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My research interests lie in the applied analysis and numerical computation of scientific problems arising from geophysical fluid, seismology, material science and biology. Specific topics include:

__Atmospheric and oceanic modeling__- Biological modeling
- Computational methods for high frequency wave propagation
- Dynamics of electrons in crystal
- Kinetic transport in heterogeneous media
- Miscellaneous research

Topic descriptions:

**Computational methods for high frequency wave propagation**

*Frozen Gaussian approximation.*We developed the frozen Gaussian approximation (FGA) for efficiently computing high frequency wave propagation ([15]). It makes use of Gaussian functions with fixed width on phase plane, and provides a stable and robust relaxation mechanism for geometric optics (GO) which helps asymptotic solution survive at caustics. This is different from Gaussian beam method (GB) where the nonlinear Riccati equation governs the relaxation (beam width) that can become either too large or small in sequential dynamics. In this sense, FGA has a more stable performance. Moreover, FGA presents better asymptotic accuracy. This method was motivated by Herman-Kluk propagator ([HK]) in quantum chemistry. The original method was in Lagrangian framework, hence suffered from the issue of divergence. This motivates us to design an efficient Eulerian FGA method which can achieve local adaptation easily ([19]). We also derived higher order asymptotic approximation and analyzed the accuracy ([18]). Especially we noticed that FGA works for wave propagation only when the initial condition is asymptotically high frequency. This shows the difference from the results by Swart and Rousse ([SR]) who proved the asymptotic accuracy for Herman-Kluk propagator.

*Eulerian Gaussian beam method.*

A novel Eulerian Gaussian beam method was developed for efficient computation of one-body Schrödinger equation in the semiclassical regime ([9]). Distinguished from traditional Gaussian beam methods, the complex Hessian function of Gaussian beams is computed by the derivatives of level set functions instead of by Riccati equation or dynamic ray tracing equations. A related preliminary work is the numerical computation of the semiclassical limit of Schrödinger equation with phase shift correction ([7]). We give a follow-up study later in [13] to deal with the easy implementation of the algorithm and generalize it to higher order. This methodology is quite general, and has been applied to some other Hamiltonian systems, for example the nonlinear Schrödinger-Poisson system ([10]) and the electron dynamics in crystal ([11]).

**Atmospheric and oceanic modeling**We studied the dynamic transition/excursion phenomena in climate systems. We built a framework using large deviation theory, in which different climate regimes are represented by the statistical most likely states and the transition is described by the most likelihood pathways connecting either metastable states or target sets in the small noise limit. Specifically we considered the energy-constrained stochastic dynamics (equilibrium statistical system), the most likely states of whose invariant measure coincide with the selective decay states. We compute the transition pathways using a constrained String method. Nonequilibrium statistical climate systems were also analyzed where the transition pathways were computed by the geometric minimum action method.

**Dynamics of electrons in crystal**We studied the dynamics of electrons in crystal by asymptotic analysis, aiming at deriving effective models for applications in nano-optics and semiconductor. In particular, we considered two time regimes which are characterized by the external field frequency. When the external field is high frequency ([14]), we started from time dependent density functional theory, and obtained effective Maxwell equations for the dynamics of interacting electrons by the method of homogenization. When the external field is low frequency ([acpt]), we focused on the Bloch dynamics of single electron in crystal. We provided a simple derivation of Berry curvature by WKB analysis and introduced the Bloch-Wigner transform for studying the macroscopic behavior of electron.

**Biological modeling**We proposed a swarming model for biological aggregation phenomenon in one dimensional space ([6]). It presents interesting dynamics that the randomly distributed animals (or bacteria) will spontaneously aggregate together and form certain patterns of traveling waves, especially including a form of soliton solution.

**Kinetic transport in heterogeneous media**This part of research was done in my master and Ph. D. thesis. By imposing correct interface conditions we solved the radiative transfer equation in heterogeneous media for both the one-scale coupling model (kinetic-kinetic coupling [5]) and the two-scale coupling model (kinetic-diffusion coupling [4]). A domain decomposition method for the two-scale coupling model was developed based on the linear response of the outgoing waves on the incoming waves in the diffusion domain, where 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 on neutron transport equation ([GJL]), followed by some numerical study in my master thesis ([3]).

**Miscellaneous research**Collaboration with mechanical engineers: spatial parallel manipulators ([2]).

Collaborators:

Weinan E /
Sergey Fomel
/ Carlos J.Garcia-Cervera / Francois Golse /
Zhongyi Huang
/
Shi Jin
/
Xiaomei
Liao /
Jianfeng Lu
/
Paul A. Milewski /
Weiqing Ren /
Richard Tsai /
Eric Vanden-Eijnden / Dongming Wei
/
Hao Wu
/ Guangwei Yuan /
Jingshan Zhao / Liang Zhu