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 lowrank 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.

Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs

“…Difficulties with the numerical stability of solvers for large L have also been noted previously in [34]. In [11,46], a reformulation as a constrained minimization problem with Volterra integral operators is proposed. It is demonstrated numerically in [11] up to L ≈ 20 to lead to improved numerical stability, compared to a direct finite difference discretization, for Poisson-type problems with D = 2 dimensions.…”

“…As a main contribution of this work, we introduce basic notions and auxiliary results for studying the representation conditioning of tensor train representations. In particular, our finding that the stiffness matrix represented in lowrank format has a representation condition number of order 2 2L explains numerical instabilities in its direct application for large L as observed in tests in [11]. We prove a new result on a BPX preconditioner for second-order elliptic problems that is tailored to our purposes, and we construct a low-rank decomposition of the preconditioned stiffness matrix with the following properties: it is well-conditioned uniformly in discretization level L as a matrix; its ranks are independent of L; and its representation condition numbers remain moderate for large L. Based on these properties, we establish an estimate for the total computational complexity of finding approximate solutions in low-rank form.…”

“…In [11,46], a reformulation as a constrained minimization problem with Volterra integral operators is proposed. It is demonstrated numerically in [11] up to L ≈ 20 to lead to improved numerical stability, compared to a direct finite difference discretization, for Poisson-type problems with D = 2 dimensions. However, in this reformulation, which so far has been studied only experimentally, the matrix condition number still grows exponentially with respect to L, and numerical stability is still observed to be lacking for larger values of L.…”