Tensor Rank bounds for Point Singularities in $\mathbb{R}^3$

Abstract

We analyze rates of approximation by quantized, tensor-structured representations of functions with isolated point singularities in ${\mathbb R}^3$. We consider functions in countably normed Sobolev spaces with radial weights and analytic- or Gevrey-type control of weighted semi-norms. Several classes of boundary value and eigenvalue problems from science and engineering are discussed whose solutions belong to the countably normed spaces. It is shown that quantized, tensor-structured approximations of functions in these classes exhibit tensor ranks bounded polylogarithmically with respect to the accuracy $\epsilon\in(0,1)$ in the Sobolev space $H^1$. We prove exponential convergence rates of three specific types of quantized tensor decompositions: quantized tensor train (QTT), transposed QTT and Tucker-QTT. In addition, the bounds for the patchwise decompositions are uniform with respect to the position of the point singularity. An auxiliary result of independent interest is the proof of exponential convergence of $hp$-finite element approximations for Gevrey-regular functions with point singularities in the unit cube $Q=(0,1)^3$. Numerical examples of function approximations and of Schr\"odinger-type eigenvalue problems illustrate the theoretical results.