The black-box fast multipole method

Fong, William; Darve, Eric

doi:10.1016/j.jcp.2009.08.031

Cited by 230 publications

(329 citation statements)

References 19 publications

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“…It was first proposed for the Laplacian kernel in 2-D, κ(x, y) = − log x − y but it has been subsequently extended to several other kernels, including radial basis functions. The original version of the fast multipole method relied on analytic expressions for the multipole expansion of the kernels but several kernel independent versions have been proposed [36,38]. For more details, the reader is referred to [60][61][62].…”

Section: Fast Multipole Methodsmentioning

confidence: 99%

“…Various fast summation schemes have been devised to provide matrix-vector products in O(N log β N) for such problems, where N is the number of unknowns and β ≥ 0 an integer depending on the method. They broadly fall under three different categories: -FFT (Fast Fourier Transform) based methods, -fast multipole methods [36][37][38][60][61][62], -hierarchical matrices [21,22,24].…”

Section: Covariance Functionsmentioning

confidence: 99%

“…Moreoever, the number of terms required for an accurate representation up to an error of ε, is O(k d ), where k = O(1/ε). Another approach is to use tensor-product interpolation, which constructs a low-rank approximation using tensor-products of interpolating polynomials such as Legendre or Chebyshev polynomials [20,36]. This approach has the advantage that it is quite general since it only uses kernel evaluations but like the Taylor series it also requires O(k d ) evaluations.…”

Section: Low Rank Approximationmentioning

confidence: 99%

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Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

Saibaba

Ambikasaran

et al. 2012

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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Nous commençons par décrire brièvement l'approche géostatistique dans le contexte d'un problème inverse linéaire, puis discutons les difficultés rencontrées dans le cadre d'une implémentation à grande échelle. Ensuite, en utilisant une approche basée sur les matrices hiérarchiques, nous montrons comment réduire le coût de calcul d'un produit matrice-vecteur de O(m 2 ) à O(m log m) dans le cas de matrices de covariance denses ; m désignant ici le nombre d'inconnues. Combinée avec un solveur de Krylov, qui ne requiert pas la construction explicite de la matrice, cette méthode conduit à un algorithme beaucoup plus rapide pour résoudre le système d'équations associé à l'approche géostatistique. Nous illustrons enfin la performance de notre algorithme dans un situation spécifique, surveiller la concentration de CO 2 à l'aide de la tomographie sismique entre puits. Abstract -Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics -Characterizing the uncertainty in the subsurface is an important step for exploration and extraction of natural resources, the storage of nuclear material and gasses such as natural gas or CO 2 . Imaging the subsurface can be posed as an inverse problem and can be solved using the geostatistical approach [Kitanidis P.K. (2007) Geophys. Monogr. Ser. 171, 19-30, doi:10.1029/171GM04;Kitanidis (2011 Kitanidis ( ) doi: 10.1002

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Section: Fast Multipole Methodsmentioning

confidence: 99%

Section: Covariance Functionsmentioning

confidence: 99%

Section: Low Rank Approximationmentioning

confidence: 99%

See 1 more Smart Citation

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

Saibaba

Ambikasaran

et al. 2012

Oil Gas Sci. Technol. – Rev. IFP Energies nouvelles

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“…The particles are located in a hierarchy of boxes. If there are k boxes on the finest grid level, the total number of boxes is approximately 8 7 k. In the first step, a multipole expansion is computed at the cost c M per particle. In the second step, multipole expansions for higher level boxes are created by applying translation operators to expansions of lower level boxes, this has the cost c M 2M per box.…”

Section: Total Gain In a Multipole Scheme By Compressionmentioning

confidence: 99%

Matrix compression by common subexpression elimination

Oppelstrup

2013

Journal of Computational Physics

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In this report a method for common subexpression elimination in matrices is explored. The method is applied to several types of matrices occurring in numerical simulations. In all cases, the cost of a matrix-vector multiplication is reduced by a significant amount. The amount of storage required for the eliminated matrices is also less than that required for the original matrices. When the proposed method is applied to the Fourier transform matrix, the output is equivalent to the fast Fourier transform. For some matrices used in the fast multipole method for dislocation dynamics, the cost of a matrix-vector multiplication is reduced from O(p 6 ) to O(p 4.5 ), where p is the expansion order. Using an expansion order of 5, one can expect a factor of four speedup of the fast multipole part of a dislocation dynamics code.

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“…The boundary integrals are accelerated using a kernel-independent fast multipole method (FMM) 23 with O(N) computational complexity. This kernel-independent FMM is an interpolation-based FMM that utilizes Chebyshev interpolation and low-rank approximation, and can also be used for kernels that are known only numerically.…”

Section: Introductionmentioning

confidence: 99%

An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

Jiang

Zhao

et al. 2016

The Journal of Chemical Physics

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we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space. In the second, we solve an electrostatic problem involving polarizable dielectric bodies in an unbounded dielectric medium. The results from these test cases show that our proposed parallel approach, which is built on a kernel-independent FMM, can enable highly efficient and accurate simulations and allow for considerable flexibility in a broad range of applications.

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The black-box fast multipole method

Cited by 230 publications

References 19 publications

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

Matrix compression by common subexpression elimination

An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

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