Singular Hamiltonian systems arise in modeling high frequency or quantum waves through interfaces or barriers, in which the potential functions become discontinuous and the underlying Hamiltonian systems have singular forces, preventing the initial value problems from being well-posed. We first introduce a notion of solutions, by building in wave transmission and reflection mechanism at the barriers, allowing one to go beyond the singular point and defines a well-posed initial value problem consistent with physical laws of refractions. This also naturally induces numerical methods that build in the interface condition into the numerical fluxes.
Such ideas have found applications from partial transmissions, reflections and even diffractions of high frequency waves, semi-classical computation of quantum tunneling through barriers, and surface hopping in which the particles can tunnel through different quantum states due to degeneracy in the eigen-structures of the potential energy surfaces. The methods allow particle simulations, with the computational cost of classical mechanics, to capture certain important quantum phenomena.
He is the Director of Institute of Natural Sciences, and Chair Professor of Mathematics, at Shanghai Jiao Tong University. He obtained his BS degree from Peking University and his Ph.D. from University of Arizona. He was a postdoc at Courant Institute, New York University, an assistant and associate professors at Georgia Institute of Technology, and full professor, department chair and Vilas Distinguished Achievement Professor at University of Wisconsin-Madison, Chair of Department of Mathematics at Shanghai Jiao Tong University.
He also serves as a co-director of the Shanghai Center of Applied Mathematics, director of Ministry of Education Key Lab on Scientific and Engineering Computing, and director of Center for Mathematical Foundation of Artificial Intelligence at Shanghai Jiao Tong University.
He received a Feng Kang Prize of Scientific Computing in 2001. He is an inaugural Fellow of the American Mathematical Society (AMS) (2012), a Fellow of Society of Industrial and Applied Mathematics (SIAM) (2013), and an Invited Speaker at the International Congress of Mathematicians in 2018.
His research interests include kinetic theory, hyperbolic conservation laws, quantum dynamics, uncertainty quantification, interacting particle systems and computational fluid dynamics, etc. He has published over 170 research papers. His paper was awarded one of the four Best Paper Awards by the Springer Journal Research in the Mathematical Sciences for its fifth year anniversary, and selected by World Scientific journal M3AS as one of the best papers published in 2019.