Accelerating fast multipole methods for the Helmholtz equation at low frequencies

Greengard, Leslie; Huang, Jingfang; Rokhlin, Vladimir; Wandzura, Stephen M.

doi:10.1109/99.714591

Cited by 216 publications

(147 citation statements)

References 13 publications

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“…This leads to an increase in the part of the matrix corresponding to the close interactions. We refer the reader to a recent article by Greengard et al [31] that addresses this problem. …”

Section: Numerical Instabilitiesmentioning

confidence: 99%

The Fast Multipole Method: Numerical Implementation

Darve¹

2000

Journal of Computational Physics

307

296

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We study integral methods applied to the resolution of the Maxwell equations where the linear system is solved using an iterative method which requires only matrix-vector products. The fast multipole method (FMM) is one of the most efficient methods used to perform matrix-vector products and accelerate the resolution of the linear system. A problem involving N degrees of freedom may be solved in CN iter N log N floating operations, where C is a constant depending on the implementation of the method. In this article several techniques allowing one to reduce the constant C are analyzed. This reduction implies a lower total CPU time and a larger range of application of the FMM. In particular, new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms. Several numerical tests are also described. These confirm the efficiency and the theoretical complexity of the FMM.

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“…This leads to an increase in the part of the matrix corresponding to the close interactions. We refer the reader to a recent article by Greengard et al [31] that addresses this problem. …”

Section: Numerical Instabilitiesmentioning

confidence: 99%

The Fast Multipole Method: Numerical Implementation

Darve¹

2000

Journal of Computational Physics

307

296

View full text Add to dashboard Cite

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“…The integrand of (29) is thus O(ξ 5 ) if x 3 + y 3 = 0, making the radial integral divergent. An alternative possibility consists in expressing the complete Green's tensor U HS in Fourier integral form, with the transformÛ HS obtained by summing contributions (19), (22) and (26) according to (8). After some manipulation, one finds that Û HS (ξ, α, y 3 ) = O(ξ −1 ), which still makes the radial integral divergent.…”

Section: Convergence Of Radial Integral the Radial Integral Inmentioning

confidence: 99%

“…Since (S · x) 3 = −x 3 > y 3 , the derivation of the Fourier transformÛ ∞ exploits (16b) and (18b), with x replaced by S ·x. The value ofÛ ∞ is then given by (19) with the following modifications: (i) replace q − a by −q + a , (ii) replace x by S ·x and y 3 by −y 3 , and (iii) right-multiply the result by S, to obtainÛ…”

Section: Derivation In the Fourier Spacementioning

confidence: 99%

“…2), and therefore bears similarities with the method used for defining low-frequency FMMs [11,19]. The partial Fourier transformû of a generic scalar, vector or tensor field u is defined bŷ…”

Section: Derivation In the Fourier Spacementioning

confidence: 99%

“…We adopt instead the methodology proposed by Rokhlin and coauthors [4,8,36], which generates generalized Gaussian quadrature (GGQ) rules for specific types of integrals and is in particular useful for implementing some forms of the FMM. For instance, GGQ-generating algorithms have been developed in [36] and [8], respectively, for integrals involved in the low-frequency [19] and wideband [7] versions of the FMM for the 3D Helmholtz equation. The tabulated GGQ rules given…”

Section: Introductionmentioning

confidence: 99%

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A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

Chaillat

Bonnet

2014

Journal of Computational Physics

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Abstract. In this article, a version of the frequency-domain elastodynamic Fast Multipole-Boundary Element Method (FM-BEM) for semi-infinite media, based on the half-space Green's tensor (and hence avoiding any discretization of the planar traction-free surface), is presented. The half-space Green's tensor is often used (in non-multipole form until now) for computing elastic wave propagation in the context of soil-structure interaction, with applications to seismology or civil engineering. However, unlike the fullspace Green's tensor, the elastodynamic half-space Green's tensor cannot be expressed using derivatives of the Helmholtz fundamental solution. As a result, multipole expansions of that tensor cannot be obtained directly from known expansions, and are instead derived here by means of a partial Fourier transform with respect to the spatial coordinates parallel to the free surface. The obtained formulation critically requires an efficient quadrature for the Fourier integral, whose integrand is both singular and oscillatory. Under these conditions, classical Gaussian quadratures would perform poorly, fail or require a large number of points. Instead, a version custom-tailored for the present needs of a methodology proposed by Rokhlin and coauthors, which generates generalized Gaussian quadrature rules for specific types of integrals, has been implemented. The accuracy and efficiency of the proposed formulation is demonstrated through numerical experiments on single-layer elastodynamic potentials involving up to about N = 6×10 5 degrees of freedom. In particular, a complexity significantly lower than that of the non-multipole version is shown to be achieved. IntroductionThe main advantage of the boundary element method (BEM) is that only the domain boundaries (and possibly interfaces) are discretized, leading to a reduction of the number of degrees of freedom (DOFs) relative to domain-discretization methods. Moreover, elastodyamic field equations and radiation conditions are exactly satisfied by the formulation [26], which avoids cumulative effects of grid dispersion and makes the BEM well suited for modeling wave propagation in unbounded domains. However, the standard BEM [3] leads to fully-populated matrices, which results in high computational complexity (O(N 2 ) per iteration using an iterative solver such as GMRES, where N denotes the number of boundary degrees of freedom (DOFs) of the BE model) and severe problem size limitations induced by the O(N 2 ) size of the influence matrix.The advent of accelerated BE methodologies has dramatically improved the capabilities of BEMs for many areas of application, in a large part owing to the rapid development of the Fast Multipole Method (FMM) over the last 15-20 years [29]. Such approaches have resulted in considerable solution speedup, memory reduction and model size increase. The FMM inherently relies on iterative solvers (usually GMRES), so as to avoid actual computation and storage of the fully-populated influence matrix, and is known to require O(N log N )...

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Multiscale electromagnetic computations using a hierarchical multilevel fast multipole algorithm

2014

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A hierarchical multilevel fast multipole method (H-MLFMM) is proposed herein to accelerate the solutions of surface integral equation methods. The proposed algorithm is particularly suitable for solutions of wideband and multiscale electromagnetic problems. As documented in Zhao and Chew (2000) that the multilevel fast multipole method (MLFMM) achieves O(N log N) computational complexity in the fixed mesh size scenario, hk = cst, where h is the mesh size and k is the corresponding wave number, for problems discretized under conventional mesh density. However, its performance deteriorates drastically for overly dense meshes where the couplings between different groups are dominated by evanescent waves or circuit physics. In the H-MLFMM algorithm, two different types of basis functions are proposed to address these two different natures of physics corresponding to the electrical size of the elements. Specifically, for the propagating wave couplings, the plane wave basis function adopted by MLFMM are effective and they are inherited by H-MLFMM. Whereas in the circuit physics and for the evanescent waves, H-MLFMM employs the so-called skeleton basis. Moreover, the proposed H-MLFMM unifies the procedures to account for the couplings using these two distinct types of basis functions. O(N) complexity is observed for both memory and CPU time from a set of numerical examples with fixed mesh sizes. Numerical results are included to demonstrate that H-MLFMM is error controllable and robust for a wide range of applications.

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Accelerating fast multipole methods for the Helmholtz equation at low frequencies

Cited by 216 publications

References 13 publications

The Fast Multipole Method: Numerical Implementation

The Fast Multipole Method: Numerical Implementation

A new Fast Multipole formulation for the elastodynamic half-space Greenʼs tensor

Multiscale electromagnetic computations using a hierarchical multilevel fast multipole algorithm

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