A robust GMRES algorithm in Tensor Train format

TL;DR

This paper introduces a robust GMRES algorithm utilizing Tensor Train format for solving high-dimensional linear systems efficiently, with a focus on maintaining accuracy through backward error analysis and applicability to parametric problems.

Contribution

The paper presents a novel GMRES algorithm in Tensor Train format that incorporates backward error analysis to ensure solution accuracy in high-dimensional tensor-structured problems.

Findings

Effective tensor approximation impacts solution accuracy.

Backward error bounds relate high-dimensional solutions to sequence quality.

Algorithm successfully applied to academic high-dimensional examples.

Abstract

We consider the solution of linear systems with tensor product structure using a GMRES algorithm. In order to cope with the computational complexity in large dimension both in terms of floating point operations and memory requirement, our algorithm is based on low-rank tensor representation, namely the Tensor Train format. In a backward error analysis framework, we show how the tensor approximation affects the accuracy of the computed solution. With the bacwkward perspective, we investigate the situations where the $(d+1)$-dimensional problem to be solved results from the concatenation of a sequence of $d$-dimensional problems (like parametric linear operator or parametric right-hand side problems), we provide backward error bounds to relate the accuracy of the $(d+1)$-dimensional computed solution with the numerical quality of the sequence of $d$-dimensional solutions that can be extracted form it. This enables to prescribe convergence threshold when solving the $(d+1)$-dimensional problem that ensures the numerical quality of the $d$-dimensional solutions that will be extracted from the $(d+1)$-dimensional computed solution once the solver has converged. The above mentioned features are illustrated on a set of academic examples of varying dimensions and sizes.