A projector-splitting integrator for dynamical low-rank approximation

TL;DR

This paper introduces a fully explicit, efficient integrator for dynamical low-rank matrix approximation that is robust and adaptable, improving numerical stability and enabling efficient rank truncation.

Contribution

It proposes a novel projector-splitting integrator for low-rank matrix dynamics, offering robustness and efficiency over standard methods.

Findings

The integrator is robust and maintains stability in numerical experiments.

It allows for adaptive rank adjustments during computations.

Efficiently facilitates low-rank matrix truncation in optimization algorithms.

Abstract

The dynamical low-rank approximation of time-dependent matrices is a low-rank factorization updating technique. It leads to differential equations for factors of the matrices, which need to be solved numerically. We propose and analyze a fully ex- plicit, computationally inexpensive integrator that is based on splitting the orthogonal projector onto the tangent space of the low-rank manifold. As is shown by theory and illustrated by numerical experiments, the integrator enjoys robustness properties that are not shared by any standard numerical integrator. This robustness can be exploited to change the rank adaptively. Another application is in optimization algorithms for low-rank matrices where truncation back to the given low rank can be done efficiently by applying a step of the integrator proposed here.