TY - JOUR
T1 - Flexibly imposing periodicity in kernel independent FMM
T2 - A multipole-to-local operator approach
AU - Yan, Wen
AU - Shelley, Michael
N1 - Funding Information:
WY thanks Dhairya Malhotra for his help in modifying his PVFMM package to implement calculation. MJS acknowledges support from NSF grants DMS-1463962 and DMS-1620331 , and NIH Grant GM104976 . WY and MJS thank Alexander Barnett and Shravan Veerapaneni for inspiring our idea of periodizing KIFMM, and for their critical readings of the manuscript.
Publisher Copyright:
© 2017 Elsevier Inc.
PY - 2018/2/15
Y1 - 2018/2/15
N2 - An important but missing component in the application of the kernel independent fast multipole method (KIFMM) is the capability for flexibly and efficiently imposing singly, doubly, and triply periodic boundary conditions. In most popular packages such periodicities are imposed with the hierarchical repetition of periodic boxes, which may give an incorrect answer due to the conditional convergence of some kernel sums. Here we present an efficient method to properly impose periodic boundary conditions using a near-far splitting scheme. The near-field contribution is directly calculated with the KIFMM method, while the far-field contribution is calculated with a multipole-to-local (M2L) operator which is independent of the source and target point distribution. The M2L operator is constructed with the far-field portion of the kernel function to generate the far-field contribution with the downward equivalent source points in KIFMM. This method guarantees the sum of the near-field & far-field converge pointwise to results satisfying periodicity and compatibility conditions. The computational cost of the far-field calculation observes the same O(N) complexity as FMM and is designed to be small by reusing the data computed by KIFMM for the near-field. The far-field calculations require no additional control parameters, and observes the same theoretical error bound as KIFMM. We present accuracy and timing test results for the Laplace kernel in singly periodic domains and the Stokes velocity kernel in doubly and triply periodic domains.
AB - An important but missing component in the application of the kernel independent fast multipole method (KIFMM) is the capability for flexibly and efficiently imposing singly, doubly, and triply periodic boundary conditions. In most popular packages such periodicities are imposed with the hierarchical repetition of periodic boxes, which may give an incorrect answer due to the conditional convergence of some kernel sums. Here we present an efficient method to properly impose periodic boundary conditions using a near-far splitting scheme. The near-field contribution is directly calculated with the KIFMM method, while the far-field contribution is calculated with a multipole-to-local (M2L) operator which is independent of the source and target point distribution. The M2L operator is constructed with the far-field portion of the kernel function to generate the far-field contribution with the downward equivalent source points in KIFMM. This method guarantees the sum of the near-field & far-field converge pointwise to results satisfying periodicity and compatibility conditions. The computational cost of the far-field calculation observes the same O(N) complexity as FMM and is designed to be small by reusing the data computed by KIFMM for the near-field. The far-field calculations require no additional control parameters, and observes the same theoretical error bound as KIFMM. We present accuracy and timing test results for the Laplace kernel in singly periodic domains and the Stokes velocity kernel in doubly and triply periodic domains.
KW - Ewald summation
KW - Kernel independent fast multipole method
KW - Periodic boundary conditions
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U2 - 10.1016/j.jcp.2017.11.012
DO - 10.1016/j.jcp.2017.11.012
M3 - Article
AN - SCOPUS:85034751580
SN - 0021-9991
VL - 355
SP - 214
EP - 232
JO - Journal of Computational Physics
JF - Journal of Computational Physics
ER -