LOW-RANK APPROXIMATION OF CONTINUOUS FUNCTIONS IN SOBOLEV SPACES WITH DOMINATING MIXED SMOOTHNESS

Let Ω<inf>i</inf> ⊂ R^{n<inf>i</inf>}, i = 1,…,m, be given domains. In this article, we study the low-rank approximation with respect to L²(Ω<inf>1</inf> × ·· · × Ω<inf>m</inf>) 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