"Walk on half-spheres" process and then application to boundary problems solution
The use of representation of boundary problems solutions for simple domains in the Monte-Carlo algorithms is well known. In particular, the representation of this type is wide spread for sphere domain [2]. It permits to formally write integral equation of the second kind for the function under consideration in arbitrary domain with regular boundary. The use of concordance conditions [1] points out a possible construction of stochastic process which "solves" the given problem. The "walk on spheres" process, which solves the first boundary problem for the Poisson equation, leads to epsilon-biased estimates; thus it is necessary to input a parameter of regularization. The authors thoroughly examine the method of "walk on half-spheres" that was suggested earlier by A. S. Sipin without detailed basis [10]. The use of the Green function for half-sphere permits to construct estimates for the first and third boundary problems and also for the problem with discontinuous derivative that turn out to be unbiased for a wide class of domains. Described algorithms possess a high degree of parallelism. Results of solving model problems are presented.