Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia2753010.22363/2658-4670-2021-29-3-260-270Research 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<p>Introduction Integrals of multivariate functions occur in many areas of physics. Here are some examples. The transfer of neutrons, photons and other particles in Belov A.A., Tintul M.A., 2021 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ the medium is described by the equation for the distribution function; this function depends on three coordinates of the medium and three components of the particle velocity vector, that is, the number of variables is six. 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