Shifted Sobol points and multigrid Monte Carlo simulation
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 conductivi...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Peoples’ Friendship University of Russia (RUDN University)
2021-09-01
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| Series: | Discrete and Continuous Models and Applied Computational Science |
| Subjects: | |
| Online Access: | http://journals.rudn.ru/miph/article/viewFile/27530/19850 |
| _version_ | 1817977543421067264 |
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| author | Aleksandr A. Belov Maxim A. Tintul |
| author_facet | Aleksandr A. Belov Maxim A. Tintul |
| author_sort | Aleksandr A. Belov |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-13T22:17:49Z |
| format | Article |
| id | doaj.art-e18609568c5e4236b84dc0c21d03d9cb |
| institution | Directory Open Access Journal |
| issn | 2658-4670 2658-7149 |
| language | English |
| last_indexed | 2024-04-13T22:17:49Z |
| publishDate | 2021-09-01 |
| publisher | Peoples’ Friendship University of Russia (RUDN University) |
| record_format | Article |
| series | Discrete and Continuous Models and Applied Computational Science |
| spelling | doaj.art-e18609568c5e4236b84dc0c21d03d9cb2022-12-22T02:27:24ZengPeoples’ Friendship University of Russia (RUDN University)Discrete and Continuous Models and Applied Computational Science2658-46702658-71492021-09-0129326027010.22363/2658-4670-2021-29-3-260-27020495Shifted Sobol points and multigrid Monte Carlo simulationAleksandr A. Belov0Maxim A. Tintul1M.V. Lomonosov Moscow State University; Peoples’ Friendship University of Russia (RUDN University)M.V. Lomonosov Moscow State UniversityMultidimensional 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.http://journals.rudn.ru/miph/article/viewFile/27530/19850multidimensional integralmonte carlo methodsobol pointsmultigrid calculationa posteriori error estimates |
| spellingShingle | Aleksandr A. Belov Maxim A. Tintul Shifted Sobol points and multigrid Monte Carlo simulation Discrete and Continuous Models and Applied Computational Science multidimensional integral monte carlo method sobol points multigrid calculation a posteriori error estimates |
| title | Shifted Sobol points and multigrid Monte Carlo simulation |
| title_full | Shifted Sobol points and multigrid Monte Carlo simulation |
| title_fullStr | Shifted Sobol points and multigrid Monte Carlo simulation |
| title_full_unstemmed | Shifted Sobol points and multigrid Monte Carlo simulation |
| title_short | Shifted Sobol points and multigrid Monte Carlo simulation |
| title_sort | shifted sobol points and multigrid monte carlo simulation |
| topic | multidimensional integral monte carlo method sobol points multigrid calculation a posteriori error estimates |
| url | http://journals.rudn.ru/miph/article/viewFile/27530/19850 |
| work_keys_str_mv | AT aleksandrabelov shiftedsobolpointsandmultigridmontecarlosimulation AT maximatintul shiftedsobolpointsandmultigridmontecarlosimulation |