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:
- Computational methods for high frequency wave propagation
- Frozen Gaussian approximation.
We developed the frozen Gaussian approximation (FGA) for
efficiently computing high frequency wave propagation ().
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 ().
We also derived higher order asymptotic approximation and analyzed the
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 ().
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 ().
We give a follow-up study later in
 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 ()
and the electron dynamics in crystal ().
- Atmospheric and oceanic
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
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 ().
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
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
) and the two-scale coupling model (kinetic-diffusion coupling
). 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 ().
- Miscellaneous research
Collaboration with mechanical engineers: spatial parallel manipulators ().
Weinan E /
/ Carlos J.Garcia-Cervera / Francois Golse /
Paul A. Milewski /
Weiqing Ren /
Richard Tsai /
Eric Vanden-Eijnden / Dongming Wei
/ Guangwei Yuan /
Jingshan Zhao / Liang Zhu