Improve the efficiency of the Cartesian tensor based fast multipole method for Coulomb interaction using the traces
Abstract
To compute the nonoscillating 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 tensorbased 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 nth 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:

 Old Dominion Univ., Norfolk, VA (United States)
 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) (SC26)
 OSTI Identifier:
 1439411
 Alternate Identifier(s):
 OSTI ID: 1582863
 Report Number(s):
 JLABACP172560; DOE/OR/231774223
Journal ID: ISSN 00219991; R&D Project: 2016LDRD7; PII: S0021999118303280; TRN: US1900604
 Grant/Contract Number:
 AC0506OR23177
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 371; Journal Issue: C; Journal ID: ISSN 00219991
 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.
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. doi:10.1016/j.jcp.2018.05.028.
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. doi:10.1016/j.jcp.2018.05.028. https://www.osti.gov/servlets/purl/1439411.
@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 nonoscillating 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 tensorbased 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 nth 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 = {2018},
month = {5}
}