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

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 Griebel and Harbrecht [IMA J. Numer. Anal. 34 (2014), pp. 28–54] and Griebel and Harbrecht [IMA J. Numer. Anal. 39 (2019), pp. 1652–1671], 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.