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

Abstract

Let Ω_<em>i</em> ⊂ R<em>^n_i</em> , <em>i</em> = 1, . . . , <em>m</em>, be given domains. In this article, we study the low-rank approximation with respect to L²(Ω₁ × · · · × Ω<em>_m</em>) 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 [13, 14], 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.