Babich’s Expansion and High-Order Eulerian Asymptotics for Point-Source Helmholtz Equations

Qian, Jianliang; Yuan, Lijun; Liu, Yuan; Luo, Songting; Burridge, Robert

doi:10.1007/s10915-015-0111-7

Cited by 19 publications

(47 citation statements)

References 23 publications

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“…This has been justified in [62] for oscillatory initial value problems of hyperbolic equations and further made rigorous in the theory of Fourier integral operators [48]. In practice, the one-term asymptotic expansion (7), namely the so-called geometric optics term, usually yields sufficiently accurate asymptotic solutions [1,2,59,65,66,85,86].…”

Section: Geometric Optics Ansatzmentioning

confidence: 99%

“…The coefficients {A l } in the asymptotic expansion (6) satisfy a recursive system of transport equations [1,2,86] which are coupled with the eikonal equation. Under the assumption that the medium is smooth and no caustic occurs, one may solve the transport equations to estimate the coefficients {A l } in different formulations [1,2,65].…”

Section: Geometric Optics Ansatzmentioning

confidence: 99%

“…Assuming that the medium is smooth and no caustic occurs, the asymptotic expansion (7) will not fail as long as the frequency parameter ω is not zero, but the resulting difference between the asymptotic expansion (7) and the exact solution may be large in the L 2 norm as the frequency approaches zero [86]. Given an inhomogeneous medium, however, it is hard to pin down how large ω should be so that the asymptotic expansion (7) is accurate up to a certain specified accuracy, as this is closely related to both fluctuations and correlation lengths of the normalized propagation speed of the medium [106] and the frequency parameter ω.…”

Section: Geometric Optics Ansatzmentioning

confidence: 99%

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Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

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We present a ray-based finite element method for the high-frequency Helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of O(ωwhere ω is the frequency parameter in the Helmholtz equation. The local basis is motivated by the geometric optics ansatz and is composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a low-frequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency wave fields iteratively. Finally, a fast solver is developed for solving the resulting linear system with an empirical complexity O(ω d ) up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.

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Section: Geometric Optics Ansatzmentioning

confidence: 99%

Section: Geometric Optics Ansatzmentioning

confidence: 99%

Section: Geometric Optics Ansatzmentioning

confidence: 99%

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Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

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“…On the other hand, the Babich's expansion [7], which is a Hankel-based asymptotic expansion, can capture source singularity and overcome the above difficulties near the source in heterogeneous media. The components of the expansion can be numerically computed by high-order Eulerian asymptotic methods [61] to yield accurate solutions in the neighborhood of the point source.…”

Section: Motivationmentioning

confidence: 99%

“…The hybrid approach presented in this paper is a natural extension of the ray-FEM [34], combining Babich's expansion [7,61] to properly address the drawbacks of the former. The ray-FEM method is able to handle high-frequency problems accurately in quasi-linear time with respect to the intrinsic degrees of freedom and it has a convergence rate of O(ω − 1 2 ).…”

Section: Introductionmentioning

confidence: 99%

A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

Fang

Qian

Zepeda-Núñez

et al. 2018

Journal of Computational Physics

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We propose a hybrid approach to solve the high-frequency Helmholtz equation with point source terms in smooth heterogeneous media. The method is based on the ray-based finite element method (ray-FEM) [34], whose original version can not handle the singularity close to point sources accurately. This pitfall is addressed by combining the ray-FEM, which is used to compute the smooth far-field of the solution accurately, with a high-order asymptotic expansion close to the point source, which is used to properly capture the singularity of the solution in the near-field. The method requires a fixed number of grid points per wavelength to accurately represent the wave field with an asymptotic convergence rate of O(ω −1/2 ), where ω is the frequency parameter in the Helmholtz equation. In addition, a fast sweepingtype preconditioner is used to solve the resulting linear system.We present numerical examples in 2D to show both accuracy and efficiency of our method as the frequency increases. In particular, we provide numerical evidence of the convergence rate, and we show empirically that the overall complexity is O(ω 2 ) up to a poly-logarithmic factor.1 The ratio between numerical error and best approximation error from a discrete finite element space is ω dependent.

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Asymptotic Solutions for High Frequency Helmholtz Equations in Anisotropic Media with Hankel Functions

Jacobs

Luo

2019

J Sci Comput

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Babich’s Expansion and High-Order Eulerian Asymptotics for Point-Source Helmholtz Equations

Cited by 19 publications

References 23 publications

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media

Asymptotic Solutions for High Frequency Helmholtz Equations in Anisotropic Media with Hankel Functions

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