Optimized M2L Kernels for the Chebyshev Interpolation based Fast Multipole MethodReport as inadecuate

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1 HiePACS - High-End Parallel Algorithms for Challenging Numerical Simulations LaBRI - Laboratoire Bordelais de Recherche en Informatique, Inria Bordeaux - Sud-Ouest 2 Mechanical Engineering Department Stanford 3 iCME - Institute for Computational and Mathematical Engineering

Abstract : A fast multipole method FMM for asymptotically smooth kernel functions 1-r, 1-r^4, Gauss and Stokes kernels, radial basis functions, etc. based on a Chebyshev interpolation scheme has been introduced in Fong et al., 2009. The method has been extended to oscillatory kernels e.g., Helmholtz kernel in Messner et al., 2012. Beside its generality this FMM turns out to be favorable due to its easy implementation and its high performance based on intensive use of highly optimized BLAS libraries. However, one of its bottlenecks is the precomputation of the multiple-to-local M2L operator, and its higher number of floating point operations flops compared to other FMM formulations. Here, we present several optimizations for that operator, which is known to be the costliest FMM operator. The most efficient ones do not only reduce the precomputation time by a factor up to 340 but they also speed up the matrix-vector product. We conclude with comparisons and numerical validations of all presented optimizations.

Keywords : Fast Multipole Method asymptotically smooth kernels oscillatory kernels black-box method Chebyshev interpolation

Author: Matthias Messner - Bérenger Bramas - Olivier Coulaud - Eric Darve -

Source: https://hal.archives-ouvertes.fr/


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