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We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrdinger equation. Traditional Gaussian beam type methods for the Schrdinger 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) SM61SM76], 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 Schrdinger equation with different initial wave functions, we only need to slightly modify the summation formula. This yields a very efficient method for computing semi

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) 77067728]. In particular, we develop implicit time discretization methods for the advectiondiffusion 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

We establish theoretical recovery guarantees of a family of Riemannian optimization algorithms for low rank matrix recovery, which is about recovering an mimes n rank mimes n matrix from mimes n number of linear measurements. The algorithms are first interpreted as iterative hard thresholding algorithms with subspace projections. Based on this connection, we show that provided the restricted isometry constant mimes n of the sensing operator is less than mimes n, the Riemannian gradient descent algorithm and a restarted variant of the Riemannian conjugate gradient algorithm are guaranteed to converge linearly to the underlying rank mimes n matrix if they are initialized by one step hard thresholding. Empirical evaluation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements necessary.

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 <i>O</i>(<i>x</i>^(<i>d</i>+1)) with <i>d</i> the dimension of the physical space. Since there are the same number of mesh points in the <i>x</i><i>t</i> space, the computational complexity of the proposed Eulerian approach is optimal in the sense that each grid point is visited for only <i>O</i>(1) time. We also extend the algorithm to compute the FTLE

We propose a fast local level set method for the inverse problem of gravimetry. The theoretical foundation for our approach is based on the following uniqueness result: if an open set D is star-shaped or x₃-convex with respect to its center of gravity, then its exterior potential uniquely determines the open set D. To achieve this purpose constructively, the first challenge is how to parametrize this open set D as its boundary may have a variety of possible shapes. To describe those different shapes we propose to use a level-set function to parametrize the unknown boundary of this open set. The second challenge is how to deal with the issue of partial data as gravimetric measurements are only made on a part of a given reference domain . To overcome this difficulty we propose a linear numerical continuation approach based on the single layer representation to find potentials on the boundary of some artificial domain

We propose the backward phase flow method to implement the FourierBrosIagolnitzer (FBI)-transform-based Eulerian Gaussian beam method for solving the Schrdinger equation in the semi-classical regime. The idea of Eulerian Gaussian beams has been first proposed in [12]. In this paper we aim at two crucial computational issues of the Eulerian Gaussian beam method: how to carry out long-time beam propagation and how to compute beam ingredients rapidly in phase space. By virtue of the FBI transform, we address the first issue by introducing the reinitialization strategy into the Eulerian Gaussian beam framework. Essentially we reinitialize beam propagation by applying the FBI transform to wavefields at intermediate time steps when the beams become too wide. To address the second issue, inspired by the original phase flow method, we propose the backward phase flow method which allows us to

We propose a simple Eulerian approach to compute the moderate to long time flow map for approximating the Lyapunov exponent of a (periodic or aperiodic) dynamical system. The idea is to generalize a recently proposed backward phase flow method which is specially designed for long time level set propagation. Unlike the original phase flow method or the backward phase flow method, which is applicable only to autonomous systems, the current approach can also be applied to any time-dependent (periodic or aperiodic) flow. We will discuss the stability of the proposed method. Numerical examples will be given to demonstrate the effectiveness of the algorithm.

We develop an efficient numerical method to compute single slit or double slit diffraction patterns from high frequency wave in inhomogeneous media. We approximate the high frequency asymptotic solution to the Helmholtz equation using the Eulerian Gaussian beam summation proposed in [20, 21]. The emitted rays from a slit are embedded in the phase space using an open segment. The evolution of this open curve is accurately computed using the recently developed Grid Based Particle Method [24] which results in a very efficient computational algorithm. Following the grid based particle method we proposed in [23, 24], we represent the open curve or the open surface by meshless Lagrangian particles sampled according to an underlying fixed Eulerian mesh. The end-points of the open curve are tracked explicitly and consistently with interior particles. To construct the overall wavefield, each of these sampling particles also carry necessary quantities that are obtained by solving advection-reaction equations. Numerical experiments show that the resulting method can model diffraction patterns in inhomogeneous media accurately, even in the occurrence of caustics.

We propose a numerical approach to solve variational problems on manifolds represented by the grid based particle method (GBPM) recently developed in Leung et al. (J. Comput. Phys. 230(7):25402561, 2011), Leung and Zhao (J. Comput. Phys. 228:77067728, 2009a, J. Comput. Phys. 228:29933024, 2009b, Commun. Comput. Phys. 8:758796, 2010). In particular, we propose a splitting algorithm for image segmentation on manifolds represented by unconnected sampling particles. To develop a fast minimization algorithm, we propose a new splitting method by generalizing the augmented Lagrangian method. To efficiently implement the resulting method, we incorporate with the local polynomial approximations of the manifold in the GBPM. The resulting method is flexible for segmentation on various manifolds including closed or open or even surfaces which are not orientable.

We propose several modifications to the grid based particle method (GBPM) [21] for moving interface modeling. There are several nice features of the proposed algorithm. The new method can significantly improve the distribution of sampling particles on the evolving interface. Unlike the original GBPM where footpoints (sampling points) tend to cluster to each other, the sampling points in the new method tend to be better separated on the interface. Moreover, by replacing the grid-based discretization using the cell-based discretization, we naturally decompose the interface into segments so that we can easily approximate surface integrals. As a possible alternative to the local polynomial least square approximation, we also study a geometric basis for local reconstruction in the resampling step. We will show that such modification can simplify the overall implementations. Numerical examples in two- and three