A space-time adaptive low-rank method for high-dimensional parabolic partial differential equations

TL;DR

This paper introduces an adaptive approach combining sparse wavelet expansions and low-rank approximations for high-dimensional parabolic PDEs, ensuring convergence, efficiency, and providing rigorous error bounds.

Contribution

It develops a novel adaptive method that integrates wavelet and low-rank techniques for high-dimensional parabolic PDEs, with proven convergence and error estimation.

Findings

Method converges and has similar complexity bounds as elliptic low-rank methods.

Provides computable rigorous a posteriori error bounds.

Numerical experiments validate the approach.

Abstract

An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and satisfy similar complexity bounds as existing adaptive low-rank methods for elliptic problems, establishing its suitability for parabolic problems on high-dimensional spatial domains. The construction also yields computable rigorous a posteriori error bounds for such problems. The results are illustrated by numerical experiments.