Robust discretization in quantized tensor train format for elliptic problems in two dimensions

Abstract: In this work we propose an efficient black-box solver for two-dimensional stationary diffusion equations, which is based on a new robust discretization scheme. The idea is to formulate an equation in a certain form without derivatives with a non-local stencil, which leads us to a linear system of equations with dense matrix. This matrix and a right-hand side are represented in a low-rank parametric representation -- the quantized tensor train (QTT-) format, and then all operations are performed with logarithmic complexity and memory consumption. Hence very fine grids can be used, and very accurate solutions with extremely high spatial resolution can be obtained. Numerical experiments show that this formulation gives accurate results and can be used up to $2^{60}$ grid points with no problems with conditioning, while total computational time is around several seconds.