Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness

TL;DR

This paper investigates low-rank approximation techniques for multivariate functions in Sobolev spaces with dominating mixed smoothness, demonstrating improved convergence and the ability to mitigate the curse of dimensionality.

Contribution

It provides new estimates for the rank of continuous singular value decompositions and applies tensor train decomposition to efficiently approximate high-dimensional functions.

Findings

Improved rank estimates for bivariate approximations.

Tensor train decomposition effectively reduces complexity in high dimensions.

Method surpasses traditional approaches in handling the curse of dimensionality.

Abstract

Let $\Omega_i\subset\mathbb{R}^{n_i}$, $i=1,\ldots,m$, be given domains. In this article, we study the low-rank approximation with respect to $L^2(\Omega_1\times\dots\times\Omega_m)$ of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare \cite{GH14,GH19}, we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.