Education:
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My current interest primarily lies in the applied analysis of the electronic structure and the computational methods for the high frequency wave equations. Some of the research results are described as below.
Electronic dynamics in crystals is studied with the help of the WKB analysis and mutliscale asymptotic expansion ([pre2]).
- A novel Eulerian Gaussian beam method was developed for the efficient computation of the one-body Schrödinger equation in the semiclassical regime ([9]). Distinguished from the traditional Gaussian beam methods, the complex Hessian information for Gaussian beams is carried by the derivatives of the level set functions instead of being described by the dynamic ray tracing equations. The related preliminary work was the numerical computation of the semiclassical limit of the Schrödinger equation with phase shift correction ([7]). A follow-up study was given later in [pre3] which dealt with the easy implementation of the algorithm and generalized it to have higher order accuracy. Some applications have been carried to other equations in quantum mechanics ([pre1, acpt1]).
- Hybrid phase flow method [pre4].
(Detailed descriptions are available at Dr. Hao Wu's website.)
This part of research is related to my master and main Ph. D. thesis. By imposing the correct interface conditions we solved the radiative transfer equation in the heterogeneous media for both the one-scale coupling model (kinetic-kinetic coupling [5]) and the two-scales coupling model (kinetic-diffusion coupling [4]). A domain decomposition method for the two-scales coupling model was developed based on the linear response between the incoming waves and outgoing waves of the diffusion domain, in which the nonlocal susceptibility was given in terms of the Chandrasekhar H-function. The original idea of this decoupling method goes back to the former work by Golse, Jin and Levermore for the neutron transport equation ([GJL]), followed by some numerical study done in my master thesis ([3]).
In collaboration with Prof. Paul A. Milewski, we set up a swarming model for the biological aggregation phenomenon in one dimensional space ([6]). It presented interesting dynamics that the randomly distributed animals (or bacteria) would spontaneously aggregate together and form certain patterns of traveling waves.
Miscellaneous research
Other researches are also done in the numerical study of weak solution selection principle for the Vlasov-Poisson equations ([8]) and some industry application project ([2]).
Publication lists (chronological order)
Conferences:
Partial Differential Equations in Kinetic Theories, Isaac Newton Institute for Mathematical Sciences, University of Cambridge, UK, Aug. 16 - Dec. 22, 2010.
"Natural Algorithms" Workshop, Princeton University, Princeton, Nov. 2-3, 2009.
Kinetic FRG Young Researchers Workshop, University of Maryland, College Park, March 2-5, 2009.