Stochastic Symplectic Methods Research Articles

Strong backward error analysis of symplectic integrators for stochastic Hamiltonian systems

Backward error analysis is a powerful tool in order to detect the long-term conservative behavior of numerical methods. In this work, we present a long-term analysis of symplectic stochastic numerical integrators, applied to Hamiltonian systems with multiplicative noise. We first compute and analyze the associated stochastic modified differential equations. Then, suitable bounds for the coefficients of such equations are provided towards the computation of long-term estimates for the Hamiltonian deviations occurring along the aforementioned numerical dynamics. This result generalizes Benettin-Giorgilli Theorem to the scenario of stochastic symplectic methods. Finally, specific numerical methods are considered, in order to provide a numerical evidence confirming the effectiveness of the theoretical investigation.

Asymptotically-Preserving Large Deviations Principles by Stochastic Symplectic Methods for a Linear Stochastic Oscillator

It is well known that symplectic methods have been rigorously shown to be superior to nonsymplectic ones especially in long-time computation, when applied to deterministic Hamiltonian systems. In this paper, we attempt to study the superiority of stochastic symplectic methods by means of the large deviations principle. We propose the concept of asymptotical preservation of numerical methods for large deviations principles associated with the exact solutions of the general stochastic Hamiltonian systems. Considering that the linear stochastic oscillator is one of the typical stochastic Hamiltonian systems, we take it as the test equation in this paper to obtain precise results about the rate functions of large deviations principles for both exact and numerical solutions. Based on the Gärtner--Ellis theorem, we first study the large deviations principles of the mean position and the mean velocity for both the exact solution and its numerical approximations. Then, we prove that stochastic symplectic methods asymptotically preserve these two large deviations principles, but nonsymplectic ones do not. This indicates that stochastic symplectic methods are able to approximate well the exponential decay speed of the “hitting probability" of the mean position and mean velocity of the stochastic oscillator. Finally, numerical experiments are performed to show the superiority of stochastic symplectic methods in computing the large deviations rate functions. To the best of our knowledge, this is the first result about applying the large deviations principle to reveal the superiority of stochastic symplectic methods compared with nonsymplectic ones in the existing literature.

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.

Stochastic symplectic methods based on the Padé approximations for linear stochastic Hamiltonian systems

In this article, we propose a kind of numerical methods based on the Padé approximations, for two kinds of stochastic Hamiltonian systems. For the general linear stochastic Hamiltonian systems, it is shown that the applied Padé approximations P(k,k) produce numerical solutions that are symplectic, and the proposed numerical schemes based on P(r,s) are of root-mean-square convergence order r+s2. For a special kind of linear stochastic Hamiltonian systems with additive noises, the numerical methods using two kinds of Padé approximations, P(rˆ,sˆ) and P(ř,1), possess root-mean-square convergence order ř+2 when rˆ+sˆ=ř+3, and are symplectic if rˆ=sˆ. These generalize the Padé approximation approaches for symplectic integration of linear Hamiltonian systems to the stochastic context.

Modified equations for weakly convergent stochastic symplectic schemes via their generating functions

In this paper, an approach of constructing modified equations of weak $$k+k'$$ order ( $$k'\ge 1$$ ) apart from the k-th order weakly convergent stochastic symplectic methods, i.e., stochastic symplectic methods with respect to weak convergence and of weak order k, is given using the underlying generating functions of them. This approach is valid for stochastic Hamiltonian systems with additive noises, and those with multiplicative noises but for which the Hamiltonian functions $$H_r(p,q),\,\,r\ge 1$$ associated to the diffusion parts depend only on p or only on q. In such cases, we find that the modified equations of the weakly convergent stochastic symplectic methods are perturbed stochastic Hamiltonian systems of the original systems.

Generating functions for stochastic symplectic methods

Symplectic integration of stochastic Hamiltonian systems is adeveloping branch of stochastic numerical analysis. In the presentpaper, a stochastic generating function approach is proposed, basedon the derivation of stochastic Hamilton-Jacobi PDEs satisfied bythe generating functions, and a method of approximating solutions tothem. Thus, a systematic approach of constructing stochasticsymplectic methods is provided. As validation, numerical tests onseveral stochastic Hamiltonian systems are performed, where some symplectic schemes are constructed via stochastic generatingfunctions. Moreover, generating functions for some known stochasticsymplectic mappings are given.