Second-order robust parallel integrators for dynamical low-rank approximation

TL;DR

This paper introduces higher-order, robust parallel integrators for dynamical low-rank approximation, improving accuracy while maintaining efficiency and robustness to manifold curvature.

Contribution

It extends a parallel low-rank integrator to higher order by basis augmentation, enhancing accuracy without sacrificing robustness or computational efficiency.

Findings

Derived a robust error bound with better dependence on normal components.

Achieved norm preservation up to small terms.

Numerical experiments confirm theoretical improvements.

Abstract

Due to its reduced memory and computational demands, dynamical low-rank approximation (DLRA) has sparked significant interest in multiple research communities. A central challenge in DLRA is the development of time integrators that are robust to the curvature of the manifold of low-rank matrices. Recently, a parallel robust time integrator that permits dynamic rank adaptation and enables a fully parallel update of all low-rank factors was introduced. Despite its favorable computational efficiency, the construction as a first-order approximation to the augmented basis-update & Galerkin integrator restricts the parallel integrator's accuracy to order one. In this work, an extension to higher order is proposed by a careful basis augmentation before solving the matrix differential equations of the factorized solution. A robust error bound with an improved dependence on normal components of the vector field together with a norm preservation property up to small terms is derived. These analytic results are complemented and demonstrated through a series of numerical experiments.