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Subspace gradient descent method for linear tensor equations

Martina Iannacito, Lorenzo Piccinini, Valeria Simoncini

TL;DR

Two new gradient-descent type methods for tensor equations that generalize the recently proposed Subspace Conjugate Gradient (SS-CG) are proposed that generalize the recently proposed Subspace Conjugate Gradient (SS-CG).

Abstract

The numerical solution of algebraic tensor equations is a largely open and challenging task. Assuming that the operator is symmetric and positive definite, we propose two new gradient-descent type methods for tensor equations that generalize the recently proposed Subspace Conjugate Gradient (SS-CG), D. Palitta et al, SIAM J. Matrix Analysis and Appl (2025). As our interest is mainly in a modest number of tensor modes, the Tucker format is used to efficiently represent low-rank tensors. Moreover, mixed-precision strategies are employed in certain subtasks to improve the memory usage, and different preconditioners are applied to enhance convergence. The potential of our strategies is illustrated by experimental results on tensor-oriented discretizations of three-dimensional partial differential equations with separable coefficients. Comparisons with the state-of-the-art Alternating Minimal Energy (AMEn) algorithm confirm the competitiveness of the proposed strategies.

Subspace gradient descent method for linear tensor equations

TL;DR

Two new gradient-descent type methods for tensor equations that generalize the recently proposed Subspace Conjugate Gradient (SS-CG) are proposed that generalize the recently proposed Subspace Conjugate Gradient (SS-CG).

Abstract

The numerical solution of algebraic tensor equations is a largely open and challenging task. Assuming that the operator is symmetric and positive definite, we propose two new gradient-descent type methods for tensor equations that generalize the recently proposed Subspace Conjugate Gradient (SS-CG), D. Palitta et al, SIAM J. Matrix Analysis and Appl (2025). As our interest is mainly in a modest number of tensor modes, the Tucker format is used to efficiently represent low-rank tensors. Moreover, mixed-precision strategies are employed in certain subtasks to improve the memory usage, and different preconditioners are applied to enhance convergence. The potential of our strategies is illustrated by experimental results on tensor-oriented discretizations of three-dimensional partial differential equations with separable coefficients. Comparisons with the state-of-the-art Alternating Minimal Energy (AMEn) algorithm confirm the competitiveness of the proposed strategies.
Paper Structure (16 sections, 2 theorems, 51 equations, 1 figure, 4 tables, 4 algorithms)

This paper contains 16 sections, 2 theorems, 51 equations, 1 figure, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Subspace gradient descent methods for multilinear system
  4. Conjugate gradients
  5. Steepest descent method
  6. Implementation
  7. The Tucker format and tensor operations
  8. Computation of the tensor coefficients
  9. The complete preconditioned gradient descent algorithms
  10. Matrix preconditioning
  11. Eigenvalue factorization
  12. Fast Fourier Transforms
  13. Inner-Outer preconditioning
  14. Numerical experiments
  15. Conclusions
  16. ...and 1 more sections

Key Result

Proposition 2.1

The minimizer $\bm{\alpha}_k\in\mathbb{R}^{\underline{\bm{s}}_k}$ of eq:min:alpha is the unique solution of where for $\underline{\bm{\ell}}, \underline{\bm{h}}\in\mathbb{R}^{\underline{\bm{s}}_k}$ and $\underline{\bm{i}}, \underline{\bm{j}}\in\mathbb{R}^{\underline{\bm{n}}}$ multilinear indices,

Figures (1)

  • Figure 1: Convergence history for $n=1000$, tol=$10^{-4}$ and ${\tt maxrank}=10$ (left) and ${\tt maxrank}=15$ (right).

Theorems & Definitions (9)

  • Proposition 2.1
  • Proof 1
  • Remark 2.2
  • Proposition 2.3
  • Proof 2
  • Example 5.1
  • Example 5.2
  • Example 5.3
  • Example 5.4