Eulerian Geometrical Optics and Fast Huygens Sweeping Methods for Three-Dimensional Time-Harmonic High-Frequency Maxwell's Equations in Inhomogeneous Media

Qian, Jianliang; Lu, Wangtao; Yuan, Lijun; Luo, Songting; Burridge, Robert

doi:10.1137/15m1013158

Cited by 19 publications

(25 citation statements)

References 47 publications

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“…In this work, we are using the version of the fast butterfly algorithm first developed in [12] and further analyzed in [16]. This fast butterfly algorithm was then adapted for Helmholtz equations [32,44] and for Maxwell's equations [43]. The significance of the fast butterfly algorithm for high-frequency wave computation was illustrated in a recent work [18].…”

Section: Related Workmentioning

confidence: 99%

“…The second implication is that when ω increases, the -numerical rank for the corresponding matrix increases as well so that no obvious low-rank structure exists when the wavefield is resolved with a fixed number of points per wavelength. To create low-rank structures in the corresponding matrix, we have to set up the two sets X and Y in an ω-dependent manner which is exactly the departure point for fast butterfly algorithms in [12,32,44,43] and in the current paper.…”

Section: Related Workmentioning

confidence: 99%

“…However , which is extremely expensive and impractical. To resolve this issue, we adopt a multilevel matrix decomposition based butterfly algorithm [12,16,32,44,43] to speed up the multiplications.…”

Section: Online Stage: Wavefields By the Huygens-kirchhoff Summationmentioning

confidence: 99%

“…Based on the decomposition (89), we can adopt the low-rank separation based butterfly algorithm [12,16,32,43] to speed up the matrix-vector products in equation (82). Some modifications can be made so as to make it more suitable to our applications.…”

Section: A Butterfly Algorithmmentioning

confidence: 99%

“…Such computational cost is far too high to make our strategy practical. To tackle this difficulty, we adapt to our application the multilevel matrix decomposition based butterfly algorithm [36,39,12,59,16,32,44,43]. The resulting butterfly algorithm allows us to carry out the required matrix-vector products with the total computational cost of O(N log N ) complexity, where the proportionality constant depends only on the specified accuracy and is independent of the frequency parameter ω.…”

Section: Introductionmentioning

confidence: 99%

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Babich's expansion and the fast Huygens sweeping method for the Helmholtz wave equation at high frequencies

Qian

Burridge

2016

Journal of Computational Physics

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In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that the Helmholtz equation can be viewed as an evolution equation in one of the spatial directions. With such applications in mind, starting from Babich's expansion, we develop a new high-order asymptotic method, which we dub the fast Huygens sweeping method, for solving point-source Helmholtz equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of this method is that we develop a new Eulerian approach to compute the asymptotics, i.e. the traveltime function and amplitude coefficients that arise in Babich's expansion, yielding a locally valid solution, which is accurate close enough to the source. The second novelty is that we utilize the Huygens-Kirchhoff integral to integrate many locally valid wavefields to construct globally valid wavefields. This automatically treats caustics and yields uniformly accurate solutions both near the source and remote from it. The third novelty is that the butterfly algorithm is adapted to accelerate the Huygens-Kirchhoff summation, achieving nearly optimal complexity O(N log N ), where N is the number of mesh points; the complexity prefactor depends on the desired accuracy and is independent of the frequency. To reduce the storage of the resulting tables of asymptotics in Babich's expansion, we use the multivariable Chebyshev series expansion to compress each table $ Qian is supported by NSF grants 1222368 and 1522249.

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Online Stage: Wavefields By the Huygens-kirchhoff Summationmentioning

confidence: 99%

Section: A Butterfly Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Babich's expansion and the fast Huygens sweeping method for the Helmholtz wave equation at high frequencies

Qian

Burridge

2016

Journal of Computational Physics

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Numerical solutions of the time‐dependent Schrödinger equation with position‐dependent effective mass

Gao

Mayfield

Luo

2023

Numerical Methods Partial

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Numerical solution of the time‐dependent Schrödinger equation with a position‐dependent effective mass is challenging to compute due to the presence of the non‐constant effective mass. To tackle the problem we present operator splitting‐based numerical methods. The wavefunction will be propagated either by the Krylov subspace method‐based exponential integration or by an asymptotic Green's function‐based time propagator. For the former, the wavefunction is given by a matrix exponential whose associated matrix–vector product can be approximated by the Krylov subspace method; and for the latter, the wavefunction is propagated by an integral with retarded Green's function that is approximated asymptotically. The methods have complexity O(NlogN)$$ O\left(N\log N\right) $$ per step with appropriate algebraic manipulations and fast Fourier transform, where N$$ N $$ is the number of spatial points. Numerical experiments are presented to demonstrate the accuracy, efficiency, and stability of the methods.

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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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Eulerian Geometrical Optics and Fast Huygens Sweeping Methods for Three-Dimensional Time-Harmonic High-Frequency Maxwell's Equations in Inhomogeneous Media

Cited by 19 publications

References 47 publications

Babich's expansion and the fast Huygens sweeping method for the Helmholtz wave equation at high frequencies

Babich's expansion and the fast Huygens sweeping method for the Helmholtz wave equation at high frequencies

Numerical solutions of the time‐dependent Schrödinger equation with position‐dependent effective mass

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

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