# Институт вычислительной математики

и математической геофизики

# The International Conference on Computational Mathematics

ICCM-2004

June, 21-25, 2004
Novosibirsk, Academgorodok

## Abstracts

*Statistic modeling and Monte Carlo methods*

## Functional algorithms of Monte Carlo method for solving boundary value problems

**Institute of Computational Mathematics and Mathematical Geophysics,**

Siberian Branch of Russian Acad. Sci (Novosibirsk)
We consider functional algorithms based on Random Walk on Spheres
[1] 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 [2].

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.

[1] 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.

[2] 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.

*Note. Abstracts are published in author's edition*

© 1996-2000, Siberian Branch of Russian Academy of Sciences, Novosibirsk

Last update: 06-Jul-2012 (11:52:06)