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Summary of ongoing and past projects:
1. Semiclassical theory, seismic tomography and deep learning; (ongoing)
2. Interface problems and edge mode in photonic graphene; (ongoing)
3. Spin-magnetization coupling in ferromagnetic materials and semiclassical limit; (ongoing)
4. Chemotaxis and mean-field models with memory effect and/or preference bias; (ongoing)
5. Frozen Gaussian approximation, with applications in quantum mechanics; (ongoing)
6. High-frequency wave computation, WKB, and Gaussian beam methods; (past)
7. Electron dynamics in crystal, density functional theory and homogenization; (past)
Meta-architectures of GeoDSegDe-2 (left) and GeoDUDe-2 (righ): Notice the presence of rungs and bridge in the UNet architecture. API GeoSeg is available at here, with which, it is easy to implement neural networks designed for more general segmentation problems of seismogram data than the case of flat interface with piecewise constant velocities. See [arXiv] for more details.
Sub-project 1: 3-D elastic wave propagation and seismic tomography is computationally challenging in large scales and high-frequency regime. In this project, we develop the frozen Gaussian approximation (FGA) to compute the 3-D elastic wave equation and use it as the forward modeling tool for seismic tomography with high-frequency data. Rather than standard ray-based methods (e.g. geometric optics and Gaussian beam method), the derivation requires to do asymptotic expansion in the week sense (integral form) so that one is able to perform integration by parts. Compared to the FGA theory for acoustic wave equation, the calculations for the derivation are much more technically involved due to the existence of both P- and S-waves, and the coupling of the polarized directions for SH- and SV-waves. In particular, we obtain the diabatic coupling terms for SH- and SV-waves, with the form closely connecting to the concept of Berry phase that is intensively studied in quantum mechanics and topology (Chern number). The accuracy and parallelizability of the FGA algorithm is illustrated by comparing to the spectral element method for 3-D elastic wave equation. With a parallel FGA solver built as a computational platform, we explore various applications in 3-D seismic tomography, including seismic traveltime tomography and full waveform inversion, respectively. Global minimization for seismic tomography is investigated based on particle swarm algorithm.
Sub-project 2: We build an efficient computational platform based on parallel FGA algorithm to train deep neural networks to detect seismic interface. We first generate the time series of synthetic seismogram data by FGA, which contains accurate traveltime information but not exact amplitude information. With the synthetic seismogram generated by the FGA, we build network models using an open source API GeoSeg, developed using Keras and Tensorflow. In particular, networks using encoder-decoder and Unet architectures, despite only being trained on FGA data, can detect an interface with a high success rate from the seismograms generated by the spectral element method. All the tests are done for P-waves (acoustic) and P- and S-waves (elastic), respectively.
Dispersion graph computed by the superconvergent method for the C-symmetry breaking case (left); Weight function of the air-hole case in the unit cell (right). More details in [JCP].
Interface problems frequently appear in the fields of fluid dynamics, materials science, and computational biology, where the background consists of rather different materials. This project has developed a series of superconvergent finite element methods for 2D elliptic interface problems using both body-fitted and Cartesian grids. The key observation is that the solution is piecewise smooth on each subdomain so that one can use the divide-and-conquer strategy to design proper superconvergent methods. Specifically, we have developed gradient recovery methods for standard finite element method based on body-fitted meshes, symmetric and consistent immersed finite element method, Petrov-Galerkin immersed finite element method, partially penalized immersed finite element method, and unfitted finite element method based on Nitsche's method. Compared to standard gradient recovery methods such as superconvergent patch recovery (SPR) and polynomial preserving recovery (PPR), these gradient recovery methods perform more reliably, i.e., on one hand, they provide superconvergent gradient at the presence of discontinuity where both SPR and PPR fail, and on the other hand, they overcome the problem of over-refinement in SPR and PPR. The gradient recovery methods have been also generalized for the computation of topological edge mode in photonic graphene with honeycomb structures.
