Статистическое моделирование и методы Монте-Карло
We consider functional algorithms based on Random Walk on Spheres  for solving Dirichlet problem for linear and nonlinear Helmholtz equations. These algorithms consist in estimating of the equation solution in grid nodes by the Monte Carlo method and using an appropriate approximation procedure to obtain the functional approximation of the solution.
There exist two parameters of the algorithm: the number of grid nodes M and the sample size N. The problem is to choose optimal relation between these parameters which minimize the computational cost of the algorithm. For this purpose the upper bound of the error depending on these parameters can be constructed and the cost function of the algorithm can be minimized under some fixed error level .
Moreover, it is known that the Monte Carlo method enables to estimate not only the solution of a problem at some points, but the partial derivatives of the solution also. The estimators of partial derivatives can be used for constructing of a more smooth approximation. And so some gain in efficiency can be attained. We consider the availability of such approach.
This work was supported by the Russian Foundation of the Basic Research (projects 02-01-00958) and youth grant of Siberian Branch of Russian Academy of Science.
 Mikhailov G.A. Minimization of computational costs of non-analogue Monte Carlo methods. Series of Soviet and East European Mathematics. Vol. 5. Singapore: World Scientific, 1991.
 Shkarupa E.V., Voytishek A.V. Optimization of discretely stochastic procedures for globally estimating the solution of an integral equation of the second kind // Rus. J. Numer. Analys. and Math. Modelling. 1997. V. 12, N 6, P. 525-546.
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