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:在这项工作中,我们证明,基于SVD的减少模式技术,以普通差异方程式为名,如正正正正正正正的正方形分解,可以推广到随机分解方程式,以降低考虑的随机系统高尺寸和大量独立的蒙特卡洛运行所产生的计算成本。我们还将适当的静脉分解方法推广到有外力和无外力的随机汉密尔顿系统,并论证说,维护基本的随机结构或变异结构,可以产生比使用非几何方法时更准确、更稳定的节能解决方案。我们用数字实验来验证我们提出的技术,以半分解非直线性非辛基方程式和Kubo振动器的半分解法。