Refs for the following sub-project: [SISC], [KRM] and [NHM]
Interface problems also exist in kinetic theory. By imposing correct interface conditions, we solve the radiative transfer equation in heterogeneous media for both the one-scale coupling model (kinetic-kinetic coupling) and the two-scale coupling model. 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.
Simulation of domain wall structures by the mean-field model in 3D domain. More details in [IEEE].
The Schr\"odinger-Poisson-Landau-Lifshitz-Gilbert (SPLLG) system is an effective microscopic model that describes the coupling between conduction electron spins and the magnetization in ferromagnetic materials. In this project, we develop a mean-field model for describing the dynamics of spin transfer torque in multilayered ferromagnetic media. Specifically, we use the techniques of Wigner transform and moment closure to connect the underlying physics at different scales, and reach a macroscopic model for the dynamics of spin coupled with the magnetization within the material. This provides a further understanding of the linear response model proposed by [Zhang, Levy, and Fert, Phys. Rev. Lett. 88 (2002), 236601], and in particular we get an extra relaxation term which helps to stabilize the system. Fully 3D numerical simulation is implemented and applied to predict current-driven domain wall motions. It shows a nonlinear dependence of the wall speed on the current density which agrees with the experiments in [Yamaguchi et.al., Phys. Rev. Lett., 96 (2006), 179904]. We rigorously prove the existence of weak solutions to SPLLG and derive the Vlasov-Poisson-Landau-Lifshitz-Glibert system as the semiclassical limit connected to the mean-field model. Diffusion limit of this semiclassical limit system is also studied.
Comparison of augmented Keller-Segel model to classic Keller-Segel model and particle simulation (SPECS). Top for density and bottom for activity function; Left: Slowly changing environment; Right: Fast changing environment. See [CMS] for more details.
This project develops mean-field models for chemotaxis based on kinetic theory, including pathway based mean-field models, augmented Keller-Segel model for E. coli chemotaxis, and an asymmetric model for biological aggregation. Mathematical derivation is given for the man-field models by taking proper moment closure of kinetic biological systems. Building biological mechanism in the models are essential to capture some interesting phenomena, for example, phase-delayed traveling wave (memory effect) and soliton solution (asymmetric sensing). Connections to the chemotaxis model proposed in [G. Si, T. Wu, Q. OuYang and Y. Tu, Phys. Rev. Lett., 109 (2012), 048101] are also studied.
Simulation of Dirac equation by FGA. Left: null reflection (low potential); Middle: total reflection (high potential); Right: no potential. More details on [ResearchGate].
In this project, we develop the frozen Gaussian approximation (FGA) for efficiently computing high-frequency wave propagation, with applications in quantum mechanics. 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 makes asymptotic solution valid at caustics. This is different from Gaussian beam methods (GB) where the nonlinear Riccati equation governs the relaxation (beam width) that can become either too large of 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 in quantum mechanics. The original method was in Lagrangian frame work, and thus 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 high order asymptotic approximation and analyzed the accuracy. 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 [T. Swart and V. Rousse, Commnum. Math. Phys., 286 (2009), 725--750] which proved the asymptotic accuracy for Herman-Kluk propagator.
High-frequency wave computation
Refs: [JSC], [CMS], [JCM], [JCP], [JCP], and [CiCP]
In this project, a novel Eulerian Gaussian beam method was developed for efficient computation of one-body Schr\"odinger equation in the semiclassical regime. Different 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. We give an easy implementation of the algorithm and generalize it to higher order. The method is quite general, and has been applied to other Hamiltonian systems, e.g., the nonlinear Schr\"odinger-Poisson system and the electron dynamics in crystal.
Dynamics of electrons in crystal
In this project, 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, 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, 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 studing the macroscopic behavior of electron.
I acknowledgement research support from National Science Foundation, Hellman Family Foundation FacultyFellowship (UCSB) and Regents Junior Faculty Fellowship (UCSB).
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