Multipole Translation Theory for the Three-Dimensional Laplace and Helmholtz Equations

Epton, Michael A.; Dembart, B.

doi:10.1137/0916051

Cited by 250 publications

(177 citation statements)

References 7 publications

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“…The table and the associated graph for the potential calculation (see 6) show that the timing of the CPU is consistent with the theoretical dependence, Time = AN 2 , while the GPU timing approaches quadratic dependence in an asymptotic way. This is also clear from the time ratio between the CPU and GPU processes shown in the table, which stabilizes at relatively large values of N .…”

Section: Direct Summation Baselinesupporting

confidence: 65%

“…Such recursions can be found elsewhere (e.g. [6,12]). Note however, that one should convert these formulae to the real basis according to the definitions (7), (11), and (12).…”

Section: Computation Of Gradientmentioning

confidence: 96%

“…This is an unnecessary expense when the terms in (1) are all real. One of the most convenient bases for translation was proposed by Epton and Dembart [6], which is basis (7) with…”

Section: Basis Functionsmentioning

confidence: 99%

“…Explicit expressions for these coefficients can be found elsewhere (see e.g. [6,4]). It is not difficult to show then that reexpansion of the basis produces the entries of the matrix translation operator, which relates expansion coefficients in different bases with different centers.…”

Section: Rcr Decompositionmentioning

confidence: 99%

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Fast multipole methods on graphics processors

Gumerov

Duraiswami

2008

Journal of Computational Physics

157

131

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The Fast Multipole Method allows the rapid evaluation of sums of radial basis functions centered at points distributed inside a computational domain at a large number of evaluation points to a specified accuracy . The method scales as O (N ) in both time and memory compared to the direct method with complexity O(N 2 ), which allows the solution of larger problems with given resources. Graphical processing units (GPU) are now increasingly viewed as data parallel compute coprocessors that can provide significant computational performance at low price. We describe acceleration of the FMM using the data parallel GPU architecture.The FMM has a complex hierarchical (adaptive) structure, which is not easily implemented on dataparallel processors. We described strategies for parallelization of all components of the FMM, develop a model to explain the performance of the algorithm on the GPU architecture; and determined optimal settings for the FMM on the GPU, which are different from those on usual CPUs. Some innovations in the FMM algorithm, including the use of modified stencils, real polynomial basis functions for the Laplace kernel, and decompositions of the translation operators, are also described.We obtained accelerations of the Laplace kernel FMM on a single NVIDIA GeForce 8800 GTX GPU in the range of 30-60 compared to a serial CPU FMM implementation. For a problem with a million sources, the summations involved are performed in approximately one second. This performance is equivalent to solving of the same problem at a 43 Teraflop rate if we use straightforward summation.

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Section: Direct Summation Baselinesupporting

confidence: 65%

“…Such recursions can be found elsewhere (e.g. [6,12]). Note however, that one should convert these formulae to the real basis according to the definitions (7), (11), and (12).…”

Section: Computation Of Gradientmentioning

confidence: 96%

“…This is an unnecessary expense when the terms in (1) are all real. One of the most convenient bases for translation was proposed by Epton and Dembart [6], which is basis (7) with…”

Section: Basis Functionsmentioning

confidence: 99%

Section: Rcr Decompositionmentioning

confidence: 99%

See 2 more Smart Citations

Fast multipole methods on graphics processors

Gumerov

Duraiswami

2008

Journal of Computational Physics

157

131

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show abstract

“…Usually the Runge approximation is not constructive, because its proof is done by using the unique continuation and Hahn-Banach theorem. However, for the Helmholtz equation, it is possible to make the Runge approximation constructive by using the translation theory (see [7]). …”

Section: ∂D = {C(t) | T = T(c D) C Is a Needle And T(c D) < 1} (31)mentioning

confidence: 99%

Recovery of multiple obstacles by probe method

et al. 2009

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Abstract. We consider an inverse scattering problem for multiple obstacles D = N j=1 D j ⊂ R 3 with different types of boundary for D j . By constructing an indicator function from the far-field pattern of the scattered wave, we can firstly reconstruct the shape of all obstacles, then identify the type of boundary for each obstacle, as well as the boundary impedance in the case that obstacles have the Robin-type boundary condition. The novelty of our probe method compared with the existing probe method is that we succeeded in identifying the type of boundary condition for multiple obstacles by analyzing the behavior of both the imaginary part and the real part of the indicator function. The numerical realizations are given to show the performance of this inversion method.

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Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D

Yoshida

Nishimura

Kobayashi

2001

Int. J. Numer. Meth. Engng.

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Fast multipole method (FMM) has been developed as a technique to reduce the computational cost and memory requirements in solving large-scale problems. This paper discusses an application of FMM to threedimensional boundary integral equation method for elastostatic crack problems. The boundary integral equation for many crack problems is discretized with FMM and Galerkin's method. The resulting algebraic equation is solved with generalized minimum residual method (GMRES). The numerical results show that FMM is more e cient than conventional methods when the number of unknowns is more than about 1200 and, therefore, can be useful in large-scale analyses of fracture mechanics.

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Multipole Translation Theory for the Three-Dimensional Laplace and Helmholtz Equations

Cited by 250 publications

References 7 publications

Fast multipole methods on graphics processors

Fast multipole methods on graphics processors

Recovery of multiple obstacles by probe method

Application of fast multipole Galerkin boundary integral equation method to elastostatic crack problems in 3D

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