Splitting-based randomized dynamical low-rank approximations for stiff matrix differential equations

TL;DR

This paper introduces splitting-based randomized dynamical low-rank methods for efficiently solving large-scale stiff matrix differential equations, combining exponential integrators and randomized approaches for improved robustness and accuracy.

Contribution

It proposes a novel splitting-based framework with randomized low-rank techniques for stiff matrix differential equations, including rank-adaptive extensions.

Findings

Achieves desired convergence orders on canonical problems

Demonstrates robustness and high accuracy in numerical experiments

Effective for spatially discretized Allen-Cahn and Riccati equations

Abstract

In the fields of control theory and machine learning, the dynamic low-rank approximation for large-scale matrices has received substantial attention. Considering large-scale semilinear stiff matrix differential equations, we propose splitting-based randomized dynamical low-rank approximations for a low-rank solution of the stiff matrix differential equation. We first split such the equation into a stiff linear subproblem and a nonstiff nonlinear subproblem. Then, a low-rank exponential integrator is applied to the linear subproblem. Two randomized low-rank approaches are employed for the nonlinear subproblem. Furthermore, we extend the proposed methods to rank-adaptation scenarios. Through rigorous validation on canonical stiff matrix differential problems, including spatially discretized Allen-Cahn equations and differential Riccati equations, we demonstrate that our methods achieve desired convergence orders. Numerical results confirm the robustness and accuracy of the proposed methods.