Multi-symplectic Problems for Stochastic Hamiltonian System

1. Multi-symplectic Problems for Stochastic Hamiltonian System Shanshan Jiang*, Jialin Hong**, Lijin Wang** *Beijing University of Chemical Technology, Beijing , China **Chinese Academy of Sciences, Beijing , China Nanjing , Dec 15 , 2012

2. Stochastic Numerical Methods for Stochastic Korteweg-de Vries Equation Outline: • Stochastic Hamiltonian ODEs and Stochastic • Symplectic Structure • Stochastic Hamiltonian PDEs and Stochastic • Multi-Symplectic Conservation law • Further Problems

3. Deterministic Hamiltonian ODEs have the form of Here, P and Q are d-dimensional variables. Proposition1[1]:The phase flows of the deterministic Hamiltonian ODEs preserve the symplectic structure:

4. Stochastic Hamiltonian ODEs are defined as Here, P and Q are d-dimensional variables, and W(t) is the standard Wiener process, and o means Stratonovich product. Proposition2[2]: The phase flow of the above system preserves the stochastic symplectic structure:

5. We have some conclusions: The above two systems are called Hamiltonian systems, both deterministic and stochastic cases. The above Hamiltonian systems possess some geometric property, i.e. the symplectic structures. Many numerical methods are investigated to simulate these systems, especially those methods which can preserve the geometric structure.

6. Properties of various ODEs systems

7. Deterministic Hamiltonian PDEs are written as Here, M and K are skew-symmetric matrices. Proposition3[3]: The system possesses the multi-symplectic conservation law , which is the local geometric structure: are differential 2-form.

8. We ask some questions: What kind of Stochastic Partial Differential Equations can be considered as the Stochastic Hamiltonian PDEs? Whether this kind of Stochastic Hamiltonian PDEs also possesses some kind of stochastic geometric properties ? This kind of Stochastic Hamiltonian system is exist or not ? How about their practical significance of application ?

9. Properties of various PDEs systems

10. We propose a kind of Stochastic Hamiltonian PDEs: is real-valued white noise, which is delta correlated in time, and either smooth or delta correlated in space. Here, M and K are two skew-symmetric matrices. • There are some mathematical expression[4]: • Define the cylindrical wiener process on , the space of square • integrable functions associated to the stochastic basis

11. Theorem 1 [5]:The stochastic Hamiltonian PDE preserves the stochastic multi-symplectic conservation law locally in any definition domain : 2. is a sequence of independent real Brownian motions, is any orthonormal basis of 3. The space-time white noise has the form

12. Deterministic Korteweg-de Vries equation Initial-boundary problem possesses infinite invariants functionals,

13. Introduce potential variable and momentum variable Set with The equation is transformed to the multi-symplectic PDE

14. Stochastic Korteweg–de Vries equation with additive noise: Further set corresponding to the deterministic case. represents the amplitude of noise source. The equation is transformed to the stochastic multi-symplectic PDE:

15. The space-time white noise Correlation function Theorem 2: The stochastic Korteweg-de Vries equation preserves the stochastic multi-symplectic conservation law locally in any domain

16. Recursion of the average invariants, We see that the global errors of the averages invariants are related to

17. Numerical Methods: Midpoint Rule Method(MP) Theorem 3: The discretization (MP) is a stochastic multi-symplectic integrator, and it can preserve thediscrete multi-sysmplectic conservation law

18. Finally get 8-pointMP Scheme:

19. Numerical Experiments The profile of numerical solution as and

20. The profile of conservation laws as

21. The profile of conservation laws as

22. Ratio of transformation

23. We get some conclusions: Korteweg-de Vries equation with additive noise can be considered as the Stochastic Hamiltonian PDE. Stochastic Hamiltonian PDEs possesses some kind of stochastic geometric properties . Multi-symplectic schemes can stably simulate the stochastic KdV equation for a long time interval, just as applied to the deterministic case.

24. Further Problems The mean square orders of discrete integrators: theoretical proof and numerical simulations. Various schemes, for example conservative schemes, for the stochastic Hamiltonian systems. Other kind of partial differential equations which are included in the field of Stochastic Hamiltonian systems exist in practical significance of application.

25. References: [1] E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations, Springer-Verlag, 2002 [2] G. Milstein, M. Tretyakov, Stochastic Numierics for Mathematical Physics, Kluwer Axcademic Publisher, 1995 [3] T. Bridges, S.Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001),184-193 [4] A. Debussche, J. Printems, Numerical Simulation of the Stochastic Korteweg-de Vries Equation, Phys. D, 134 (1999) 200-226 [5] S. Jiang, L. Wang, J. Hong, Stochastic Multi-symplectic Integrator for Stochastic Nonlinear Schrodinger Equation, Comm. Comput. Phys. (2013 accepted)

26. Thanks for your attention!