Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations

TL;DR

This paper establishes the existence, stability, and convergence of dynamical low-rank tensor approximations for high-dimensional parabolic problems, accommodating various tensor formats like tensor train and hierarchical tensors.

Contribution

It provides a rigorous framework proving existence, stability, and convergence of low-rank tensor methods for complex high-dimensional PDEs, extending previous approaches.

Findings

Proves existence of solutions for dynamical low-rank tensor approximations.

Shows stability of solutions under perturbations.

Demonstrates convergence of spatial discretizations in the tensor framework.

Abstract

We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability with respect to perturbations to the problem data, we show convergence of spatial discretizations. Our framework accommodates various standard low-rank tensor formats for multivariate functions, including tensor train and hierarchical tensors.