Fast Multipole Methods and Applications

We propose a method of a parallel distribution of densely populated matrices arising in boundary element discretizations of partial differential equations. In our method the underlying boundary element mesh consisting of n elements is decomposed into N submeshes. The related N ×N submatrices are assigned to N concurrent processes to be assembled. Additionally we require each process to hold exactly one diagonal submatrix, since its assembling is typically most time consuming when applying fast boundary elements. We obtain a class of such optimal parallel distributions of the submeshes and corresponding submatrices by cyclic decompositions of undirected complete graphs. It results in a method the theoretical complexity of which is O((n/ √ N) log(n/ √ N)) in terms of time for the setup, assembling, matrix action, as well as memory consumption per process. Nevertheless, numerical experiments up to n = 2744832 and N = 273 on a real-world geometry document that the method exhibits superior parallel scalability O((n/N) log n) of the overall time, while the memory consumption scales accordingly to the theoretical estimate.

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“…For details see e.g. [14]. The overall algorithmic complexity for the approximate assembling and an action of V is O(p 2 n log n).…”

Section: Fast Multipole Methodsmentioning

confidence: 99%

A parallel fast boundary element method using cyclic graph decompositions

et al. 2015

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“…Since the fundamental solution can be represented as the fundamental solution of the Poisson equation and its derivatives, the FMM can be applied in an analogous way. The boundary integral operators for the system of linear elasticity were realized, for example, in [14,15]; see also the references given therein. In a straightforward manner, those techniques and the techniques described in section 3 can be combined for a fast evaluation of Newton potentials in linear elasticity.…”

Section: Linear Elasticity Problemsmentioning

confidence: 99%

“…For an efficient evaluation of the Newton potential N 0 f by using the FMM [7,8,14], the kernel of the volume integral is approximated by the truncated series expansion…”

Section: Fast Multipole Evaluation Of Newton Potentialsmentioning

confidence: 99%

“…We propose the use of the FMM [7,8,14] for an efficient evaluation of the Newton potentials in the symmetric formulation of boundary integral equations. In contrast to the literature, we treat the normal derivative of the Newton potential as used within the hypersingular boundary integral equation, analyze the complexity of the FMM for volume integrals, and present an error analysis of the approximation by the FMM.…”

mentioning

confidence: 99%

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Fast Evaluation of Volume Potentials in Boundary Element Methods

Of¹,

Steinbach²,

Urthaler³

2010

SIAM J. Sci. Comput.

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The solution of inhomogeneous partial differential equations by boundary element methods requires the evaluation of volume potentials. A direct standard computation of the classical Newton potentials is possible but expensive. Here, a fast evaluation of the Newton potentials by using the fast multipole method is described and analyzed. In particular, an approximation by the fast multipole method is investigated and related error estimates are given. Furthermore, an indirect evaluation of the normal derivative of the Newton potential is presented. A numerical analysis is presented for all approaches mentioned above. Numerical results are presented for the Poisson equation and for the system of linear elastostatics.

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“…Nonadmissible blocks are assembled in the standard way as full rank matrices. The fast multipole method (FMM) is based on the approximation of the system matrices by the multipole series expansion [3][4][5], whereas the adaptive cross approximation (ACA) assembles the low rank approximation from an algebraic point of view [6,2].…”

Section: Introductionmentioning

confidence: 99%