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

TL;DR

This paper demonstrates that QTT tensor approximation efficiently and stably solves singularly perturbed 1D PDEs with accuracy independent of perturbation scale, showing exponential convergence and robustness without prior scale knowledge.

Contribution

It establishes polylogarithmic rank bounds for QTT solutions of singularly perturbed PDEs and introduces a preconditioning strategy for stability across scales.

Findings

QTT solutions converge exponentially with respect to the number of parameters.

Rank bounds are independent of the perturbation scale.

A preconditioning strategy ensures stability at all scales.

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.