Shingyu Leung

Eulerian Gaussian beams for high-frequency wave propagation

We design an Eulerian Gaussian beam summation method for solving Helmholtz equations in the high-frequency regime. The traditional Gaussian beam summation method is based on Lagrangian ray tracing and local ray-centered coordinates. We propose a new Eulerian formulation of Gaussian beam theory which adopts global Cartesian coordinates, level sets, and Liouville equations, yielding uniformly distributed Eulerian traveltimes and amplitudes in phase space simultaneously for multiple sources. The time harmonic wavefield can be constructed by summing up Gaussian beams with ingredients provided by the new Eulerian formulation. The conventional Gaussian beam summation method can be derived from the proposed method. There are three advantages of the new method: (1) We have uniform resolution of ray distribution. (2) We can obtain wavefields excited at different sources by varying only source locations in the summation formula. (3) We can obtain wavefields excited at different frequencies by varying only frequencies in the summation formula. Numerical experiments indicate that the Gaussian beam summation method yields accurate asymptotic wavefields even at caustics. The new method may be used for seismic modeling and migration.

Eulerian Gaussian beams for Schrödinger equations in the semi-classical regime

We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrodinger equation. Traditional Gaussian beam type methods for the Schrodinger equation are based on the Lagrangian ray tracing. Based on the first Eulerian Gaussian beam framework proposed in Leung et al. [S. Leung, J. Qian, R. Burridge, Eulerian Gaussian beams for high frequency wave propagation, Geophysics 72 (2007) SM61-SM76], we develop a new Eulerian Gaussian beam method which uses global Cartesian coordinates, level-set based implicit representation and Liouville equations. The resulting method gives uniformly distributed phases and amplitudes in phase space simultaneously. To obtain semi-classical solutions to the Schrodinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi-classical solutions to the Schrodinger equation. For instance, in the one-dimensional case the proposed algorithm requires only O(sNm2) operations to compute s different solutions with s different initial wave functions under the influence of the same potential, where N=O(1/), is the Planck constant, and m N is the number of computed beams which depends on weakly. Numerical experiments indicate that this Eulerian Gaussian beam approach yields accurate semi-classical solutions even at caustics.

A grid based particle method for moving interface problems

We propose a novel algorithm for modeling interface motions. The interface is represented and is tracked using quasi-uniform meshless particles. These particles are sampled according to an underlying grid such that each particle is associated to a grid point which is in the neighborhood of the interface. The underlying grid provides an Eulerian reference and local sampling rate for particles on the interface. It also renders neighborhood information among the meshless particles for local reconstruction of the interface. The resulting algorithm, which is based on Lagrangian tracking using meshless particles with Eulerian reference grid, can naturally handle/control topological changes. Moreover, adaptive sampling of the interface can be achieved easily through local grid refinement with simple quad/oct-tree data structure. Extensive numerical examples are presented to demonstrate the capability of our new algorithm.

A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion

We develop numerical methods for solving partial differential equations (PDE) defined on an evolving interface represented by the grid based particle method (GBPM) recently proposed in [S. Leung, H.K. Zhao, A grid based particle method for moving interface problems, J. Comput. Phys. 228 (2009) 7706-7728]. In particular, we develop implicit time discretization methods for the advection-diffusion equation where the time step is restricted solely by the advection part of the equation. We also generalize the GBPM to solve high order geometrical flows including surface diffusion and Willmore-type flows. The resulting algorithm can be easily implemented since the method is based on meshless particles quasi-uniformly sampled on the interface. Furthermore, without any computational mesh or triangulation defined on the interface, we do not require remeshing or reparametrization in the case of highly distorted motion or when there are topological changes. As an interesting application, we study locally inextensible flows governed by energy minimization. We introduce tension force via a Lagrange multiplier determined by the solution to a Helmholtz equation defined on the evolving interface. Extensive numerical examples are also given to demonstrate the efficiency of the proposed approach.

An Eulerian approach for computing the finite time Lyapunov exponent

We propose efficient Eulerian methods for approximating the finite-time Lyapunov exponent (FTLE). The idea is to compute the related flow map using the Level Set Method and the Liouville equation. There are several advantages of the proposed approach. Unlike the usual Lagrangian-type computations, the resulting method requires the velocity field defined only at discrete locations. No interpolation of the velocity field is needed. Also, the method automatically stops a particle trajectory in the case when the ray hits the boundary of the computational domain. The computational complexity of the algorithm is O(@Dx^-^(^d^+^1^)) with d the dimension of the physical space. Since there are the same number of mesh points in the x-t space, the computational complexity of the proposed Eulerian approach is optimal in the sense that each grid point is visited for only O(1) time. We also extend the algorithm to compute the FTLE on a co-dimension one manifold. The resulting algorithm does not require computation on any local coordinate system and is simple to implement even for an evolving manifold.

Guarantees of Riemannian Optimization for Low Rank Matrix Recovery

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an

The backward phase flow and FBI-transform-based Eulerian Gaussian beams for the Schrödinger equation

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A grid based particle method for evolution of open curves and surfaces

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Transmission traveltime tomography based on paraxial Liouville equations and level set formulations

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Global minimization of the active contour model with TV-Inpainting and two-phase denoising

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