TT-GMRES: on solution to a linear system in the structured tensor format

TL;DR

This paper introduces TT-GMRES, a tensor-structured GMRES method for solving high-dimensional linear systems in the Tensor Train format, demonstrating improved convergence and performance over existing methods in numerical experiments.

Contribution

The paper develops and analyzes a tensor-structured GMRES method tailored for the TT format, including relaxation techniques to enhance convergence and performance.

Findings

TT-GMRES shows comparable performance to ALS and DMRG methods.

Relaxation techniques improve convergence of the TT-GMRES method.

TT-GMRES outperforms ALS solver with a good preconditioner in high-dimensional problems.

Abstract

A adapted tensor-structured GMRES method for the TT format is proposed and investigated. The Tensor Train (TT) approximation is a robust approach to high-dimensional problems. One class of problems is solution of a linear system. In this work we study the convergence of the GMRES method in the presence of tensor approximations and provide relaxation techniques to improve its performance. Several numerical examples are presented. The method is also compared with a projection TT linear solver based on the ALS and DMRG methods. On a particular sPDE (high-dimensional parametric) problem, these methods manifest comparable performance, with a good preconditioner the TT-GMRES overcomes the ALS solver.