论文标题
随机汉密尔顿系统的数据驱动结构的模型降低
Data-driven structure-preserving model reduction for stochastic Hamiltonian systems
论文作者
论文摘要
在这项工作中，我们证明了基于SVD的模型还原技术已知的普通微分方程（例如适当的正交分解）可以扩展到随机微分方程，以减少由所考虑的随机系统的高维度和大量独立蒙特卡洛运行所产生的计算成本。我们还将适当的符号分解方法扩展到随机的汉密尔顿系统，无论是有或没有外部强迫，并认为保留基本的符号或变分结构会导致更准确，更稳定的溶液，从而比使用非几何方法更好地保存能量。我们通过数值实验验证了我们提出的技术，以对随机非线性Schrödinger方程和Kubo振荡器进行半差异。
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 Schrödinger equation and the Kubo oscillator.
