Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2753010.22363/2658-4670-2021-29-3-260-270Research ArticleShifted Sobol points and multigrid Monte Carlo simulationBelovAleksandr A.<p>Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics of Peoples’ Friendship University of Russia (RUDN University); Researcher of Faculty of Physics, M.V. Lomonosov Moscow State University</p>aa.belov@physics.msu.ruhttps://orcid.org/0000-0002-0918-9263TintulMaxim A.<p>Master’s degree student of Faculty of Physics</p>maksim.tintul@mail.ruhttps://orcid.org/0000-0002-5466-1221M.V. Lomonosov Moscow State UniversityPeoples’ Friendship University of Russia (RUDN University)3009202129326027030092021Copyright © 2021, Belov A.A., Tintul M.A.2021<p style="text-align: justify;">Multidimensional integrals arise in many problems of physics. For example, moments of the distribution function in the problems of transport of various particles (photons, neutrons, etc.) are 6-dimensional integrals. When calculating the coefficients of electrical conductivity and thermal conductivity, scattering integrals arise, the dimension of which is equal to 12. There are also problems with a significantly large number of variables. The Monte Carlo method is the most effective method for calculating integrals of such a high multiplicity. However, the efficiency of this method strongly depends on the choice of a sequence that simulates a set of random numbers. A large number of pseudo-random number generators are described in the literature. Their quality is checked using a battery of formal tests. However, the simplest visual analysis shows that passing such tests does not guarantee good uniformity of these sequences. The magic Sobol points are the most effective for calculating multidimensional integrals. In this paper, an improvement of these sequences is proposed: the shifted magic Sobol points that provide better uniformity of points distribution in a multidimensional cube. This significantly increases the cubature accuracy. A significant difficulty of the Monte Carlo method is a posteriori confirmation of the actual accuracy. In this paper, we propose a multigrid algorithm that allows one to find the grid value of the integral simultaneously with a statistically reliable accuracy estimate. Previously, such estimates were unknown. Calculations of representative test integrals with a high actual dimension up to 16 are carried out. The multidimensional Weierstrass function, which has no derivative at any point, is chosen as the integrand function. These calculations convincingly show the advantages of the proposed methods.</p>multidimensional integralMonte Carlo methodSobol pointsmultigrid calculationa posteriori error estimatesмногомерный интегралметод Монте-Карлоточки Соболямногосеточный расчетапостериорные оценки точности[I. M. Sobol, Numerical Monte-Carlo methods [Chislennyye metody MonteKarlo]. Moscow: Nauka, 1973, In Russian.][D. E. Knuth, The art of computer programming, 3rd ed. Reading, Massachusetts: Addison-Wesley, 1997, vol. 2.][G. S. Fishman, Monte Carlo: concepts, algorithms and applications. Berlin: Springer, 1996. DOI: 10.1007/978-1-4757-2553-7.][M. Matsumoto and T. Nishimura, “Mersenne twister: a 623dimensionally equidistributed uniform pseudo-random number generator,” ACM Transactions on Modeling and Computer Simulation (TOMACS), vol. 8, no. 1, pp. 3-30, 1998. DOI: 10.1145/272991.272995.][T. Nishimura, “Tables of 64-bit Mersenne twisters,” ACM Transactions on Modeling and Computer Simulation, vol. 10, no. 4, pp. 348-357, 2000. DOI: 10.1145/369534.369540.][“Mersenne Twister Home Page.” (2021), [Online]. Available: http:// www.math.sci.hiroshima-u.ac.jp/m-mat/MT/emt.html.][S. K. Park and K. W. Miller, “Random number generators: good ones are hard to find,” Communications of the ACM, vol. 31, no. 10, pp. 1192-1201, 1998. DOI: 10.1145/63039.63042.][M. Mascagni and A. Srinivasan, “Parameterizing parallel multiplicative Lagged-Fibonacci generators,” Parallel Computing, vol. 30, pp. 899-916, 2004. DOI: 10.1016/j.parco.2004.06.001.][P. L’Ecuyer, “Good parameter sets for combined multiple recursive random number generators,” Operations Research, vol. 47, no. 1, pp. 159-164, 1999. DOI: 10.1287/opre.47.1.159.][J. K. Salmon, M. A. Moraes, R. O. Dror, and D. E. Shaw, “Parallel random numbers: as easy as 1, 2, 3,” 2011 International Conference for High Performance Computing, Networking, Storage and Analysis (SC), pp. 1-12, 2011. DOI: 10.1145/2063384.2063405.][G. Marsaglia and W. W. Tsang, “The ziggurat method for generating random variables,” Journal of Statistical Software, vol. 5, pp. 1-7, 2000. DOI: 10.18637/jss.v005.i08.][G. Marsaglia and A. Zaman, “A new class of random number generators,” Annals of Applied Probability, vol. 1, no. 3, pp. 462-480, 1991. DOI: 10.1214/aoap/1177005878.][B. A. Wichmann and I. D. Hill, “An efficient and portable pseudorandom number generator,” Applied Statistics, vol. 31, no. 2, pp. 188-190, 1982. DOI: 10.2307/2347988.][E. A. Tsvetkov, “Empirical tests for statistical properties of some pseudorandom number generators,” Mathematical Models and Computer Simulations, vol. 3, pp. 697-705, 2011. DOI: 10.1134/S207004821106010X.][“The Marsaglia Random Number CDROM including the Diehard Battery of Tests of Randomness.” (2021), [Online]. Available: http:// ftpmirror.your.org/pub/misc/diehard/.][P. L’Ecuyer and R. Simard, “TestU01: A C library for empirical testing of random number generators,” ACM Transactions on Mathematical Software (TOMS), vol. 33, no. 4, pp. 1-40, 2007. DOI: 10.1145/1268776.1268777.][L. E. Bassham, A. L. Rukhin, J. Soto, J. R. Nechvatal, M. E. Smid, S. D. Leigh, M. Levenson, M. Vangel, N. A. Heckert, and D. L. Banks, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” National Institute of Standards and Technology, NIST Special Publication, Gaithersburg, MD, Tech. Rep., 2010.][A. A. Belov, N. N. Kalitkin, and M. A. Tintul, “Visual verification of pseudo-random number generators [Vizual’naya verifikatsiya generatorov psevdosluchaynykh chisel],” Keldysh IAM Preprints, Moscow, Tech. Rep. 137, 2019, In Russian. DOI: 10.20948/prepr-2019-137.][A. A. Belov, N. N. Kalitkin, and M. A. Tintul, “Unreliability of pseudorandom number generators,” Computational Mathematics and Mathematical Physics, vol. 60, no. 11, pp. 1747-1753, 2020. DOI: 10.1134/S0965542520110044.][I. M. Sobol, “Uniformly distributed sequences with additional uniformity properties,” USSR Computational Mathematics and Mathematical Physics, vol. 16, no. 5, pp. 236-242, 1976. DOI: 10.1016/0041-5553(76)90154-3.]