The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and have been successfully applied to a variety of problems. The dynamical low- \rank approximation is a recent technique for solving high-dimensional differential equations by means of low-rank approximations. However, the domain is lacking numerical methods for stiff equations since existing methods are either not robust-to-stiffness or have unreasonably large hidden constants. In this paper, we focus on solving large-scale stiff matrix differential equations with a Sylvester-like structure, that admit good low-rank approximations. We propose two new methods that have good convergence properties, small memory footprint and that are fast to compute. The theoretical analysis shows that the new methods have order one and two, respectively. We also propose a practical implementation based on Krylov techniques. The approximation error is analyzed, leading to a priori error bounds and, therefore, a mean for choosing the size of the Krylov space. Numerical experiments are performed on several examples, confirming the theory and showing good speedup in comparison to existing techniques.

翻译：刚性方程的数值积分是一个具有挑战性的问题，需要采用专门的数值方法来解决。指数积分器作为其中一类流行方法，因其对刚性问题具有理论上的鲁棒性而被成功应用于多种问题。动态低秩逼近是近年来通过低秩近似求解高维微分方程的一项技术，然而该领域仍缺乏针对刚性方程的数值方法——现有方法要么对刚性问题不够鲁棒，要么隐含常数过大且不合理。本文聚焦于求解具有Sylvester结构的大规模刚性矩阵微分方程，这类方程具有良好的低秩近似特性。我们提出两种新方法，具有良好的收敛性、较小的内存占用和更快的计算速度。理论分析表明，新型方法分别具有一阶和二阶精度。我们还提出了基于Krylov技术的实用实现方案，并分析了近似误差，从而给出先验误差界以及选择Krylov空间大小的依据。数值实验在多个算例上验证了理论分析，并与现有技术相比显示出良好的加速效果。