Fast electrostatic force calculation on parallel computer clusters

The fast multipole method (FMM) and smooth particle mesh Ewald (SPME) are well known fast algorithms to evaluate long range electrostatic interactions in molecular dynamics and other fields. FMM is a multi-scale method which reduces the computation cost by approximating the potential due to a group of particles at a large distance using few multipole functions. This algorithm scales like O(N) for N particles. SPME algorithm is an O(NlnN) method which is based on an interpolation of the Fourier space part of the Ewald sum and evaluating the resulting convolutions using fast Fourier transform (FFT). Those algorithms suffer from relatively poor efficiency on large parallel machines especially for mid-size problems around hundreds of thousands of atoms. A variation of the FMM, called PWA, based on plane wave expansions is presented in this paper. A new parallelization strategy for PWA, which takes advantage of the specific form of this expansion, is described. Its parallel efficiency is compared with SPME through detail time measurements on two different computer clusters.

[1]  Rajiv K. Kalia,et al.  Scalable and portable implementation of the fast multipole method on parallel computers , 2003 .

[2]  Berend Smit,et al.  Understanding molecular simulation: from algorithms to applications , 1996 .

[3]  Jiming Song,et al.  Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects , 1997 .

[4]  Anoop Gupta,et al.  Load Balancing and Data locality in Adaptive Hierarchical N-Body Methods: Barnes-Hut, Fast Multipole, and Rasiosity , 1995, J. Parallel Distributed Comput..

[5]  T. Darden,et al.  Particle mesh Ewald: An N⋅log(N) method for Ewald sums in large systems , 1993 .

[6]  R. Valdarnini.  Parallelization of a treecode , 2003 .

[7]  L. Greengard,et al.  A new version of the fast multipole method for screened Coulomb interactions in three dimensions , 2002 .

[8]  Weng Cho Chew,et al.  A multilevel algorithm for solving a boundary integral equation of wave scattering , 1994 .

[9]  Jianming Jin,et al.  Fast solution methods in electromagnetics , 1997 .

[10]  Jakub Kurzak,et al.  Communications overlapping in fast multipole particle dynamics methods , 2005 .

[11]  L. Greengard,et al.  Regular Article: A Fast Adaptive Multipole Algorithm in Three Dimensions , 1999 .

[12]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[13]  Jakub Kurzak,et al.  Massively parallel implementation of a fast multipole method for distributed memory machines , 2005, J. Parallel Distributed Comput..

[14]  T. Darden,et al.  A smooth particle mesh Ewald method , 1995 .

[15]  Martin Head-Gordon,et al.  Derivation and efficient implementation of the fast multipole method , 1994 .

[16]  John K. Salmon,et al.  Parallel hierarchical N-body methods , 1992 .

[17]  John Dubinski.  A parallel tree code , 1996 .

[18]  S. Velamparambil,et al.  Analysis and performance of a distributed memory multilevel fast multipole algorithm , 2005, IEEE Transactions on Antennas and Propagation.

[19]  Eric Darve,et al.  The Fast Multipole Method , 2000 .

[20]  Eric Darve,et al.  The Fast Multipole Method I: Error Analysis and Asymptotic Complexity , 2000, SIAM J. Numer. Anal..

[21]  L. Greengard,et al.  A new version of the Fast Multipole Method for the Laplace equation in three dimensions , 1997, Acta Numerica.

[22]  Piet Hut,et al.  A hierarchical O(N log N) force-calculation algorithm , 1986, Nature.

[23]  H. G. Petersen.  Accuracy and efficiency of the particle mesh Ewald method , 1995 .

[24]  Shang-Hua Teng,et al.  Provably Good Partitioning and Load Balancing Algorithms for Parallel Adaptive N-Body Simulation , 1998, SIAM J. Sci. Comput..

[25]  Christian Holm,et al.  How to mesh up Ewald sums. I. A theoretical and numerical comparison of various particle mesh routines , 1998 .

[26]  William Gropp,et al.  A Parallel Version of the Fast Multipole Method-Invited Talk , 1987, PPSC.

[27]  R W Hockney,et al.  Computer Simulation Using Particles , 1966 .

[28]  Eric Darve,et al.  Efficient fast multipole method for low-frequency scattering , 2004 .

[29]  Eric F Darve.  Regular ArticleThe Fast Multipole Method: Numerical Implementation , 2000 .

[30]  L Greengard,et al.  Fast Algorithms for Classical Physics , 1994, Science.

[31]  Eric F Darve.  The Fast Multipole Method , 2000 .

[32]  A. Edelman,et al.  Building Blocks and Excluded Sums , 2005 .

[33]  L. Greengard,et al.  Accelerating fast multipole methods for the Helmholtz equation at low frequencies , 1998 .

[34]  Norman Yarvin,et al.  Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators , 1998, SIAM J. Sci. Comput..