A weight Monte Carlo estimation of fluctuations in branching processes
-
Vladimir Uchaikin
and
Elena Kozhemiakina
Abstract
It is well known that shortened modeling of particle trajectories with the use multiplicative statistical weights, as a rule, increases the efficiency of the program (in terms of accuracy/time ratio). This trick is often used in non-branching schemes simulating transfer processes without multiplication (for example, the transfer of X-ray radiation), in which it is sufficient to confine ourselves to studying only the average values of the field characteristics. With an increase in energy, however, multiplication processes begin to play a significant role (the production of electron-photon pairs by gamma quanta with energies above 1.022 MeV, etc.), when the resulting trajectory is not just a broken curve in the phase space, but a branched tree. This technique is also applicable to this process, but only if the study of statistical fluctuations and correlations is not the purpose of the calculation. The present review contains the basic concepts of the Monte Carlo method as applied to the theory of particle transport, demonstration of the weighting method in non-branching processes, and ends with a discussion of unbiased estimates of the second moment and covariance of additive functionals.
Funding source: Russian Science Foundation
Award Identifier / Grant number: 23-79-30017
Award Identifier / Grant number: 075-15-2021-581
Funding statement: The financial support from the Russian Science Foundation, grant no. 23-79-30017 (transport process in random media) and the Ministry of Science and Higher Education of the Russian Federation, grant no. 075-15-2021-581 (branching process and detection).
References
[1] S. M. Ermakov, Variance of the simplest Monte Carlo estimators in the sign-changing case, Monte Carlo Methods Appl. 17 (2011), no. 4, 411–417. 10.1515/mcma.2011.017Search in Google Scholar
[2] S. M. Ermakov and A. A. Zhiglyavskii, The Monte Carlo method for estimation of functionals of characteristic measures of linear integral operators, USSR Comput. Math. Math. Phys. 25 (1985), no. 3, 15–23. 10.1016/0041-5553(85)90068-0Search in Google Scholar
[3] P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, New York, 2004. 10.1007/978-0-387-21617-1Search in Google Scholar
[4] J. Guo, The progress of three astrophysics simulation methods: Monte-Carlo, PIC and MHD, J. Phys. Conf. Ser. 2012 (2021), no. 1, Article ID 012136. 10.1088/1742-6596/2012/1/012136Search in Google Scholar
[5] B. C. Kiedrowski, Review of early 21st-century Monte Carlo perturbation and sensitivity techniques for 𝑘-eigenvalue radiation transport calculations, Nuclear Science and Engineering, 185 (2017), no. 3, 426–444. 10.1080/00295639.2017.1283153Search in Google Scholar
[6] A. M. Kolchuzhkin and V. V. Uchaikin, Introduction in the Theory of Particles Penetration through a Matter (in Russian), Atomizdat, Moscow, 1978. Search in Google Scholar
[7] D. P. Kroese, T. Brereton, T. Taimre and Z. I. Botev, Why the Monte Carlo method is so important today, Wiley Interdiscip. Rev. Comput. Stat. 6 (2014), no. 6,386–392. 10.1002/wics.1314Search in Google Scholar
[8] G. I. Marchuk, G. A. Mikhailov, M. A. Nazareliev, R. A. Darbinjan, B. A. Kargin and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics, Springer, Berlin, 2013. Search in Google Scholar
[9] G. A. Mikhailov and K. K. Sabelfeld, Optimization of Weighted Monte Carlo Methods, Springer, Berlin, 1992. 10.1007/978-3-642-75981-9Search in Google Scholar
[10] C. P. Robert and G. Casella, Monte Carlo Statistical Methods, Springer, New York, 2004. 10.1007/978-1-4757-4145-2Search in Google Scholar
[11] K. K. Sabelfeld, Expansion of random boundary excitations for some elliptic PDEs, Monte Carlo Methods Appl. 13 (2007), no. 5–6, 403–451. 10.1515/mcma.2007.022Search in Google Scholar
[12] K. Sabelfeld, I. Shalimova and A. Levykin, Stochastic simulation method for a 2D elasticity problem with random loads, Probab. Engrg. Mech. 24 (2009), no. 1, 2–15. 10.1016/j.probengmech.2007.12.020Search in Google Scholar
[13] K. K. Sabelfeld and N. A. Simonov, Random Walks on Boundary for Solving PDFs, VSP, Utrecht, 1994. 10.1515/9783110942026Search in Google Scholar
[14] G. K. Stuart, S. E. Minkoff and F. Pereira, A two-stage Markov chain Monte Carlo method for seismic inversion and uncertainty quantification, Geophysics, 84 (2019), no. 6, R1003–R1020. 10.1190/geo2018-0893.1Search in Google Scholar
[15] V. V. Uchaikin, Monte Carlo simulation of particle transport with multiplication, USSR Comput. Math. Math. Phys. 16 (1976), no. 3, 209–217. 10.1016/0041-5553(76)90216-0Search in Google Scholar
[16] V. V. Uchaikin, Electron-Photon Cascades, Springer, Heidelberg, in press. Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Likelihood and decoding problems for mixed space hidden Markov model
- A weight Monte Carlo estimation of fluctuations in branching processes
- Choice of a constant in the expression for the error of the Monte Carlo method
- Estimation in shape mixtures of skew-normal linear regression models via ECM coupled with Gibbs sampling
- An improved unbiased particle filter
- Partitions for stratified sampling
- A gradient method for high-dimensional BSDEs
- On bias reduction in parametric estimation in stage structured development models
Articles in the same Issue
- Frontmatter
- Likelihood and decoding problems for mixed space hidden Markov model
- A weight Monte Carlo estimation of fluctuations in branching processes
- Choice of a constant in the expression for the error of the Monte Carlo method
- Estimation in shape mixtures of skew-normal linear regression models via ECM coupled with Gibbs sampling
- An improved unbiased particle filter
- Partitions for stratified sampling
- A gradient method for high-dimensional BSDEs
- On bias reduction in parametric estimation in stage structured development models
