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A tensor Alternating Anderson-Richardson method for solving multilinear systems with M-tensors

Jing Niu, Lei Du, Tomohiro Sogabe, Shao-Liang Zhang

TL;DR

The paper tackles solving multilinear systems $\mathcal{A}\boldsymbol{x}^{m-1}=\boldsymbol{b}$ where $\mathcal{A}$ is a nonsingular $\mathcal{M}$-tensor. It extends the Alternating Anderson--Richardson framework to tensors, introducing the Tensor Richardson (TR), Tensor Anderson--Richardson (TAR), and Tensor Alternating Anderson--Richardson (TAAR) methods, and embeds TAR periodically to form TAAR. Through theoretical formulation and extensive experiments, TAAR demonstrates at least an order-of-magnitude acceleration over existing tensor-splitting methods while maintaining accuracy, with favorable computational cost profiles. The results support TAAR as an effective, scalable approach for high-dimensional tensor equations arising in applications like PDE discretizations and data-driven tensor problems.

Abstract

It is well-known that a multilinear system with a nonsingular M-tensor and a positive right-hand side has a unique positive solution. Tensor splitting methods generalizing the classical iterative methods for linear systems have been proposed for finding the unique positive solution. The Alternating Anderson-Richardson (AAR) method is an effective method to accelerate the classical iterative methods. In this study, we apply the idea of AAR for finding the unique positive solution quickly. We first present a tensor Richardson method based on tensor regular splittings, then apply Anderson acceleration to the tensor Richardson method and derive a tensor Anderson-Richardson method, finally, we periodically employ the tensor Anderson-Richardson method within the tensor Richardson method and propose a tensor AAR method. Numerical experiments show that the proposed method is effective in accelerating tensor splitting methods.

A tensor Alternating Anderson-Richardson method for solving multilinear systems with M-tensors

TL;DR

The paper tackles solving multilinear systems where is a nonsingular -tensor. It extends the Alternating Anderson--Richardson framework to tensors, introducing the Tensor Richardson (TR), Tensor Anderson--Richardson (TAR), and Tensor Alternating Anderson--Richardson (TAAR) methods, and embeds TAR periodically to form TAAR. Through theoretical formulation and extensive experiments, TAAR demonstrates at least an order-of-magnitude acceleration over existing tensor-splitting methods while maintaining accuracy, with favorable computational cost profiles. The results support TAAR as an effective, scalable approach for high-dimensional tensor equations arising in applications like PDE discretizations and data-driven tensor problems.

Abstract

It is well-known that a multilinear system with a nonsingular M-tensor and a positive right-hand side has a unique positive solution. Tensor splitting methods generalizing the classical iterative methods for linear systems have been proposed for finding the unique positive solution. The Alternating Anderson-Richardson (AAR) method is an effective method to accelerate the classical iterative methods. In this study, we apply the idea of AAR for finding the unique positive solution quickly. We first present a tensor Richardson method based on tensor regular splittings, then apply Anderson acceleration to the tensor Richardson method and derive a tensor Anderson-Richardson method, finally, we periodically employ the tensor Anderson-Richardson method within the tensor Richardson method and propose a tensor AAR method. Numerical experiments show that the proposed method is effective in accelerating tensor splitting methods.
Paper Structure (19 sections, 5 theorems, 75 equations, 3 figures, 6 tables, 3 algorithms)

This paper contains 19 sections, 5 theorems, 75 equations, 3 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notions
  4. Tensor splitting methods
  5. Tensor splitting method 1
  6. Tensor splitting method 2
  7. Tensor splitting method 3
  8. Alternating Anderson--Richardson method
  9. Alternating Anderson--Richardson method
  10. Richardson method
  11. Anderson--Richardson method
  12. Alternating Anderson--Richardson method
  13. Tensor Alternating Anderson--Richardson method
  14. Tensor Richardson method
  15. Tensor Anderson--Richardson method
  16. ...and 4 more sections

Key Result

Theorem 2.6

(r22, Theorem 2 and Theorem 3). If $\mathcal{A} \in \mathbb{R}^{[m\text{,}n]}$ is a $\mathcal{Z}$-tensor, the following conditons are equivalent

Figures (3)

  • Figure 1: The comparison of computational cost between the TAAR method with PF and other splitting methods. $x$-axis: the sum of flops up to the current iteration step.
  • Figure 2: The results for Experiment \ref{['emp4']}.
  • Figure 3: The results for Experiment \ref{['emp5']}.

Theorems & Definitions (14)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 4 more