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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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