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Keywords = von-Neumann–Ulam scheme

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19 pages, 663 KB  
Article
Monte Carlo Algorithms for the Extracting of Electrical Capacitance
by Andrei Kuznetsov and Alexander Sipin
Mathematics 2021, 9(22), 2922; https://doi.org/10.3390/math9222922 - 17 Nov 2021
Cited by 5 | Viewed by 2515
Abstract
We present new Monte Carlo algorithms for extracting mutual capacitances for a system of conductors embedded in inhomogeneous isotropic dielectrics. We represent capacitances as functionals of the solution of the external Dirichlet problem for the Laplace equation. Unbiased and low-biased estimators for the [...] Read more.
We present new Monte Carlo algorithms for extracting mutual capacitances for a system of conductors embedded in inhomogeneous isotropic dielectrics. We represent capacitances as functionals of the solution of the external Dirichlet problem for the Laplace equation. Unbiased and low-biased estimators for the capacitances are constructed on the trajectories of the Random Walk on Spheres or the Random Walk on Hemispheres. The calculation results show that the accuracy of these new algorithms does not exceed the statistical error of estimators, which is easily determined in the course of calculations. The algorithms are based on mean value formulas for harmonic functions in different domains and do not involve a transition to a difference problem. Hence, they do not need a lot of storage space. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications II)
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10 pages, 233 KB  
Article
Monte Carlo Algorithms for the Parabolic Cauchy Problem
by Alexander Sipin
Mathematics 2019, 7(2), 177; https://doi.org/10.3390/math7020177 - 15 Feb 2019
Viewed by 2207
Abstract
New Monte Carlo algorithms for solving the Cauchy problem for the second order parabolic equation with smooth coefficients are considered. Unbiased estimators for the solutions of this problem are constructed. Full article
(This article belongs to the Special Issue Stochastic Processes: Theory and Applications)
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