The paper deals with the mixed boundary-value problem for the Poisson equation. Random walks inside the domain are constructed and unbiased estimators for the solution of the boundary-value problem are defined on their trajectories. The finiteness of the estimator variance is proved. The possibility of practical realization of the estimators by the Monte Carlo method is studied.
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S. M. Ermakov and A. S. Sipin, Monte Carlo Method and Parametric Separability of Algorithms [in Russian], St.Petersburg Univ. (2014).
K. K. Sabelfeld, Monte Carlo Methods in Boundary-Value Problems, Springer-Verlag, Berlin, Heidelberg (1991).
S. M. Ermakov, V. V. Nekrutkin, and A. S. Sipin, Random Processes for Classical Equations of Mathematical Physics, Kluwer Academic Publishers, Dordrecht (1989).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 442, 2015, pp. 133–142.
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Sipin, A.S. On Stochastic Algorithms for Solving Boundary-Value Problems for the Laplace Operator. J Math Sci 225, 812–817 (2017). https://doi.org/10.1007/s10958-017-3497-y
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DOI: https://doi.org/10.1007/s10958-017-3497-y