Robust solver in a quantized tensor format for three-dimensional elliptic problems

2019. No. 30.

Rakhuba M.

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 ill-conditioning of discretized differential operators, such fine grids lead to numerical instabilities. The idea to obtain a robust solver is to 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 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: tensor networks, elliptic pdes, ADI method, preconditioning, robust computations