In this work we demonstrate that SVD-based model reduction techniques known for ordinary differential equations, such as the proper orthogonal decomposition, can be extended to stochastic differential equations in order to reduce the computational cost arising from both the high dimension of the considered stochastic system and the large number of independent Monte Carlo runs. We also extend the proper symplectic decomposition method to stochastic Hamiltonian systems, both with and without external forcing, and argue that preserving the underlying symplectic or variational structures results in more accurate and stable solutions that conserve energy better than when the non-geometric approach is used. We validate our proposed techniques with numerical experiments for a semi-discretization of the stochastic nonlinear Schr\"odinger equation and the Kubo oscillator.

翻译：本文证明，针对常微分方程的基于奇异值分解的模型降阶技术（如本征正交分解）可推广至随机微分方程，以降低由随机系统的高维度和大量独立蒙特卡洛模拟所导致的计算成本。我们还将本征辛分解方法拓展至带外力与不带外力的随机哈密顿系统，并论证保持底层辛或变分结构可得到比非几何方法更精确、更稳定的解，且能量守恒性更优。通过随机非线性薛定谔方程半离散化和久保振荡器的数值实验，我们验证了所提出技术的有效性。