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Title: An O( N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

Abstract

Large classes of materials systems in physics and engineering are governed by magnetic and electrostatic interactions. Continuum or mesoscale descriptions of such systems can be cast in terms of integral equations, whose direct computational evaluation requires O( N 2) operations, where N is the number of unknowns. Such a scaling, which arises from the many-body nature of the relevant Green's function, has precluded wide-spread adoption of integral methods for solution of large-scale scientific and engineering problems. In this work, a parallel computational approach is presented that relies on using scalable open source libraries and utilizes a kernel-independent Fast Multipole Method (FMM) to evaluate the integrals in O( N) operations, with O( N) memory cost, thereby substantially improving the scalability and efficiency of computational integral methods. We demonstrate the accuracy, efficiency, and scalability of our approach in the context of two examples. In the first, we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space. In the second, we solve an electrostatic problem involving polarizable dielectric bodies in an unbounded dielectric medium. Lastly, the results from these test cases show that our proposed parallel approach, which is built on a kernel-independent FMM, can enable highly efficient and accuratemore » simulations and allow for considerable flexibility in a broad range of applications.« less

Authors:
 [1];  [2]; ORCiD logo [3];  [4];  [5];  [6];  [7]; ORCiD logo [8]
  1. Argonne National Lab. (ANL),Lemont, IL (United States)
  2. Univ. of Chicago, Chicago, IL (United States)
  3. Argonne National Lab. (ANL), Lemont, IL (United States)
  4. Stanford Univ., Stanford, CA (United States)
  5. Argonne National Lab. (ANL), Lemont, IL (United States); Univ. of Chicago, Chicago, IL (United States); KCG Holdings, Inc. (United States)
  6. Univ. of Chicago, Chicago, IL (United States); Univ. Nacional de Colombia-Medellin, Medellin (Columbia)
  7. Argonne National Lab. (ANL),Lemont, IL (United States); Univ. of Chicago, Chicago, IL (United States)
  8. Argonne National Lab. (ANL),Lemont, IL (United States); Northwestern-Argonne Institute for Science and Engineering, Evanston, IL (United States)
Publication Date:
Research Org.:
Argonne National Lab. (ANL), Argonne, IL (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences and Engineering Division; USDOE
OSTI Identifier:
1339573
Grant/Contract Number:
AC02-06CH11357
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 145; Journal Issue: 6; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats

Jiang, Xikai, Li, Jiyuan, Zhao, Xujun, Qin, Jian, Karpeev, Dmitry, Hernandez-Ortiz, Juan, de Pablo, Juan J., and Heinonen, Olle. An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles. United States: N. p., 2016. Web. doi:10.1063/1.4960436.
Jiang, Xikai, Li, Jiyuan, Zhao, Xujun, Qin, Jian, Karpeev, Dmitry, Hernandez-Ortiz, Juan, de Pablo, Juan J., & Heinonen, Olle. An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles. United States. doi:10.1063/1.4960436.
Jiang, Xikai, Li, Jiyuan, Zhao, Xujun, Qin, Jian, Karpeev, Dmitry, Hernandez-Ortiz, Juan, de Pablo, Juan J., and Heinonen, Olle. 2016. "An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles". United States. doi:10.1063/1.4960436. https://www.osti.gov/servlets/purl/1339573.
@article{osti_1339573,
title = {An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles},
author = {Jiang, Xikai and Li, Jiyuan and Zhao, Xujun and Qin, Jian and Karpeev, Dmitry and Hernandez-Ortiz, Juan and de Pablo, Juan J. and Heinonen, Olle},
abstractNote = {Large classes of materials systems in physics and engineering are governed by magnetic and electrostatic interactions. Continuum or mesoscale descriptions of such systems can be cast in terms of integral equations, whose direct computational evaluation requires O(N2) operations, where N is the number of unknowns. Such a scaling, which arises from the many-body nature of the relevant Green's function, has precluded wide-spread adoption of integral methods for solution of large-scale scientific and engineering problems. In this work, a parallel computational approach is presented that relies on using scalable open source libraries and utilizes a kernel-independent Fast Multipole Method (FMM) to evaluate the integrals in O(N) operations, with O(N) memory cost, thereby substantially improving the scalability and efficiency of computational integral methods. We demonstrate the accuracy, efficiency, and scalability of our approach in the context of two examples. In the first, we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space. In the second, we solve an electrostatic problem involving polarizable dielectric bodies in an unbounded dielectric medium. Lastly, the results from these test cases show that our proposed parallel approach, which is built on a kernel-independent FMM, can enable highly efficient and accurate simulations and allow for considerable flexibility in a broad range of applications.},
doi = {10.1063/1.4960436},
journal = {Journal of Chemical Physics},
number = 6,
volume = 145,
place = {United States},
year = 2016,
month = 8
}

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  • We present in this paper a new kernel-independent fast multipole method (FMM), named as FKI-FMM, for pairwise particle interactions with translation-invariant kernel functions. FKI-FMM creates, using numerical techniques, sufficiently accurate and compressive representations of a given kernel function over multi-scale interaction regions in the form of a truncated Fourier series. It provides also economic operators for the multipole-to-multipole, multipole-to-local, and local-to-local translations that are typical and essential in the FMM algorithms. The multipole-to-local translation operator, in particular, is readily diagonal and does not dominate in arithmetic operations. FKI-FMM provides an alternative and competitive option, among other kernel-independent FMM algorithms, formore » an efficient application of the FMM, especially for applications where the kernel function consists of multi-physics and multi-scale components as those arising in recent studies of biological systems. We present the complexity analysis and demonstrate with experimental results the FKI-FMM performance in accuracy and efficiency.« less
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  • No abstract prepared.
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