On low-rank approximability of solutions to high-dimensional operator equations and eigenvalue problems

TL;DR

This paper investigates the conditions under which solutions to high-dimensional linear systems and eigenvalue problems can be efficiently approximated using low-rank tensor methods, ensuring scalability in complex applications.

Contribution

It develops a constructive framework that guarantees low-rank approximability in high dimensions under broad conditions, assuming the operator admits a low-rank tensor representation.

Findings

Low-rank approximability can be maintained as dimension increases.

Constructive error estimates are derived for low-rank truncations.

Applicable to a wide class of high-dimensional problems.

Abstract

Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems and eigenvalue problems on Hilbert spaces. Although this question is central to the success of all existing solvers based on low-rank tensor techniques, very few of the results available so far allow to draw meaningful conclusions for higher dimensions. In this work, we develop a constructive framework to study low-rank approximability. One major assumption is that the involved linear operator admits a low-rank representation with respect to the chosen tensor format, a property that is known to hold in a number of applications. Simple conditions, which are shown to hold for a fairly general problem class, guarantee that our derived low-rank truncation error estimates do not deteriorate as the dimensionality increases.