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

TL;DR

This paper demonstrates that tensor-structured approximations of functions with point singularities in three-dimensional space can achieve polylogarithmic rank bounds and exponential convergence rates, with uniform bounds regardless of singularity position.

Contribution

It establishes polylogarithmic tensor rank bounds and exponential convergence rates for quantized tensor decompositions of functions with point singularities, including uniform bounds for boundary problems.

Findings

Tensor ranks are polylogarithmically bounded with respect to accuracy.

Exponential convergence of QTT, transposed QTT, and Tucker-QTT decompositions.

Uniform bounds for tensor approximations regardless of singularity location.

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.