Low rank tensor approximation of singularly perturbed partial differential equations in one dimension

Abstract

We derive rank bounds on the quantized tensor train (QTT) compressed approximation of singularly perturbed reaction diffusion partial differential equations (PDEs) in one dimension. Specifically, we show that, independently of the scale of the singular perturbation parameter, a numerical solution with accuracy $0<\epsilon<1$ can be represented in QTT format with a number of parameters that depends only polylogarithmically on $\epsilon$. In other words, QTT compressed solutions converge exponentially to the exact solution, with respect to a root of the number of parameters. We also verify the rank bound estimates numerically, and overcome known stability issues of the QTT based solution of PDEs by adapting a preconditioning strategy to obtain stable schemes at all scales. We find, therefore, that the QTT based strategy is a rapidly converging algorithm for the solution of singularly perturbed PDEs, which does not require prior knowledge on the scale of the singular perturbation and on the shape of the boundary layers.