HODLR3D: hierarchical matrices for N-body problems in three dimensions

This article introduces HODLR3D, a class of hierarchical matrices arising out of N -body problems in three dimensions. HODLR3D relies on the fact that certain off-diagonal matrix sub-blocks arising out of the N -body problems in three dimensions are numerically low rank. For the Laplace kernel in 3D...

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Vydané v:	Numerical algorithms Ročník 97; číslo 4; s. 1635 - 1672
Hlavní autori:	A, Kandappan V., Gujjula, Vaishnavi, Ambikasaran, Sivaram
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.12.2024 Springer Nature B.V
Predmet:	Algebra Algorithms Complexity Computer Science Distributed memory Many body problem Numeric Computing Numerical Analysis Original Paper Representations Theory of Computation 68Q25 Hierarchical matrices 68U20 45B05 body problems HODLR 68U05 68R10
ISSN:	1017-1398, 1572-9265
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Abstract	This article introduces HODLR3D, a class of hierarchical matrices arising out of N -body problems in three dimensions. HODLR3D relies on the fact that certain off-diagonal matrix sub-blocks arising out of the N -body problems in three dimensions are numerically low rank. For the Laplace kernel in 3D, which is widely encountered, we prove that all the off-diagonal matrix sub-blocks are rank deficient in finite precision. We also obtain the growth of the rank as a function of the size of these matrix sub-blocks. For other kernels in three dimensions, we numerically illustrate a similar scaling in rank for the different off-diagonal sub-blocks. We leverage this hierarchical low-rank structure to construct HODLR3D representation, with which we accelerate matrix-vector products. The storage and computational complexity of the HODLR3D matrix-vector product scales almost linearly with system size. We demonstrate the computational performance of HODLR3D representation through various numerical experiments. Further, we explore the performance of the HODLR3D representation on distributed memory systems. HODLR3D, described in this article, is based on a weak admissibility condition. Among the hierarchical matrices with different weak admissibility conditions in 3D, only in HODLR3D did the rank of the admissible off-diagonal blocks not scale with any power of the system size. Thus, the storage and the computational complexity of the HODLR3D matrix-vector product remain tractable for N -body problems with large system sizes.
AbstractList	This article introduces HODLR3D, a class of hierarchical matrices arising out of N -body problems in three dimensions. HODLR3D relies on the fact that certain off-diagonal matrix sub-blocks arising out of the N -body problems in three dimensions are numerically low rank. For the Laplace kernel in 3D, which is widely encountered, we prove that all the off-diagonal matrix sub-blocks are rank deficient in finite precision. We also obtain the growth of the rank as a function of the size of these matrix sub-blocks. For other kernels in three dimensions, we numerically illustrate a similar scaling in rank for the different off-diagonal sub-blocks. We leverage this hierarchical low-rank structure to construct HODLR3D representation, with which we accelerate matrix-vector products. The storage and computational complexity of the HODLR3D matrix-vector product scales almost linearly with system size. We demonstrate the computational performance of HODLR3D representation through various numerical experiments. Further, we explore the performance of the HODLR3D representation on distributed memory systems. HODLR3D, described in this article, is based on a weak admissibility condition. Among the hierarchical matrices with different weak admissibility conditions in 3D, only in HODLR3D did the rank of the admissible off-diagonal blocks not scale with any power of the system size. Thus, the storage and the computational complexity of the HODLR3D matrix-vector product remain tractable for N -body problems with large system sizes. This article introduces HODLR3D, a class of hierarchical matrices arising out of N-body problems in three dimensions. HODLR3D relies on the fact that certain off-diagonal matrix sub-blocks arising out of the N-body problems in three dimensions are numerically low rank. For the Laplace kernel in 3D, which is widely encountered, we prove that all the off-diagonal matrix sub-blocks are rank deficient in finite precision. We also obtain the growth of the rank as a function of the size of these matrix sub-blocks. For other kernels in three dimensions, we numerically illustrate a similar scaling in rank for the different off-diagonal sub-blocks. We leverage this hierarchical low-rank structure to construct HODLR3D representation, with which we accelerate matrix-vector products. The storage and computational complexity of the HODLR3D matrix-vector product scales almost linearly with system size. We demonstrate the computational performance of HODLR3D representation through various numerical experiments. Further, we explore the performance of the HODLR3D representation on distributed memory systems. HODLR3D, described in this article, is based on a weak admissibility condition. Among the hierarchical matrices with different weak admissibility conditions in 3D, only in HODLR3D did the rank of the admissible off-diagonal blocks not scale with any power of the system size. Thus, the storage and the computational complexity of the HODLR3D matrix-vector product remain tractable for N-body problems with large system sizes.
Author	Gujjula, Vaishnavi A, Kandappan V. Ambikasaran, Sivaram
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References_xml	– reference: CoulierPDarveEEfficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole methodComput. Methods Appl. Mech. Eng.2016308286309352227910.1016/j.cma.2016.05.029 – reference: GreengardLRokhlinVA new version of the fast multipole method for the Laplace equation in three dimensionsActa Numer19976229269148925710.1017/S0962492900002725 – reference: Khan, R., Kandappan, V., Ambikasaran, S.: Numerical rank of singular kernel functions. arXiv:2209.05819 (2022) – reference: BeatsonRGreengardLA short course on fast multipole methodsWavelets, multilevel methods and elliptic PDEs.199711371600672 – reference: Gray, A., Moore, A.: N-body’ problems in statistical learning. Advances in neural information processing systems 13 (2000) – reference: VandebrilRVan BarelMMastronardiNA note on the representation and definition of semiseparable matricesNumerical Linear Algebra with Applications.2005128839858217268110.1002/nla.455 – reference: TyrtyshnikovEIncomplete cross approximation in the mosaic-skeleton methodComputing2000644367380178346810.1007/s006070070031 – reference: Barnes, J., Hut, P.: A hierarchical O (N log N) force-calculation algorithm. Nature. 324(6096), 446–449 (1986) – reference: Yokota, R., Ibeid, H., Keyes, D.: Fast multipole method as a matrix-free hierarchical low-rank approximation. In: International Workshop on Eigenvalue Problems: Algorithms, Software and Applications in Petascale Computing, pp. 267–286 (2015). Springer – reference: GrasedyckLHackbuschWConstruction and arithmetics of H-matricesComputing2003704295334201141910.1007/s00607-003-0019-1 – reference: Ambikasaran, S., Darve, E.: The inverse fast multipole method. arXiv:1407.1572 (2014) – reference: Hackbusch, W.: A sparse matrix arithmetic based on H-matrices. part i:Introduction to H-matrices. Computing. 62(2), 89–108 (1999) – reference: AmbikasaranSDarveEAn O(nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\cal{O} (n \log n)$$\end{document}-fast direct solver for partial hierarchically semi-separable matricesJ. Sci. Comput.2013573477501312355410.1007/s10915-013-9714-z – reference: Amestoy, P., Buttari, A., l’Excellent, J.-Y., Mary, T.: On the complexity of the block low-rank multifrontal factorization. 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SubjectTerms	Algebra Algorithms Complexity Computer Science Distributed memory Many body problem Numeric Computing Numerical Analysis Original Paper Representations Theory of Computation
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