motzkin-Kaczmarz methods for solving linear systems