Robust Alternating Direction Implicit Solver in Quantized Tensor Formats for a Three-Dimensional Elliptic PDE

Maxim V. Rakhuba

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, such 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 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 Schrödinger's equation.