A Scalable Parallel Wideband MLFMA for Efficient Electromagnetic Simulations on Large Scale Clusters

The development of the multilevel fast multipole algorithm (MLFMA) and its multiscale variants have enabled the use of integral equation (IE) based solvers to compute scattering from complicated structures. Development of scalable parallel algorithms, to extend the reach of these solvers, has been a topic of intense research for about a decade. In this paper, we present a new algorithm for parallel implementation of IE solver that is augmented with a wideband MLFMA and scalable on large number of processors. The wideband MLFMA employed here, to handle multiscale problems, is a hybrid combination of the accelerated Cartesian expansion (ACE) and the classical MLFMA. The salient feature of the presented parallel algorithm is that it is implicitly load balanced and exhibits higher performance. This is achieved by developing a strategy to partition the MLFMA tree, and hence the associated computations, in a self-similar fashion among the parallel processors. As detailed in the paper, the algorithm employs both spatial and direction partitioning approaches in a flexible manner to ensure scalable performance. Plethora of results are presented here to exhibit the scalability of this algorithm on 512 and more processors.

[1]  J.L. Volakis,et al.  Massively Parallel Fast Multipole Method Solutions of Large Electromagnetic Scattering Problems , 2007, IEEE Transactions on Antennas and Propagation.

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

[3]  B. Shanker,et al.  A Novel Wideband FMM for Fast Integral Equation Solution of Multiscale Problems in Electromagnetics , 2009, IEEE Transactions on Antennas and Propagation.

[4]  Ö. Ergül,et al.  Fast and accurate solutions of extremely large integral-equation problems discretised with tens of millions of unknowns , 2007 .

[5]  R. Coifman,et al.  The fast multipole method for the wave equation: a pedestrian prescription , 1993, IEEE Antennas and Propagation Magazine.

[6]  L. Gurel,et al.  Efficient Parallelization of the Multilevel Fast Multipole Algorithm for the Solution of Large-Scale Scattering Problems , 2008, IEEE Transactions on Antennas and Propagation.

[7]  J.M. Song,et al.  ScaleME: a portable scaleable multipole engine for electromagnetic and acoustic integral equation solvers , 1998, IEEE Antennas and Propagation Society International Symposium. 1998 Digest. Antennas: Gateways to the Global Network. Held in conjunction with: USNC/URSI National Radio Science Meeting (Cat. No.98CH36.

[8]  J. Applequist,et al.  Traceless cartesian tensor forms for spherical harmonic functions: new theorems and applications to electrostatics of dielectric media , 1989 .

[9]  Jan Fostier,et al.  Provably scalable parallel multilevel fast multipole algorithm , 2008 .

[10]  Carlos Delgado,et al.  An Incomplete Review of Fast Multipole Methods–from Static to Wideband–as Applied to Problems in Computational Electromagnetics , 2009 .

[11]  D. Wilton,et al.  Electromagnetic scattering by surfaces of arbitrary shape , 1980 .

[12]  L. Greengard.  The Rapid Evaluation of Potential Fields in Particle Systems , 1988 .

[13]  Srinivas Aluru,et al.  Parallel accelerated cartesian expansions for particle dynamics simulations , 2009, 2009 IEEE International Symposium on Parallel & Distributed Processing.

[14]  S. Velamparambil,et al.  10 million unknowns: is it that big? [computational electromagnetics] , 2003, IEEE Antennas and Propagation Magazine.

[15]  Balasubramaniam Shanker,et al.  Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein-Gordon potentials , 2010, J. Comput. Phys..

[16]  Srinivas Aluru,et al.  A provably optimal, distribution-independent parallel fast multipole method , 2000, Proceedings 14th International Parallel and Distributed Processing Symposium. IPDPS 2000.

[17]  Jianming Jin,et al.  A higher order parallelized multilevel fast multipole algorithm for 3-D scattering , 2001 .

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

[19]  Weng Cho Chew,et al.  Three-dimensional multilevel fast multipole algorithm from static to electrodynamic , 2000 .

[20]  Mingyu Lu,et al.  Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm , 2003 .

[21]  L. Gurel,et al.  A Hierarchical Partitioning Strategy for an Efficient Parallelization of the Multilevel Fast Multipole Algorithm , 2009, IEEE Transactions on Antennas and Propagation.

[22]  Michael S. Warren,et al.  Astrophysical N-body simulations using hierarchical tree data structures , 1992, Proceedings Supercomputing '92.

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

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

[25]  Weng Cho Chew,et al.  MLFMA for solving integral equations of 2-D electromagnetic problems from static to electrodynamic , 1999 .

[26]  Pangfeng Liu,et al.  Experiences with parallel N-body simulation , 1994, SPAA '94.

[27]  Michael Griebel,et al.  Numerical Simulation in Molecular Dynamics: Numerics, Algorithms, Parallelization, Applications , 2007 .

[28]  S. Rao Kosaraju,et al.  A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields , 1995, JACM.

[29]  Jiming Song,et al.  Error Analysis for the Numerical Evaluation of the Diagonal Forms of the Scalar Spherical Addition Theorem , 1999 .

[30]  M. Nilsson.  Stability of the High Frequency Fast Multipole Method for Helmholtz’ Equation in Three Dimensions , 2004 .

[31]  Jian-Ming Jin,et al.  Fast and Efficient Algorithms in Computational Electromagnetics , 2001 .

[32]  Zydrunas Gimbutas,et al.  A wideband fast multipole method for the Helmholtz equation in three dimensions , 2006, J. Comput. Phys..

[33]  V. Rokhlin.  Rapid solution of integral equations of classical potential theory , 1985 .

[34]  Balasubramaniam Shanker,et al.  Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE) , 2007, J. Comput. Phys..

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

[36]  R. Mittra,et al.  Computational Methods for Electromagnetics , 1997 .

[37]  Srinivas Aluru,et al.  A Scalable Parallel Fast Multipole Method for Analysis of Scattering from Perfect Electrically Conducting Surfaces , 2002, ACM/IEEE SC 2002 Conference (SC'02).

[39]  He Huang,et al.  Accelerated Cartesian expansions - A fast method for computing of potentials of the form R-ν for all real ν , 2007, J. Comput. Phys..

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

[41]  Bradley K. Alpert,et al.  A Fast Spherical Filter with Uniform Resolution , 1997 .

[42]  M. S. Warren,et al.  A parallel hashed Oct-Tree N-body algorithm , 1993, Supercomputing '93.

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