Data-driven structure-preserving model reduction for stochastic Hamiltonian systems

TL;DR

The paper addresses the computational bottlenecks of high-dimensional stochastic systems and large numbers of Monte Carlo realizations by extending POD to SDEs and PSD to stochastic Hamiltonian systems. It develops both DEIM and SDEIM to handle nonlinear terms while preserving geometric structure through cotangent-lift symplectic projections and appropriate time integrators. Numerical experiments on a semi-discretized stochastic nonlinear Schrödinger equation and the Kubo oscillator demonstrate that structure-preserving reductions yield superior energy conservation and long-time accuracy, with substantial speedups over full-order simulations. These results validate the utility of geometry-aware model reduction for stochastic PDEs and motivate extensions to other stochastic PDEs and kinetic/plasma models.

Abstract

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.

Figures (16)

• Figure 5.1: The decay of the singular values for the POD and PSD reductions, and for the DEIM approximation of the nonlinear term for the empirical data ensemble for the stochastic NLS equation.
• Figure 5.2: The solution $|\psi(x,t)|=|q(x,t)+ip(x,t)|$ of the stochastic NLS equation with the parameters $\beta=0.15$ and $\epsilon=1$ at times $t=100$ (Left) and $t=200$ (Right) obtained with the help of the reduced models integrated with the stochastic midpoint method using the time step $\Delta t=0.01$. While at $t=100$ all simulations resolve the soliton relatively well, at $t=200$ the PSD models yield a more accurate solution than the POD models of the same dimension. The POD+DEIM and PSD+SDEIM simulations capture the propagation of the soliton, but create some spurious oscillations in its tail.
• Figure 5.3: The relative error $e_1(t)$ as a function of time for several example reduced model simulations of the stochastic NLS equation using the stochastic midpoint method with the time step $\Delta t=0.01$.
• Figure 5.4: The relative error $e_2$ for the reduced model simulations of the stochastic NLS equation using the stochastic midpoint method with the time step $\Delta t=0.01$ is depicted as a function of the dimension of the reduced system. Recall that the dimension of the reduced model is equal to $k$ for the POD methods, and $2k$ for the PSD methods.
• Figure 5.5: The relative error of the Hamiltonian as a function of time for several example reduced model simulations of the stochastic NLS equation using the stochastic midpoint method with the time step $\Delta t=0.01$ is depicted, where $H_0$ denotes the initial value of the Hamiltonian. The PSD simulations preserve the Hamiltonian nearly exactly, in contrast to the POD simulations. Although the PSD reduced models in combination with the SDEIM approximation do not show such a good behavior, they still preserve the Hamiltonian better than the POD models of the same dimension in combination with the DEIM approximation. Note that the plots for the PSD method with $k=95$ and $k=105$ overlap very closely and are therefore indistinguishable.

Theorems & Definitions (2)

• Theorem 3.1
• proof