Iterative thresholding low-rank time integration

TL;DR

This paper introduces an adaptive low-rank time integration method using iterative soft thresholding, effectively balancing accuracy and computational efficiency for time-dependent PDEs like Schrödinger and parabolic equations.

Contribution

It presents a novel iterative thresholding scheme that adaptively adjusts approximation ranks while maintaining accuracy, applicable to various time-dependent PDEs.

Findings

Effective rank adaptation in Schrödinger equation simulations

Maintains proportionality to best approximation ranks

Numerical tests confirm accuracy and efficiency

Abstract

We develop time integration methods in low-rank representation that can adaptively adjust approximation ranks to achieve a prescribed accuracy, while ensuring that these ranks remain proportional to the corresponding best approximation ranks. Our approach relies on an iterative scheme combined with soft thresholding of the iterates. A model case of a time-dependent Schr\"odinger equation with low-rank matrix approximation is analyzed in detail, and the required modifications for second-order parabolic problems are described. Numerical tests illustrate the results for both cases.