Drift-preserving numerical integrators for stochastic Poisson systems

Abstract

We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments.

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

“…Furthermore, it would be interesting to extend the methods discussed in our work to stochastic non-canonical systems. For instance, one could consider structure-preserving model reduction techniques for stochastic Poisson systems by combining the method developed in [76] with the stochastic geometric integrators presented in [48] and [50]. This would be of great interest for systems appearing in gyrokinetic and guiding-center theories (see [14], [23], [28], [29], [39], [124], [125], [140], [148], [152]).…”

“…However, since these models are Hamiltonian, it is advisable to integrate them using structure-preserving methods. Stochastic symplectic integrators, similar to their deterministic counterparts, preserve the symplecticity of the Hamiltonian flow and demonstrate good energy behavior in long-time simulations (see [7], [8], [9], [10], [27], [35], [37], [45], [48], [50], [51], [60], [79], [81], [83], [94], [106], [107], [111], [112], [113], [142], [153], [154], [156], [159] and the references therein). In this work we will focus on two stochastic symplectic Runge-Kutta methods, namely the stochastic midpoint method (2.10), which is symplectic when applied to a Hamiltonian system, and the stochastic Störmer-Verlet method ( [81], [94], [107]).…”

“…In particular, their phase space flows (almost surely) preserve the canonical symplectic structure. We will argue that maintaining this property in model reduction is beneficial because the resulting reduced systems can then be integrated using stochastic symplectic methods (see [7], [8], [9], [10], [27], [35], [37], [45], [48], [50], [51], [60], [79], [81], [83], [94], [106], [107], [111], [112], [113], [142], [153], [154], [156], [159]), and consequently more accurate numerical solutions can be obtained. This goal can be achieved by an appropriate adaptation of the PSD method in the stochastic setting.…”

Splitting integrators for stochastic Lie--Poisson systems

“…To conclude this subsection, we would like to remark that the above analysis of the preservation properties of stochastic Poisson systems is only valid when considering stochastic differential equations with multiplicative noise interpreted in the Stratonovich sense. Other behaviours are observed for Itô SDEs, see for instance [14,18], where Hamiltonian and Poisson Itô SDEs and their numerical discretisations are studied. Indeed, if the noise is interpreted in the Itô sense, the Hamiltonian function H or the Casimir functions C are not preserved.…”

“…The main contribution of this manuscript is the analysis of a class of explicit stochastic Poisson integrators, see equation (18), based on a splitting strategy. The splitting strategy is often applicable for stochastic Lie-Poisson systems, which have a structure matrix B(y) which depends linearly on y.…”