Fast Hyigens sweeping methods for Schrodinger equations in the semi-classical regime

Agraïments: This paper is dedicated to Prof. Stan Osher on the occasion of his 70th birthday. Leung is supported in part by the Hong Kong RGC under Grant GRF603011. Qian is supported by NS.We propose fast Huygens sweeping methods for Schrodinger equations in the semi-classical regime by incorporating short-time Wentzel-Kramers-Brillouin-Jeffreys (WKBJ) propagators into Huygens' principle. Even though the WKBJ solution is valid only for a short time period due to the occurrence of caustics, Huygens' principle allows us to construct the global-in-time semi-classical solution. To improve the computational efficiency, we develop analytic approximation formulas for the short-time WKBJ propagator by using the Taylor expansion in time. These analytic formulas allow us to develop two classes of fast Huygens sweeping methods, among which one is posed in the momentum space, and the other is posed in the position space, and both of these methods are of computational complexity O(N log N ) for each time step, where N is the total number of sampling points in the d-dimensional position space. To further speed up these methods, we also incorporate the soft-thresholding sparsification strategy into our new algorithms so that the computational cost can be further reduced. The methodology can also be extended to nonlinear Schrodinger equations. One, two, and three dimensional examples demonstrate the performance of the new algorithms

The method of polarized traces for the 2D Helmholtz equation

We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as O(NL), where N is the number of volume unknowns, and L is the number of processors, as long as L grows at most like a small fractional power of N. The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define "polarized traces", i.e., up- and down-going waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses second-order finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations. While the parallelism in this paper stems from decomposing the domain, we do not explore the alternative of parallelizing the systems solves with distributed linear algebra routines. Keywords: Domain decomposition; Helmholtz equation; Integral equations; High-frequency; Fast methodsUnited States. Air Force Office of Scientific Research (Grant FA9550-15-1-0078)United States. Office of Naval Research (Grant N00014-13-1-0403)National Science Foundation (U.S.) (Grant DMS-1255203

Approximate inversion of generalized Radon transforms

Generalized Radon transforms (GRT) serve, for instance, as linear models for seismic imaging in the acoustic regime. They occur when the corresponding inverse problem is linearized about a known background compression wave speed (Born approximation). The resulting GRT is completely determined by this background velocity. In this work, we present an implementation of approximate inversion formulas for this class of GRTs proposed and analyzed in [Inverse Problems, 34 (2018), 014002, 114001], where we restrict ourselves to layered background velocities in 2D. In a series of numerical experiments, we intensively test our implementation, reproducing theoretical predictions. Further, we drive the validity of the linearization to its limits

Direct and inverse scattering problems for domains with multiple corners

Direct and inverse scattering problems have wide applications in geographical exploration, radar, sonar, medical imaging and non-destructive testing. In many applications, the obstacles are not smooth. Corner singularity challenges the design of a forward solver. Also, the nonlinearity and ill-posedness of the inverse problem challenge the design of an efficient, robust and accurate imaging method. This dissertation presents numerical methods for solving the direct and inverse scattering problems for domains with multiple corners. The acoustic wave is sent from the transducers, scattered by obstacles and received by the transducers. This forms the response matrix data. The goal for the direct scattering problem is to compute the response matrix data using the knowledge of the shape of the obstacles. The goal for the inverse scattering problem is to image the location and geometry of the obstacles based on the response matrix data. Both the near field and far field cases are considered. For the direct problem, the challenges of logarithmic singularity from Green\u27s functions and corner singularity are both taken care of. For the inverse problem, an efficient and robust direct imaging method similar to the Multiple Signal Classification algorithm is proposed. Multiple frequency data are combined to capture details while not losing robustness. The near field and far field response matrices are compared and their singular value patterns are compared as well. The singular value perturbation is carefully studied. Extensive numerical results demonstrate that our forward solver is capable of handling domains with multiple corners by solving a linear system with low condition numbers generated from a boundary integral equation, that our inverse problem solver is efficient, accurate and robust. It could handle response matrix data with noise

A direction preserving discretization for computing phase-space densities

Ray flow methods are an efficient tool to estimate vibro-acoustic or electromagnetic energy transport in complex domains at high-frequencies. Here, a Petrov--Galerkin discretization of a phase-space boundary integral equation for transporting wave energy densities on two-dimensional surfaces is proposed. The directional dependence of the energy density is approximated at each point on the boundary in terms of a finite local set of directions propagating into the domain. The direction of propagation can be preserved for transport across multicomponent domains when the directions within the local set are inherited from a global direction set. The range of applicability and computational cost of the method will be explored through a series of numerical experiments, including wave problems from both acoustics and elasticity in both single and multicomponent domains. The domain geometries considered range from both regular and irregular polygons to curved surfaces, including a cast aluminium shock tower from a Range Rover car