Improve the efficiency of the Cartesian tensor based fast multipole method for Coulomb interaction using the traces

Huang, He; Luo, Li -Shi; Li, Rui; Chen, Jie; Zhang, He

doi:10.1016/j.jcp.2018.05.028

Title: Improve the efficiency of the Cartesian tensor based fast multipole method for Coulomb interaction using the traces

Journal Article · Thu May 17 00:00:00 EDT 2018 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2018.05.028· OSTI ID:1439411

Huang, He ^[1]; Luo, Li -Shi ^[1]; Li, Rui ^[2]; Chen, Jie ^[2];

^[2]

Old Dominion Univ., Norfolk, VA (United States)
Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)

To compute the non-oscillating mutual interaction for a systems with N points, the fast multipole method (FMM) has an efficiency that scales linearly with the number of points. Specifically, for Coulomb interaction, FMM can be constructed using either the spherical harmonic functions or the totally symmetric Cartesian tensors. In this paper, we will present that the effciency of the Cartesian tensor-based FMM for the Coulomb interaction can be significantly improved by implementing the traces of the Cartesian tensors in calculation to reduce the independent elements of the n-th rank totally symmetric Cartesian tensor from (n + 1)(n + 2)=2 to 2n + 1. The computation complexity for the operations in FMM are analyzed and expressed as polynomials of the highest rank of the Cartesian tensors. For most operations, the complexity is reduced by one order. Numerical examples regarding the convergence and the effciency of the new algorithm are demonstrated. As a result, a reduction of computation time up to 50% has been observed for a moderate number of points and rank of tensors.

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Research Organization:: Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Nuclear Physics (NP)

Grant/Contract Number:: AC05-06OR23177

OSTI ID:: 1439411

Alternate ID(s):: OSTI ID: 1582863

Report Number(s):: JLAB-ACP-17-2560; DOE/OR/23177-4223; R&D Project: 2016-LDRD-7; PII: S0021999118303280; TRN: US1900604

Journal Information:: Journal of Computational Physics, Vol. 371, Issue C; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Cited by: 2 works

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References (13)

An Efficient Program for Many-Body Simulation Appel, Andrew W. SIAM Journal on Scientific and Statistical Computing, Vol. 6, Issue 1 https://doi.org/10.1137/0906008	journal	January 1985
A fast algorithm for particle simulations Greengard, L.; Rokhlin, V. Journal of Computational Physics, Vol. 73, Issue 2 https://doi.org/10.1016/0021-9991(87)90140-9	journal	December 1987
A new version of the Fast Multipole Method for the Laplace equation in three dimensions Greengard, Leslie; Rokhlin, Vladimir Acta Numerica, Vol. 6 https://doi.org/10.1017/S0962492900002725	journal	January 1997
A Fast Adaptive Multipole Algorithm for Particle Simulations Carrier, J.; Greengard, L.; Rokhlin, V. SIAM Journal on Scientific and Statistical Computing, Vol. 9, Issue 4 https://doi.org/10.1137/0909044	journal	July 1988
A Fast Adaptive Multipole Algorithm in Three Dimensions Cheng, H.; Greengard, L.; Rokhlin, V. Journal of Computational Physics, Vol. 155, Issue 2 https://doi.org/10.1006/jcph.1999.6355	journal	November 1999
A New Version of the Fast Multipole Method for Screened Coulomb Interactions in Three Dimensions Greengard, Leslie F.; Huang, Jingfang Journal of Computational Physics, Vol. 180, Issue 2 https://doi.org/10.1006/jcph.2002.7110	journal	August 2002
An Implementation of the Fast Multipole Method without Multipoles Anderson, Christopher R. SIAM Journal on Scientific and Statistical Computing, Vol. 13, Issue 4 https://doi.org/10.1137/0913055	journal	July 1992
A kernel-independent adaptive fast multipole algorithm in two and three dimensions Ying, Lexing; Biros, George; Zorin, Denis Journal of Computational Physics, Vol. 196, Issue 2 https://doi.org/10.1016/j.jcp.2003.11.021	journal	May 2004
The black-box fast multipole method Fong, William; Darve, Eric Journal of Computational Physics, Vol. 228, Issue 23 https://doi.org/10.1016/j.jcp.2009.08.031	journal	December 2009
Accelerated Cartesian expansions – A fast method for computing of potentials of the form R−ν for all real ν Shanker, B.; Huang, H. Journal of Computational Physics, Vol. 226, Issue 1 https://doi.org/10.1016/j.jcp.2007.04.033	journal	September 2007
Cartesian polytensors Applequist, Jon Journal of Mathematical Physics, Vol. 24, Issue 4 https://doi.org/10.1063/1.525770	journal	April 1983
Fundamental relationships in the theory of electric multipole moments and multipole polarizabilities in static fields Applequist, Jon Chemical Physics, Vol. 85, Issue 2 https://doi.org/10.1016/0301-0104(84)85039-9	journal	April 1984
Traceless cartesian tensor forms for spherical harmonic functions: new theorems and applications to electrostatics of dielectric media Applequist, J. Journal of Physics A: Mathematical and General, Vol. 22, Issue 20 https://doi.org/10.1088/0305-4470/22/20/011	journal	October 1989

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A New Relatively Simple Approach to Multipole Interactions in Either Spherical Harmonics or Cartesians, Suitable for Implementation into Ewald Sums Burnham, Christian J.; English, Niall J. International Journal of Molecular Sciences, Vol. 21, Issue 1 https://doi.org/10.3390/ijms21010277	journal	December 2019