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

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.