2022/01/31 by Tomasz M. Tyranowski, Tyranowski, Tomasz M. · 1 citation
Engineering · Mathematics · Physics and Astronomy · #35R60 #37M15 #53Z50 #60H10 #60H15 #60H35 #65C30 #65P10 #68T09 #Advanced Numerical Methods in Computational Mathematics #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2201.13391
openalex publication_date 2022/01/31 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28
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.