Frozen Gaussian Approximation for General Linear Strictly Hyperbolic Systems: Formulation and Eulerian Methods

Lu, Jianfeng; Yang, Xu

doi:10.1137/10081068x

Cited by 26 publications

(38 citation statements)

References 22 publications

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“…This becomes a drawback when the solution spreads [23,28,32].The Herman-Kluk propagator [11,21,22] was proposed for Schrödinger equation without the oscillatory periodic background potential. The method was rigorously analyzed in [35,36] and further extended as the frozen Gaussian approximation (FGA) for general high frequency wave propagation in [23][24][25]. The FGA method uses Gaussian functions with fixed widths, instead of using those that might spread over time, to approximate the wave solution.…”

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confidence: 99%

Frozen Gaussian approximation for high frequency wave propagation in periodic media

Delgadillo

Yang

2018

ASY

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Propagation of high-frequency wave in periodic media is a challenging problem due to the existence of multiscale characterized by short wavelength, small lattice constant and large physical domain size. Conventional computational methods lead to extremely expensive costs, especially in high dimensions. In this paper, based on Bloch decomposition and asymptotic analysis in the phase space, we derive the frozen Gaussian approximation for high-frequency wave propagation in periodic media and establish its converge to the true solution. The formulation leads to efficient numerical algorithms, which are presented in a companion paper [5].1 2 RICARDO DELGADILLO, JIANFENG LU, AND XU YANG avoid the boundary effects. Therefore the total number of grid points is huge, which usually leads to unaffordable computational cost, especially in high (d > 1) dimensions.An alternative efficient approach is to solve (1.1) asymptotically by the Bloch decomposition and modified WKB methods [3,4,7], which lead to eikonal and transport equations in the semi-classical regime. An advantage of this method is that the computational cost is independent of ε. However, the eikonal equation can develop singularities which make the method break down at caustics. The Gaussian beam method (GBM) [31] was then introduced by Popov to overcome this drawback at caustics. The idea is to allow the phase function to be complex and choose the imaginary part properly so that the solution has a Gaussian profile; see [15-20, 28, 37, 38] for recent developments. Similar ideas can be also found in the Hagedorn wave packet method [8,9]. Unlike the geometric optics based method, the Gaussian beam method allows for accurate computation of wave function around caustics [6,33]. But the problem is that the constructed beam must stay near the geometric rays to maintain accuracy. This becomes a drawback when the solution spreads [23,28,32].The Herman-Kluk propagator [11,21,22] was proposed for Schrödinger equation without the oscillatory periodic background potential. The method was rigorously analyzed in [35,36] and further extended as the frozen Gaussian approximation (FGA) for general high frequency wave propagation in [23][24][25]. The FGA method uses Gaussian functions with fixed widths, instead of using those that might spread over time, to approximate the wave solution. Despite its superficial similarity with the Gaussian beam method, it is different at a fundamental level. FGA is based on phase plane analysis, while GBM is based on the asymptotic solution to a wave equation with Gaussian initial data. In FGA, the solution to the wave equation is approximated by a superposition of Gaussian functions living in phase space, and each function is not necessarily an asymptotic solution, while GBM uses Gaussian functions (called beams) in physical space, with each individual beam being an asymptotic solution to the wave equation. The main advantage of FGA over GBM is that the problem of beam spreading no longer exists.In this paper, we extend FGA for computation of hi...

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confidence: 99%

Frozen Gaussian approximation for high frequency wave propagation in periodic media

Delgadillo

Yang

2018

ASY

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“…Note that the presentation of FGA here has been tailored and simplified for the purposes of this work. For more details, such as the asymptotic derivation, the error estimates, the validity at caustics, and generalization to other strictly hyperbolic systems, we refer the readers to [23][24][25]42]. …”

Section: Frozen Gaussian Approximationmentioning

confidence: 99%

“…The method was originally motivated by Herman-Kluk propagator for solving the Schrödinger equation in quantum chemistry [14,19,20], and later generalized to linear strictly hyperbolic systems [23][24][25]. The main idea of FGA is to use Gaussian functions with fixed widths to approximate the solution of wave equation.…”

Section: Introductionmentioning

confidence: 99%

A frozen Gaussian approximation-based multi-level particle swarm optimization for seismic inversion

Lin

Yang

2015

Journal of Computational Physics

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In this paper, we propose a frozen Gaussian approximation (FGA)-based multi-level particle swarm optimization (MLPSO) method for seismic inversion of high-frequency wave data. The method addresses two challenges in it: First, the optimization problem is highly nonconvex, which makes hard for gradient-based methods to reach global minima. This is tackled by MLPSO which can escape from undesired local minima. Second, the character of high-frequency of seismic waves requires a large number of grid points in direct computational methods, and thus renders an extremely high computational demand on the simulation of each sample in MLPSO. We overcome this difficulty by three steps: First, we use FGA to compute high-frequency wave propagation based on asymptotic analysis on phase plane; Then we design a constrained full waveform inversion problem to prevent the optimization search getting into regions of velocity where FGA is not accurate; Last, we solve the constrained optimization problem by MLPSO that employs FGA solvers with different fidelity. The performance of the proposed method is demonstrated by a twodimensional full-waveform inversion example of the smoothed Marmousi model.

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“…Specifically, one decomposes the initial wave function into localized wave packets (Gaussian beams) which are then evolved individually along particle trajectories and finally summed up to construct the solution at a later time. It was first studied rigorously in [25], and has seen many recent developments in both Eulerian and Lagrangian frameworks [13,14,15,17,20,21], error estimates [2,19], and fast Gaussian wave decompositions [1,24]. A related approach, known as the Hagedorn wave packet method, was studied in [8,6].…”

Section: Introductionmentioning

confidence: 99%

A Gaussian Beam Method for High Frequency Solution of Symmetric Hyperbolic Systems with Polarized Waves

Jefferis¹,

Jin²

2015

Multiscale Model. Simul.

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Symmetric hyperbolic systems include many physically relevant systems of partial differential equations (PDEs) such as Maxwell's equations, the elastic wave equations and the acoustic equations [26]. In this paper we extend the Gaussian beam method to efficiently compute the high frequency solutions to such systems with constant degeneracy that corresponds to polarized waves, in which the dispersion matrix of the hyperbolic system has eigenvalues with constant degeneracy over the domain of computation. The new results in this paper include new Gaussian beam equations in the presence of eigenvalue degeneracy, improved error estimates for Gaussian beam summation and a new multi-directional Eulerian summation formula which maintains accuracy after the formation of caustics.

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Frozen Gaussian Approximation for General Linear Strictly Hyperbolic Systems: Formulation and Eulerian Methods

Cited by 26 publications

References 22 publications

Frozen Gaussian approximation for high frequency wave propagation in periodic media

Frozen Gaussian approximation for high frequency wave propagation in periodic media

A frozen Gaussian approximation-based multi-level particle swarm optimization for seismic inversion

A Gaussian Beam Method for High Frequency Solution of Symmetric Hyperbolic Systems with Polarized Waves

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