Convergence of a low-rank Lie--Trotter splitting for stiff matrix differential equations

TL;DR

This paper introduces a robust numerical integrator for large-scale stiff matrix differential equations that efficiently computes low-rank solutions, with proven error bounds and demonstrated effectiveness on Lyapunov and Riccati equations.

Contribution

It develops a low-rank Lie--Trotter splitting method for stiff matrix differential equations with independent error analysis and practical algorithms for Lyapunov and Riccati problems.

Findings

Error bounds are independent of stiffness.

Method effectively handles small singular values.

Numerical experiments confirm theoretical results.

Abstract

We propose a numerical integrator for determining low-rank approximations to solutions of large-scale matrix differential equations. The considered differential equations are semilinear and stiff. Our method consists of first splitting the differential equation into a stiff and a non-stiff part, respectively, and then following a dynamical low-rank approach. We conduct an error analysis of the proposed procedure, which is independent of the stiffness and robust with respect to possibly small singular values in the approximation matrix. Following the proposed method, we show how to obtain low-rank approximations for differential Lyapunov and for differential Riccati equations. Our theory is illustrated by numerical experiments.