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

Abstract

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.

Authors:
 [1];  [1];  [2];  [2]; ORCiD logo [2]
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  1. Old Dominion Univ., Norfolk, VA (United States)
  2. Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)
Publication Date:
Research Org.:
Thomas Jefferson National Accelerator Facility (TJNAF), Newport News, VA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Nuclear Physics (NP)
OSTI Identifier:
1439411
Alternate Identifier(s):
OSTI ID: 1582863
Report Number(s):
JLAB-ACP-17-2560; DOE/OR/23177-4223
Journal ID: ISSN 0021-9991; R&D Project: 2016-LDRD-7; PII: S0021999118303280; TRN: US1900604
Grant/Contract Number:  
AC05-06OR23177
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 371; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Fast multipole method; Cartesian tensor; Coulomb interaction

Citation Formats

Huang, He, Luo, Li -Shi, Li, Rui, Chen, Jie, and Zhang, He. Improve the efficiency of the Cartesian tensor based fast multipole method for Coulomb interaction using the traces. United States: N. p., 2018. Web. doi:10.1016/j.jcp.2018.05.028.
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Huang, He, Luo, Li -Shi, Li, Rui, Chen, Jie, & Zhang, He. Improve the efficiency of the Cartesian tensor based fast multipole method for Coulomb interaction using the traces. United States. https://doi.org/10.1016/j.jcp.2018.05.028
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Huang, He, Luo, Li -Shi, Li, Rui, Chen, Jie, and Zhang, He. Thu . "Improve the efficiency of the Cartesian tensor based fast multipole method for Coulomb interaction using the traces". United States. https://doi.org/10.1016/j.jcp.2018.05.028. https://www.osti.gov/servlets/purl/1439411.
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@article{osti_1439411,
title = {Improve the efficiency of the Cartesian tensor based fast multipole method for Coulomb interaction using the traces},
author = {Huang, He and Luo, Li -Shi and Li, Rui and Chen, Jie and Zhang, He},
abstractNote = {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.},
doi = {10.1016/j.jcp.2018.05.028},
journal = {Journal of Computational Physics},
number = C,
volume = 371,
place = {United States},
year = {Thu May 17 00:00:00 EDT 2018},
month = {Thu May 17 00:00:00 EDT 2018}
}
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Journal Article:
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Publisher's Version of Record
https://doi.org/10.1016/j.jcp.2018.05.028
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Cited by: 2 works
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Figures / Tables:

Table 1 Table 1: Sorting of elements in a tensor of rank 3
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Works referenced in this record:

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    Works referencing / citing this record:

    A New Relatively Simple Approach to Multipole Interactions in Either Spherical Harmonics or Cartesians, Suitable for Implementation into Ewald Sums
    journal, December 2019

    • Burnham, Christian J.; English, Niall J.
    • International Journal of Molecular Sciences, Vol. 21, Issue 1
    • DOI: 10.3390/ijms21010277

    A New Relatively Simple Approach to Multipole Interactions in Either Spherical Harmonics or Cartesians, Suitable for Implementation into Ewald Sums
    journal, December 2019

    • Burnham, Christian J.; English, Niall J.
    • International Journal of Molecular Sciences, Vol. 21, Issue 1
    • DOI: 10.3390/ijms21010277
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      Figures / Tables found in this record:

      Table 1 (p. 8)table
      Table 2 (p. 10)table
      Figure 1 (p. 11)figure
      Table 3 (p. 11)table
      Table 4 (p. 12)table
      Table 5 (p. 12)table
      Figure 2 (p. 13)figure
      Table 6 (p. 14)table
      Table 7 (p. 14)table
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