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Tensor Rank bounds for Point Singularities in R³

by C. Marcati and M. Rakhuba and Ch. Schwab

(Report number 2019-68)

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 Schrödinger-type eigenvalue problems illustrate the theoretical results.

Keywords: Quantized Tensor Train, tensor networks, low-rank approximation, exponential convergence, Schrödinger equation

@Techreport{MRS19_872,
  author = {C. Marcati and M. Rakhuba and Ch. Schwab},
  title = {Tensor Rank bounds for Point Singularities in R³},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-68},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-68.pdf },
  year = {2019}
}

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