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

Toru Takahashi1, Cris Cecka, William Fong, Eric Darve
Department of Mechanical Science and Engineering, Nagoya University, Nagoya 464-8603, Japan
International Journal for Numerical Methods in Engineering, 2011


   title={Optimizing the multipole-to-local operator in the fast multipole method for graphical processing units},

   author={Takahashi, T. and Cecka, C. and Fong, W. and Darve, E.},

   journal={International Journal for Numerical Methods in Engineering},


   publisher={Wiley Online Library}


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This paper presents a number of algorithms to run the fast multipole method (FMM) on NVIDIA CUDA-capable 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 200 Gflop/s on one NVIDIA Tesla C1060 GPU (Nvidia Corporation, Sta. Clara, CA, USA). This was compared against two quad-core Intel Xeon E5345 processors (Intel Corporation, Sta. Clara, CA, USA) running at 2.33 GHz, for a combined peak performance of 149 Gflop/s for single precision. For the low FMM accuracy case, the observed performance of the CPU code was 37 Gflop/s, whereas for the high FMM accuracy case, the performance was about 8.5 Gflop/s, most likely because of a higher frequency of cache misses. We also present benchmarks on an NVIDIA C2050 GPU (a Fermi processor)(Nvidia Corporation, Sta. Clara, CA, USA) in single and double precision.
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