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.

翻译：在控制理论与机器学习领域，大规模矩阵的动态低秩逼近方法已受到广泛关注。针对大规模半线性刚性矩阵微分方程，本文提出基于分裂的随机化动态低秩逼近方法以获取刚性矩阵微分方程的低秩解。首先将原方程分裂为刚性线性子问题与非刚性非线性子问题。对线性子问题采用低秩指数积分器求解，对非线性子问题则采用两种随机化低秩逼近方法进行处理。进一步地，我们将所提方法扩展至秩自适应场景。通过对典型刚性矩阵微分问题（包括空间离散化Allen-Cahn方程和微分Riccati方程）的严格验证，证明了所提方法能够达到预期的收敛阶数。数值实验结果证实了该方法的鲁棒性与计算精度。