# Shifted Sobol points and multigrid Monte Carlo simulation

**Authors:**Belov A.A.^{1}^{,2}, Tintul M.A.^{1}-
**Affiliations:**- M.V. Lomonosov Moscow State University
- Peoples’ Friendship University of Russia (RUDN University)

**Issue:**Vol 29, No 3 (2021)**Pages:**260-270**Section:**Articles**URL:**https://journals.rudn.ru/miph/article/view/27530**DOI:**https://doi.org/10.22363/2658-4670-2021-29-3-260-270

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## Full Text

## Abstract

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.

## Full Text

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. To determine the coefficients of thermal conductivity or electrical conductivity of a medium, it is necessary to calculate the collision integrals; they include components of the velocity vectors before the moment of collision and after the moment of collision. The total number of variables in such an integral is twelve. Problems also arise with a significantly larger number of variables. In the simplest formulation, the calculation of the integral in the unit

## About the authors

### Aleksandr A. Belov

M.V. Lomonosov Moscow State University; Peoples’ Friendship University of Russia (RUDN University)
**Author for correspondence.**

Email: aa.belov@physics.msu.ru

ORCID iD: 0000-0002-0918-9263

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

1, bld. 2, Leninskie Gory, Moscow, 119991, Russian Federation; 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation### Maxim A. Tintul

M.V. Lomonosov Moscow State University
Email: maksim.tintul@mail.ru

ORCID iD: 0000-0002-5466-1221

Master’s degree student of Faculty of Physics

1, bld. 2, Leninskie Gory, Moscow, 119991, Russian Federation## References

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