Subspace gradient descent method for linear tensor equations

TL;DR

This paper introduces new gradient descent methods for solving symmetric positive definite tensor equations, utilizing Tucker format and mixed-precision strategies, with demonstrated efficiency on PDE discretizations.

Contribution

The paper develops two novel gradient descent algorithms for tensor equations that extend existing methods, incorporating low-rank tensor representations and mixed-precision techniques.

Findings

Methods outperform state-of-the-art algorithms in experiments.

Efficient low-rank tensor representations reduce computational costs.

Preconditioning improves convergence rates.

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.