Dr. Lijin Wang, Professor, School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

Biography: Dr. Lijin Wang is currently an associate Professor at the School of Mathematical Sciences of University of Chinese Academy of Sciences. She received her Ph.D. in 2007 under joint education between the AMSS of the Chinese Academy of Sciences and the Karlsruhe Institute of Technology of Germany, and then she did postdoctoral research from 2007 to 2009 at the Tsinghua University of China. She has been working at the University of Chinese Academy of Sciences since 2009. Her research is mostly focused on numerical methods for stochastic differential equations, especially stochastic structure-preserving numerical methods for stochastic Hamiltonian systems, stochastic Poisson systems, etc..

Speech Title: Structure-preserving numerical methods for stochastic Poisson systems

Abstract: Poisson systems form a class of important mechanical systems whose long history dates back to the 19th century. As a generalization of the Hamiltonian systems which are defined on even-dimensional symplectic manifolds, the Poisson systems possess similar but extended structural properties, and can be defined on Poisson manifolds of arbitrary dimensions. They have a large scope of applications, such as in astronomy, robotics, fluid mechanics, electrodynamics, quantum mechanics, nonlinear waves, and so on. Numerical simulations of deterministic Poisson systems have been developed during the last decades. It is in recent years that numerical simulations analysis arise for certain stochastic Poisson systems (SPSs), i.e., Poisson systems under certain stochastic disturbances. In this talk, we propose a class of numerical integration methods for stochastic Poisson systems of arbitrary dimensions. Based on the Darboux-Lie theorem, we transform the SPSs to their canonical form, the generalized stochastic Hamiltonian systems (SHSs), via canonical coordinate transformations found by solving certain PDEs defined by the Poisson brackets of the SPSs. A generating function approach is then used to create symplectic discretizations of the SHSs, which are then transformed back by the inverse coordinate transformation to numerical integrators for the SPSs. These integrators are proved to preserve both the Poisson structure and the Casimir functions of the SPSs. Applications to a three-dimensional stochastic rigid body system and a three-dimensional stochastic Lotka-Volterra system show efficiency of the proposed methods.

Keywords: Stochastic Poisson systems, Poisson structure, Casimir functions, Poisson integrators, symplectic integrators, generating functions, stochastic rigid body system, stochastic Lotka-Volterra system.