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Efficient tensor-based approach to solving linear systems involving Kronecker sum of matrices

Ahmad Y. Al-Dweik, Abdallah Sayyed-Ahmad

TL;DR

The paper develops a tensor-based direct solver for linear systems formed by Kronecker sums, establishing a link to Sylvester and Sylvester tensor equations. By leveraging eigendecompositions or Schur factorizations, the authors derive a compact formula that computes the solution as a tensor transform of the right-hand side, scaled by an explicit inverse weight tensor C. They provide explicit solutions for 2D and 3D discrete Poisson problems and a 2D convection–diffusion system, including detailed complexity analyses and numerical experiments demonstrating superior scalability over traditional methods. The approach offers a principled, high-dimensional solver with guaranteed uniqueness under spectral-sum conditions and broad applicability to Dirichlet-posed Poisson-like problems.

Abstract

A novel tensor-based formula for solving the linear systems involving Kronecker sum is proposed. Such systems are directly related to the matrix and tensor forms of Sylvester equation. The new tensor-based formula demonstrates the well-known fact that a Sylvester tensor equation has a unique solution if the sum of spectra of the matrices does not contain zero. We have showcased the effectiveness of the method by efficiently solving the 2D and 3D discretized Poisson equations, as well as the 2D steady-state convection-diffusion equation, on a rectangular domain with Dirichlet boundary conditions. The results suggest that this approach is well-suited for high-dimensional problems.

Efficient tensor-based approach to solving linear systems involving Kronecker sum of matrices

TL;DR

The paper develops a tensor-based direct solver for linear systems formed by Kronecker sums, establishing a link to Sylvester and Sylvester tensor equations. By leveraging eigendecompositions or Schur factorizations, the authors derive a compact formula that computes the solution as a tensor transform of the right-hand side, scaled by an explicit inverse weight tensor C. They provide explicit solutions for 2D and 3D discrete Poisson problems and a 2D convection–diffusion system, including detailed complexity analyses and numerical experiments demonstrating superior scalability over traditional methods. The approach offers a principled, high-dimensional solver with guaranteed uniqueness under spectral-sum conditions and broad applicability to Dirichlet-posed Poisson-like problems.

Abstract

A novel tensor-based formula for solving the linear systems involving Kronecker sum is proposed. Such systems are directly related to the matrix and tensor forms of Sylvester equation. The new tensor-based formula demonstrates the well-known fact that a Sylvester tensor equation has a unique solution if the sum of spectra of the matrices does not contain zero. We have showcased the effectiveness of the method by efficiently solving the 2D and 3D discretized Poisson equations, as well as the 2D steady-state convection-diffusion equation, on a rectangular domain with Dirichlet boundary conditions. The results suggest that this approach is well-suited for high-dimensional problems.
Paper Structure (13 sections, 7 theorems, 66 equations, 3 figures, 2 tables)

This paper contains 13 sections, 7 theorems, 66 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transform the linear system involving Kronecker sum of matrices into matrix and tensor form
  4. Sylvester equation
  5. Sylvester tensor equation
  6. Main Theorems
  7. Kronecker sum of normal matrices
  8. Kronecker sum of matrices
  9. Application
  10. Explicit solution of 2D discrete Poisson equation
  11. Explicit solution of 3D discrete Poisson equation
  12. Solution of the 2D steady state convection-diffusion equation
  13. Conclusion

Key Result

Lemma 3.1

(Vectorization with $n$-mode product of a 3rd-order tensor with matrices). Given a tensor ${\cal T} \in \mathbb{C}^{d_1 \times d_2 \times d_3 }$ and matrices $\mathbf{A},\mathbf{B}$ and $\mathbf{C}$ in $\mathbb{C}^{k_1 \times d_1 } ,\mathbb{C}^{k_2 \times d_2 } ,\mathbb{C}^{k_3 \times d_3 },$ r

Figures (3)

  • Figure 1: Comparison of the numerical and exact solutions at different resolutions for the $2D$ Poisson equation on the domain $[-1,1]\times [-1,1]$ with zero Dirichlet boundary conditions . The load term is $-200 \pi^2 \sin(10\pi x) \sin(10\pi y)$. Results are shown for different grid resolutions.
  • Figure 2: Performance analysis showcasing the execution time for solving $2D$ and $3D$ Poisson equations at various grid resolutions.
  • Figure 3: The relative error between the numerical and analytical solutions of the 2D convection-diffusion equation as a function of the number of iterations for various grid resolutions.

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Remark 2.8
  • Remark 2.9
  • Lemma 3.1
  • ...and 8 more