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

F. Bouyghf, M. El Guide, A. El Ichi

TL;DR

The paper tackles the problem of solving large Sylvester tensor equations of the form $\mathscr{A}\star\mathscr{X}-\mathscr{X}\star\mathscr{B}=\mathscr{C}$ by extending matrix Krylov methods to third-order tensors through the $T$-product. It develops tensor projection algorithms, including tArnoldi, tFOM, tGMRES, and tubal-Block Krylov variants, along with a small-scale t-Bartels-Stewart method, and a large-scale TBAS framework based on Tubal-Block-Arnoldi, with restart strategies. The work provides algorithmic formulations, theoretical properties, and numerical experiments showing that TBAS(m) generally delivers favorable speed-accuracy trade-offs compared to alternative approaches. Overall, the results demonstrate the practical viability of tensor Krylov methods for large STEs and extend classical Sylvester techniques to the tensor setting with efficient FFT-based computations.

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.

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

TL;DR

The paper tackles the problem of solving large Sylvester tensor equations of the form by extending matrix Krylov methods to third-order tensors through the -product. It develops tensor projection algorithms, including tArnoldi, tFOM, tGMRES, and tubal-Block Krylov variants, along with a small-scale t-Bartels-Stewart method, and a large-scale TBAS framework based on Tubal-Block-Arnoldi, with restart strategies. The work provides algorithmic formulations, theoretical properties, and numerical experiments showing that TBAS(m) generally delivers favorable speed-accuracy trade-offs compared to alternative approaches. Overall, the results demonstrate the practical viability of tensor Krylov methods for large STEs and extend classical Sylvester techniques to the tensor setting with efficient FFT-based computations.

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.
Paper Structure (12 sections, 10 theorems, 82 equations, 1 table, 8 algorithms)

This paper contains 12 sections, 10 theorems, 82 equations, 1 table, 8 algorithms.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Definitions and properties of the T-product
  4. Tensor tFOM and tGMRES algorithms
  5. The tArnoldi method
  6. The tFOM method
  7. The tGMRES method
  8. t-Bartels-Stewart method for Sylvester tensor equations of small size
  9. Tensor Krylov methods via T-product for solving large Sylvester tensor equations
  10. Tubal Block Arnoldi method
  11. Tubal-Block-Arnoldi method for solving large Sylvester tensor equations (TBAS)
  12. Numerical experiments

Key Result

Proposition 2.1

Let $\mathscr{A},\mathscr{A}_1 \in {\mathbb R}^{n \times s \times n_{3}}$, $\mathscr{B},\mathscr{B}_1\in {\mathbb R}^{n \times p \times n_{3}}$, $\mathscr{A}_2 \in {\mathbb R}^{\ell\times s \times n_{3}}$, $\mathscr{B}_2\in {\mathbb R}^{\ell\times p \times n_{3}}$, $\mathscr{C} \in {\mathbb R

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.1
  • Definition 2.6
  • Theorem 2.1
  • Proposition 3.1
  • Proposition 3.2
  • ...and 7 more