Abstract
Representations of solutions of boundary value problems for simple domains in the Monte Carlo algorithms are widely distributed [2]. In particular, widespread use is made of such a representation for the ball. It allows one to formally write an integral equation of the second kind for the required function in an arbitrary domain with regular boundary. Moreover, with the involvement of the joining conditions [1], one can picture a possible construction of a random process to “solve” the problem. However, the “walk in spheres” process, which solves the first boundary value problem for the Poisson equation, results in ɛ-biased estimators, and so the introduction of a regularization parameter is required.
The authors investigate in detail the “walk in hemispheres” method, which has been proposed earlier by A. S. Sipin [10] without full justification. The use of the Green’s function for the hemisphere makes it possible to obtain estimators for the first and the third boundary value problems, as well as for the problem with discontinuous derivative; for a broad class of domains, these estimators are shown to be unbiased. The algorithms proposed feature a high degree of parallelism. Results of solving model problems are presented.
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Original Russian Text © S.M. Ermakov, A.S. Sipin, 2009, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2009, No. 3, pp. 9–18.
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Ermakov, S.M., Sipin, A.S. The “walk in hemispheres” process and its applications to solving boundary value problems. Vestnik St.Petersb. Univ.Math. 42, 155–163 (2009). https://doi.org/10.3103/S1063454109030029
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DOI: https://doi.org/10.3103/S1063454109030029