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The Journal of Supercomputing

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01-04-2021

Multidimensional adaptative and deterministic integration in CUDA and OpenMP

Authors: R. Quintero-Monsebaiz, A. Meneses-Viveros, F. Carranza, C. G. Cortés, A. González-Zamudio, A. Vela

Published in: The Journal of Supercomputing | Issue 10/2021

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Abstract

Parallelization schemes on many-core architectures, in this case, CUDA and OpenMP, are used to accelerate and improve the accuracy of adaptive multidimensional integration algorithms. The one-dimensional Gauss–Kronrod adaptive method is generalized to 3, 4, 5 and 6 dimensions. The implementation of the traditional tensor product construction of the grid and weights for multidimensional integration is revisited and reformulated taking advantages of the multi and many-core architectures. Tests performed in a set of benchmark functions show that the algorithm is numerically accurate, with accelerations as high as 800X in CUDA and 300X in the OpenMP implementation both compared to a sequential multidimensional integration algorithm.

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Doucet A, de Freitas N, Gordon N (2001) An introduction to sequential Monte Carlo methods. In: Doucet A, de Freitas N, Gordon N (eds) Sequential Monte Carlo methods in practice. statistics for engineering and information science. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3437-9_1CrossRef

Abramowitz M (1974) Handbook of mathematical functions, with formulas, graphs, and mathematical tables. Dover Publications Inc, New YorkMATH

Ammar GS, Calvetti D, Reichel L (1999) Computation of gauss-kronrod quadrature rules with non-positive weights. Electron Trans Num Anal 9:26–38MathSciNetMATH

Arfken G (1985) Mathematical methods for physicists, 3rd edn. Academic Press Inc, San DiegoMATH

Arumugam K, Godunov A, Ranjan D, Terzic B, Zubair M (2013) An efficient deterministic parallel algorithm for adaptive multidimensional numerical integration on gpus. In: Parallel processing (ICPP), 2013 42nd International Conference on, pp. 486–491. IEEE

Atisattapong W, Marupanthorn P (2017) A 1/t algorithm with the density of two states for estimating multidimensional integrals. Comput Phys Commun 220:122–128MathSciNetCrossRef

Becke AD (1988) A multicenter numerical integration scheme for polyatomic molecules. J Chem Phys 88(4):2547–2553CrossRef

Berntsen J, Espelid TO, Genz A (1991) An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans Math Softw (TOMS) 17(4):437–451MathSciNetCrossRef

Blinder S, House J (eds) (2019) Slater and Gaussian basis functions and computation of molecular integrals developments in physical and theoretical chemistry. Elsevier, Amsterdam

10.

Boys SF, Egerton AC (1950) Electronic wave functions - i. a general method of calculation for the stationary states of any molecular system. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 200(1063)

11.

Brass S, Kilian W, Reuter J (2019) Parallel adaptive Monte Carlo integration with the event generator WHIZARD. Eur Phys J C 79(4):1–25CrossRef

12.

Bungartz HJ, Dirnstorfer S (2003) Multivariate quadrature on adaptive sparse grids. Computing 71:89–114MathSciNetCrossRef

13.

Cheng VB, Suzukawa HH, Wolfsberg M (1973) Investigations of a nonrandom numerical method for multidimensional integration. J Chem Phys 59(8):3992–3999MathSciNetCrossRef

14.

Dahlquist G, Björck Å (2008) Numerical methods in scientific computing, vol 1. Soc Ind Appl Math

15.

Dixit AK, Sherrerd JJ et al (1990) Optimization in economic theory. Oxford University Press on Demand, Oxford

16.

Fuller SH, Millett LI (2011) Computing performance: game over or next level? Computer 44(1):31–38CrossRef

17.

G S (1975) vol. 23, fourth edn. American Mathematical Society, Rhode Island

18.

Genz A (1972) An adaptive multidimensional quadrature procedure. Comput Phys Commun 4(1):11–15CrossRef

19.

Genz A (1986) Fully symmetric interpolatory rules for multiple integrals. SIAM 23:1273–1283MathSciNetMATH

20.

Gibbs D (1916) A course in interpolation and numerical integration for the mathematical laboratory. J Inst Actuar 50(1):67–68MathSciNetCrossRef

21.

Grammer G Jr, Kinoshita T (1974) The two-photon process for particle production in colliding beam experiments. Nucl Phys B 80(3):461–490CrossRef

22.

Hahn T (2005) Cuba-a library for multidimensional numerical integration. Comput Phys Commun 176:78–95. https://doi.org/10.1016/j.cpc.2005.01.010MathSciNetCrossRefMATH

23.

Intriligator MD (2002) Mathematical optimization and economic theory. SIAM, PhiladelphiaCrossRef

24.

