Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

This article analyzes an explicit temporal splitting numerical scheme for the stochastic Allen-Cahn equation driven by additive noise, in a bounded spatial domain with smooth boundary in dimension $d\le 3$. The splitting strategy is combined with an exponential Euler scheme of an auxiliary problem. 
When $d=1$ and the driving noise is a space-time white noise, we first show some a priori estimates of this splitting scheme. Using the monotonicity of the drift nonlinearity, we then prove that under very mild assumptions on the initial data, this scheme achieves the optimal strong convergence rate $\OO(\delta t^{\frac 14})$. When $d\le 3$ and the driving noise possesses some regularity in space, we study exponential integrability properties of the exact and numerical solutions. Finally, in dimension $d=1$, these properties are used to prove that the splitting scheme has a strong convergence rate $\OO(\delta t)$.

Strong convergence rates of semi-discrete splitting approximations for stochastic Allen--Cahn equation

Strong convergence rate of splitting schemes for stochastic nonlinear Schrödinger equations

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.

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Stochastic discrete Hamiltonian variational integrators

The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel Monte Carlo estimators are studied. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.

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Drift-preserving numerical integrators for stochastic Hamiltonian systems

Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Stochastic symplectic and multi-symplectic methods for nonlinear Schrdinger equation with white noise dispersion

In this paper, we consider the numerical methods preserving single or multiple conserved quantities, and these methods are able to reach high order of strong convergence simultaneously based on some kinds of projection methods. The mean-square convergence orders of these methods under certain conditions are given, which can reach order 1.5 or even 2 according to the supporting methods embedded in the projection step. Finally, three numerical experiments are taken into account to show the superiority of the projection methods.

Projection methods for stochastic differential equations with conserved quantities

In this paper, we propose a conformal momentum-preserving method to solve a damped nonlinear Schrodinger (DNLS) equation. Based on its damped multi-symplectic formulation, the DNLS system can be split into a Hamiltonian part and a dissipative part. For the Hamiltonian part, the average vector field (AVF) method and implicit midpoint method are employed in spatial and temporal discretizations, respectively. For the dissipative part, we can solve it exactly. The proposed method conserves the conformal momentum conservation law in any local time–space region. With periodic boundary conditions, this method also preserves the total conformal momentum and the dissipation rate of momentum exactly. Numerical experiments are presented to demonstrate the conservative properties of the proposed method.

https://iopscience.iop.org/article/10.1088/1674-1056/25/11/110201/pdf

Weien Zhou

Papers

Strong convergence rate of splitting schemes for stochastic nonlinear Schrödinger equations

Stochastic symplectic Runge–Kutta methods for the strong approximation of Hamiltonian systems with additive noise

Stochastic symplectic and multi-symplectic methods for nonlinear Schrdinger equation with white noise dispersion

Projection methods for stochastic differential equations with conserved quantities

Conformal structure-preserving method for damped nonlinear Schrödinger equation*