Computational Mathematics and Mathematical Physics

, Volume 58, Issue 11, pp 1828–1837 | Cite as

Monte Carlo Methods for Estimating the Probability Distributions of Criticality Parameters of Particle Transport in a Randomly Perturbed Medium

  • G. A. Mikhailov
  • G. Z. LotovaEmail author


Parallelizable Monte Carlo algorithms are developed for estimating the probability moments of criticality parameters for transport of particles with multiplication in a random medium. For this purpose, new iterative estimates of the multiplication factor and recurrence representations of statistical estimates of moments are constructed by applying the double randomization method and the randomized projection method. The practical efficiency of the proposed approaches is confirmed by test results obtained using special randomized homogenization with an improved diffusion approximation for a multilayered ball.


Monte Carlo method statistical modeling transport theory effective particle multiplication factor 



This work was supported in part by the Russian Foundation for Basic Research, project nos. 18-01-00599, 18-01-00356, 17-01-00823, 16-01-00530, and 16-01-00145.


  1. 1.
    B. Davison, Neutron Transport Theory (Clarendon, Oxford, 1957).zbMATHGoogle Scholar
  2. 2.
    G. Z. Lotova and G. A. Mikhailov, “Moments of the critical parameters of the transport of particles in a random medium,” Comput. Math. Math. Phys. 48 (12), 2254–2265 (2008).MathSciNetCrossRefGoogle Scholar
  3. 3.
    G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, et al., The Monte Carlo Methods in Atmospheric Optics (Nauka, Novosibirsk, 1976; Springer-Verlag, Berlin, 1980).Google Scholar
  4. 4.
    Yu. I. Ershov and S. B. Shikhov, Mathematical Fundamentals of Transfer Theory (Energoatomizdat, Moscow, 1985), Vol. 1 [in Russian].Google Scholar
  5. 5.
    V. S. Vladimirov, “Monte Carlo methods as applied to the calculation of the lowest eigenvalue and the associated eigenfunction of a linear integral equation,” Probab. Theory Appl. 1 (1), 101–116 (1956).CrossRefGoogle Scholar
  6. 6.
    L. V. Kantorovich and G. P. Akilov, Functional Analysis (Pergamon, Oxford, 1982; Nauka, Moscow, 1984).Google Scholar
  7. 7.
    G. A. Mikhailov, “Efficient Monte Carlo algorithms for evaluating the correlation characteristics of conditional mathematical expectations,” USSR Comput. Math. Math. Phys. 17 (1), 244–247 (1977).CrossRefGoogle Scholar
  8. 8.
    S. M. Ermakov and G. A. Mikhailov, Statistical Modeling (Nauka, Moscow, 1982) [in Russian].Google Scholar
  9. 9.
    I. M. Sobol, Numerical Monte Carlo Methods (Nauka, Moscow, 1973) [in Russian].zbMATHGoogle Scholar
  10. 10.
    A. Yu. Ambos and G. A. Mikhailov, “Effective averaging of stochastic radiative models based on Monte Carlo simulation,” Comput. Math. Math. Phys. 56 (5), 881–893 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    N. Ya. Vilenkin, Combinatorics (Nauka, Moscow, 1969; Academic, New York, 1971).Google Scholar
  12. 12.
    S. A. Brednikhin, I. N. Medvedev, and G. A. Mikhailov, “Estimation of the criticality parameters of branching processes by the Monte Carlo method,” Comput. Math. Math. Phys. 50 (2), 345–356 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    A. Yu. Ambos, “Numerical models of mosaic homogeneous isotropic random fields and problems of radiative transfer,” Numer. Anal. Appl. 9 (1), 12–23 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    E. S. Wentzel, Probability Theory (Nauka, Moscow, 1969) [in Russian].Google Scholar
  15. 15.
    G. I. Marchuk, Methods of Calculating Nuclear Reactors (Atomizdat, Moscow, 1961) [in Russian].Google Scholar
  16. 16.
    G. Z. Lotova and G. A. Mikhailov, “Estimates of the fluctuations of criticality parameters in the particle transport process in a random medium,” Russ. J. Numer. Anal. Math. Model. 19 (2), 173–183 (2004).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    V. G. Zolotukhin and L. V. Maiorov, Estimation of Critical Reactor Parameters by Monte Carlo Methods (Energoatomizdat, Moscow, 1984) [in Russian].Google Scholar
  18. 18.
    G. Z. Lotova and G. A. Mikhailov, “New Monte Carlo Methods for the solution of nonstationary problems in the radiation transfer theory,” Russ. J. Numer. Anal. Math. Model. 15 (3–4), 285–295 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    G. Z. Lotova and G. A. Mikhailov, “New Monte Carlo methods for estimating time dependences in radiative transfer process,” Comput. Math. Math. Phys. 42 (4), 544–554 (2002).MathSciNetzbMATHGoogle Scholar
  20. 20.
    Yu. A. Romanov, “Exact solutions of one-velocity equation and their application in the computation of diffusion problems (improved diffusion method),” in Investigation of Critical Parameters of Reactor Systems (Gosatomizdat, Moscow, 1960), pp. 3–26 [in Russian].Google Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Computational Mathematics and Mathematical Geophysics, Siberian Branch, Russian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk State UniversityNovosibirskRussia

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