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Robust alternating direction implicit solver in quantized tensor formats for a three-dimensional elliptic PDE

by M. Rakhuba

(Report number 2019-30)

Abstract
The aim of this paper is to propose a robust numerical solver, which is capable of efficiently solving a three-dimensional elliptic problem in a data-sparse quantized tensor format. In particular, we use the combined Tucker and quantized tensor train format (TQTT), which allows us to use astronomically large grid sizes. However, due to the ill-conditioning of discretized differential operators, so fine grids lead to numerical instabilities. To obtain a robust solver, we utilize the well-known alternating direction implicit method and modify it to avoid multiplication by differential operators. So as to make the method efficient, we derive an explicit TQTT representation of the iteration matrix and quantized tensor train (QTT) representations of the inverses of symmetric tridiagonal Toeplitz matrices as an auxiliary result. As an application, we consider accurate solution of elliptic problems with singular potentials arising in electronic Schroedinger's equation.

Keywords: Elliptic problems, QTT decomposition, tensor networks, robust PDE solver, ADI iteration

@Techreport{R19_834,
  author = {M. Rakhuba},
  title = {Robust alternating direction implicit solver in quantized tensor formats for a three-dimensional elliptic PDE},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-30},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-30.pdf },
  year = {2019}
}

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