Tensor Krylov subspace methods via the T-product for large Sylvester tensor equations

TL;DR

This paper introduces novel tensor Krylov subspace methods using the T-product to efficiently solve large Sylvester tensor equations, expanding tensor algebra tools and demonstrating their effectiveness through numerical experiments.

Contribution

It develops new tensor Krylov subspace algorithms based on the T-product, including tensor FOM, GMRES, and Arnoldi methods, with theoretical properties and numerical validation.

Findings

New tensor products and algebraic properties introduced.

Development of tensor Krylov subspace methods for Sylvester equations.

Numerical experiments demonstrate the effectiveness of the proposed methods.

Abstract

In the present paper, we introduce new tensor krylov subspace methods for solving large Sylvester tensor equations. The proposed method uses the well-known T-product for tensors and tensor subspaces. We introduce some new tensor products and the related algebraic properties. These new products will enable us to develop third-order the tensor FOM (tFOM), GMRES (tGMRES), tubal Block Arnoldi and the tensor tubal Block Arnoldi method to solve large Sylvester tensor equation. We give some properties related to these method and present some numerical experiments.