Two-step parameterized tensor-based iterative methods for solving $\mathcal{A}_{*M}\mathcal{X}_{*M}\mathcal{B}=\mathcal{C}$

TL;DR

This paper introduces two-step tensor-based iterative methods with preconditioning and parameter optimization to efficiently solve tensor equations, including applications to Sylvester equations and image deblurring.

Contribution

It proposes novel two-step parameterized tensor iterative methods with convergence analysis and practical algorithms for solving complex tensor equations.

Findings

Enhanced convergence properties demonstrated through numerical experiments

Effective algorithms for parameter selection and tensor equation solutions

Applications to Sylvester equations and image deblurring problems

Abstract

Iterative methods based on tensors have emerged as powerful tools for solving tensor equations, and have significantly advanced across multiple disciplines. In this study, we propose two-step tensor-based iterative methods to solve the tensor equations $\mathcal{A}_{*M}\mathcal{X}_{*M}\mathcal{B}=\mathcal{C}$ by incorporating preconditioning techniques and parametric optimization to enhance convergence properties. The theoretical results were complemented by comprehensive numerical experiments that demonstrated the computational efficiency of the proposed two-step parametrized iterative methods. The convergence criterion for parameter selection has been studied and a few numerical experiments have been conducted for optimal parameter selection. Effective algorithms were proposed to compute iterative methods based on two-step parameterized tensors, and the results are promising. In addition, we discuss the solution of the Sylvester equations and a regularized least-squares solution for image deblurring problems.