J. McNamee FS (1967) Construction of fully symmetric numerical integration formulas of fully symmetric numerical integration formulas. Numerische Mathematik 10:327–344MathSciNetCrossRef

25.

Keister BD (1996) Multidimensional quadrature algorithms. Comput Phys 10(2):119–128MathSciNetCrossRef

26.

Kronrod AS (1964) Nodes and weights for quadrature formulas. Nauka pp. 597–598

27.

Lepage GP (1978) A new algorithm for adaptive multidimensional integration. J Comput Phys 27(2):192–203MathSciNetCrossRef

28.

Lytvyn O, Nechuiviter O (2014) Approximate calculation of triple integrals of rapidly oscillating functions with the use of lagrange polynomial interflation. Cyber Syst Anal 50(3):410–418MathSciNetCrossRef

29.

Mattson TG, Sanders B, Massingill B (2004) Patterns for parallel programming. Pearson Education, New YorkMATH

30.

Nechuiviter O, Iarmosh O, Kovalchuk K (2021) Numerical calculation of multidimensional integrals depended on input information about the function in mathematical modelling of technical and economic processes. In: IOP Conference Series: Materials Science and Engineering, vol. 1031, p. 012059. IOP Publishing

31.

Notaris SE (2016) Gauss-kronrod quadrature formulae a survey if fifty years of research. Electron Trans Num Anal 45(1):371–404MATH

32.

Notaris SE (2016) Gauss-kronrod quadrature formulae a survey of fifty years of research. Electron Trans Num Anal 45:371–4004MathSciNetMATH

33.

Papageorgiou A, Traub JF (1997) Faster evaluation of multidimensional integrals. Comput Phys 11(6):574–579CrossRef

34.

Patterson T (1968) The optimum addition of points to quadrature formulae. Math Comput 22:847–856MathSciNetCrossRef

35.

Piessens R, Branders M (1974) A note on the optimal addition of abscissas to quadrature formulas of gauss and lobatto type. Math Comput 28:344–347MathSciNetCrossRef

36.

Piessens R, de Doncker-Kapenga E, Überhuber CW, Kahaner DK (1983) Special applications of QUADPACK. Quadpack. Springer series in computational mathematics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61786-7_5CrossRefMATH

37.

Putteneers K, Brosens F (2012) Monte Carlo implementation of density-functional theory. Physical Review B 86(8)CrossRef

38.

S. Ninomiya ST (1996) Toward real-time pricing of complex financial derivatives. Appl Math Fin 3(1):1–20CrossRef

39.

Sasaki T (1978) Multidimensional Monte Carlo integration based on factorized approximation functions. SIAM J Num Anal 15(5):938–952MathSciNetCrossRef

40.

Sellier JM, Dimov I (2014) The many-body Wigner Monte Carlo method for time-dependent ab-initio quantum simulations. J Comput Phys 273:589–597MathSciNetCrossRef

41.

Sergienko I, Lytvyn O (2018) New information operators in mathematical modeling (a review). Cyber Syst Anal 54(1):21–30CrossRef

42.

Huzinaga S (1965) Gaussian type functions for polyatomic systems I. J chem Phys 42(4):1293–1302CrossRef

43.

Smolyak SA (1963) Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math Doklady 4:240–243MATH

44.

Szegö G (1935) Über gewisse orthogonale polynome, die zu einer oszillierenden belegungsfunktion gehören. Mathematische Annalen 110(1):501–513MathSciNetCrossRef

45.

Todorov V, Apostolov S, Dimov I, Fidanova S, Poryazov S, Dimitrov Y (2020) An optimal monte carlo algorithm for a class of multidimensional integrals. In: Communication Papers of The Federal Conference on Computer Science and Information System, vol 23, pp. 17–20. ACSIS

46.

Todorov V, Dimov I (2019) Efficient stochastic approaches for multidimensional integrals in bayesian statistics. In: International Conference on Large-scale Scientific Computing, pp. 454–462. Springer

47.

Todorov V, Dimov I, Ostromsky T, Zlatev Z (2019) Advanced quasi-monte carlo algorithms for multidimensional integrals in air pollution modelling. In: International Conference on Variability of the Sun and Sun-Like Stars: From Asteroseismology to Space Weather, pp. 155–167. Springer

Title: Multidimensional adaptative and deterministic integration in CUDA and OpenMP
Authors: R. Quintero-Monsebaiz
A. Meneses-Viveros
F. Carranza
C. G. Cortés
A. González-Zamudio
A. Vela
Publication date: 01-04-2021
Publisher: Springer US
Published in: The Journal of Supercomputing / Issue 10/2021
Print ISSN: 0920-8542
Electronic ISSN: 1573-0484
DOI: https://doi.org/10.1007/s11227-021-03752-1

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