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Restarted Nonnegativity Preserving Tensor Splitting Methods via Relaxed Anderson Acceleration for Solving Multi-linear Systems

Dongdong Liu Ting Hua nd Xifu Liu

Abstract

Multilinear systems play an important role in scientific calculations of practical problems. In this paper, we consider a tensor splitting method with a relaxed Anderson acceleration for solving multilinear systems. The new method preserves nonnegativity for every iterative step and improves the existing ones. Furthermore, the convergence analysis of the proposed method is given. The new algorithm performs effectively for numerical experiments.

Restarted Nonnegativity Preserving Tensor Splitting Methods via Relaxed Anderson Acceleration for Solving Multi-linear Systems

Abstract

Multilinear systems play an important role in scientific calculations of practical problems. In this paper, we consider a tensor splitting method with a relaxed Anderson acceleration for solving multilinear systems. The new method preserves nonnegativity for every iterative step and improves the existing ones. Furthermore, the convergence analysis of the proposed method is given. The new algorithm performs effectively for numerical experiments.
Paper Structure (6 sections, 9 theorems, 37 equations, 1 figure, 5 tables)

This paper contains 6 sections, 9 theorems, 37 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor splitting methods with Anderson acceleration
  4. Convergence analysis
  5. Numerical examples
  6. Conclusion

Key Result

Lemma 2.3

wei2 Let $\mathcal{A}$ be a strong $\mathcal{M}$-tensor and ${\bf{b}}\in \mathbb{R}^{n}_{++}$. Thus, the multilinear systems ax=b have a unique positive solution.

Figures (1)

  • Figure 1: The relationship between the residual and the running time with different algorithms.

Theorems & Definitions (23)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 3.1
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Remark 4.3
  • ...and 13 more