High performance BLAS formulation of the adaptive Fast Multipole Method

Coulaud, Olivier; Fortin, Pierre; Roman, Jean

doi:10.1016/j.mcm.2009.08.039

Cited by 11 publications

(11 citation statements)

References 19 publications

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“…We have already extended this BLAS scheme to the adaptive version of the FMM [39], and we are currently parallelizing it on shared and distributed memory architectures. In the near future, our BLAS implementation may appear even more appealing in the context of heterogeneous high performance computing with specialpurpose hardware, such as Graphics Processing Units (GPU) or Cell processors.…”

Section: Resultsmentioning

confidence: 99%

High performance BLAS formulation of the multipole-to-local operator in the fast multipole method

Coulaud

Fortin

Roman

2008

Journal of Computational Physics

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The multipole-to-local (M2L) operator is the most time-consuming part of the far field computation in the fast multipole method for Laplace equation. Its natural expression, though commonly used, does not respect a sharp error bound: we here first prove the correctness of a second expression. We then propose a matrix formulation implemented with basic linear algebra subprograms (BLAS) routines in order to speed up its computation for these two expressions. We also introduce special data storages in memory to gain greater computational efficiency. This BLAS scheme is finally compared, for uniform distributions, to other M2L improvements such as block FFT, FFT with polynomial scaling, rotations and plane wave expansions. When considering runtime, extra memory storage, numerical stability and common precisions for Laplace equation, the BLAS version appears as the best one.

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Section: Resultsmentioning

confidence: 99%

High performance BLAS formulation of the multipole-to-local operator in the fast multipole method

Coulaud

Fortin

Roman

2008

Journal of Computational Physics

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“…This results in a modified adaptive algorithm with proven linear operation count without any assumption about the particle distribution (see Nabors et al 1994). In the context of parallel computing in multi-process mode, this second algorithm also yields more predictable communication patterns (Coulaud et al 2010).…”

Section: The Different Algorithmsmentioning

confidence: 99%

“…The FMB code (Fast Multipole with BLAS 2 ) is an implementation of the traditional fast multipole method (Coulaud et al 2007(Coulaud et al , 2008(Coulaud et al , 2010. This code is based on the algorithm of Nabors et al (1994): the octree height H is optimally set by the user to minimize the computation time.…”

Section: The Different Algorithmsmentioning

confidence: 99%

“…Usually, the H value must roughly balance the near field and far field computations, which depends on the particle distribution and the P value used. An octree with indirection is used as the octree data structure to manage a highly unbalanced octree in non-uniform particle distributions (Coulaud et al 2010). For the relatively low precisions required in galactic dynamics, we use P = 4 in this paper, and the degrees of the M2L operator terms are limited to P as in multipole and local expansions; as presented in Coulaud et al (2008), this approach is indeed more efficient for low P values.…”

Section: The Different Algorithmsmentioning

confidence: 99%

“…In addition, FMB can compute efficiently some expansion operations thanks to a matrix formulation with BLAS routines (Coulaud et al 2008). However no BLAS computation will be performed here since, as shown in Coulaud et al (2010), the BLAS routines are mainly efficient for greater P values or within large uniform areas, which are not present in particle distributions representing galaxies or groups of galaxies.…”

Section: The Different Algorithmsmentioning

confidence: 99%

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Comparisons of different codes for galacticN-body simulations

Fortin

Athanassoula²,

Lambert³

2011

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Context. N-body simulations are widely used for galactic dynamics studies. Several different algorithms have been developed and a number of authors have made their codes public. Aims. We wish to help potential users of these codes, particularly less experienced ones, to select the appropriate code by giving them relevant information. Methods. We consider different implementations of three different algorithms, namely the Barnes-Hut tree-code algorithm (implemented both in GADGET-2 and as a NEMO program), the fast multipole method (implemented in the FMB code -Fast Multipole with BLAS) and W. Dehnen's algorithm (implemented in falcON -force algorithm with complexity O(N)). We compare extensively the timing and memory comparisons of these different implementations and discuss other advantages and disadvantages of these codes.Results. In terms of serial executions, falcON is clearly the fastest code, in many cases by a wide margin. In addition, it is much less sensitive to concentration than other codes, it conserves linear momentum, its computational costs depends roughly linearly on the number of particles, and it allows for manipulators. GADGET-2 is the second fastest and, being parallel, can outperform falcON if a sufficient number of processors is used, the number of which, in our tests is between three and ten. Its public version includes an SPH representation of the gas component. FMB is faster than the implementation of the Barnes-Hut algorithm in NEMO, as long as the particle distribution is not too concentrated, but remains considerably slower than falcON and GADGET-2. It has, nevertheless, the highest parallel efficiency and memory scalability. Conclusions. We recommend falcON for pure N-body simulations with a relatively restricted number of particles (up to a few million). In cases of a larger number of particles, a number of processors being available, or studies including gas and its physics, we recommend GADGET-2. FMB provides the highest parallel efficiency and memory scalability, but is considerably slower in execution time and is ill-adapted to the concentrated distributions encountered in galactic work.

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Optimizing the multipole‐to‐local operator in the fast multipole method for graphical processing units

Takahashi

Cecka

Fong

et al. 2011

Numerical Meth Engineering

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SUMMARYThis paper presents a number of algorithms to run the fast multipole method (FMM) on NVIDIA CUDAcapable graphical processing units (GPUs) (Nvidia Corporation, Sta. Clara, CA, USA). The FMM is a class of methods to compute pairwise interactions between N particles for a given error tolerance and with computational cost of O.N /. The methods described in the paper are applicable to any FMMs in which the multipole-to-local (M2L) operator is a dense matrix and the matrix is precomputed. This is the case for example in the black-box fast multipole method (bbFMM), which is a variant of the FMM that can handle large class of kernels. This example will be used in our benchmarks.In the FMM, two operators represent most of the computational cost, and an optimal implementation typically tries to balance those two operators. One is the nearby interaction calculation (direct sum calculation, line 29 in Listing 1), and the other is the M2L operation. We focus on the M2L. By combining multiple M2L operations and reordering the primitive loops of the M2L so that CUDA threads can reuse or share common data, these approaches reduce the movement of data in the GPU. Because memory bandwidth is the primary bottleneck of these methods, significant performance improvements are realized. Four M2L schemes are detailed and analyzed in the case of a uniform tree.The four schemes are tested and compared with an optimized, OpenMP parallelized, multi-core CPU code. We consider high and low precision calculations by varying the number of Chebyshev nodes used in the bbFMM. The accuracy of the GPU codes is found to be satisfactory and achieved performance over

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High performance BLAS formulation of the adaptive Fast Multipole Method

Cited by 11 publications

References 19 publications

High performance BLAS formulation of the multipole-to-local operator in the fast multipole method

High performance BLAS formulation of the multipole-to-local operator in the fast multipole method

Comparisons of different codes for galacticN-body simulations

Optimizing the multipole‐to‐local operator in the fast multipole method for graphical processing units

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