Hierarchical tensor approximation of high-dimensional functions of isotropic and anisotropic Sobolev smoothness

In this article, we study the hierarchical tensor decomposition of functions from both smoothness classes, isotropic Sobolev spaces and anisotropic Sobolev spaces. For this purpose, we consider the known rank estimates in case of bivariate approximation, which can be found in [14] and [16], and successively apply them to analyze the truncated hierarchical tensor decomposition. In comparison to the isotropic case, we obtain improved results with respect to anisotropic Sobolev spaces. Indeed, the associated ranks of the truncated hierarchical tensor decomposition stay essentially bounded which beats the curse of dimension that is observed in the isotropic case.