Adaptive Low-Rank Methods: Problems on Sobolev Spaces

TL;DR

This paper develops an adaptive iterative solver using low-rank tensor formats for high-dimensional elliptic problems, with proven convergence, complexity bounds, and demonstrated practical efficiency.

Contribution

It introduces a novel adaptive low-rank tensor method with preconditioning for elliptic problems, capable of automatically detecting low-rank structures and ensuring convergence.

Findings

Convergence to the continuous solution with guaranteed error reduction.

Complexity bounds including tensor rank estimates during iteration.

Numerical experiments confirm practical efficiency in high dimensions.

Abstract

This paper is concerned with the development and analysis of an iterative solver for high-dimensional second-order elliptic problems based on subspace-based low-rank tensor formats. Both the subspaces giving rise to low-rank approximations and corresponding sparse approximations of lower-dimensional tensor components are determined adaptively. A principal obstruction to a simultaneous control of rank growth and accuracy turns out to be the fact that the underlying elliptic operator is an isomorphism only between spaces that are not endowed with cross norms. Therefore, as central part of this scheme, we devise a method for preconditioning low-rank tensor representations of operators. Under standard assumptions on the data, we establish convergence to the solution of the continuous problem with a guaranteed error reduction. Moreover, for the case that the solution exhibits a certain low-rank structure and representation sparsity, we derive bounds on the computational complexity, including in particular bounds on the tensor ranks that can arise during the iteration. We emphasize that such assumptions on the solution do not enter in the formulation of the scheme, which in fact is shown to detect them automatically. Our findings are illustrated by numerical experiments that demonstrate the practical efficiency of the method in high spatial dimensions.