5 research outputs found
Data-driven structure-preserving model reduction for stochastic Hamiltonian systems
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.Comment: 43 pages, 16 figure
Stochastic discrete Hamiltonian variational integrators
Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge–Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods