An unconventional robust integrator for dynamical low-rank approximation

TL;DR

This paper introduces a novel robust numerical integrator for low-rank matrix and tensor approximations that improves stability and parallelism over existing methods, especially for dissipative problems.

Contribution

A new low-rank integrator that differs from the projector-splitting method, maintaining robustness, avoiding backward steps, and enhancing parallelism for matrix and tensor differential equations.

Findings

Retains robustness to small singular values

Avoids backward time integration substep

Offers increased parallelism

Abstract

We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation. Furthermore, the integrator is extended to the approximation of time-dependent tensors by Tucker tensors of fixed multilinear rank. The proposed low-rank integrator is different from the known projector-splitting integrator for dynamical low-rank approximation, but it retains the important robustness to small singular values that has so far been known only for the projector-splitting integrator. The new integrator also offers some potential advantages over the projector-splitting integrator: It avoids the backward time integration substep of the projector-splitting integrator, which is a potentially unstable substep for dissipative problems. It offers more parallelism, and it preserves symmetry or anti-symmetry of the matrix or tensor when the differential equation does. Numerical experiments illustrate the behaviour of the proposed integrator.