Информационная система "Конференции"

Abstracts

This paper addresses several problems of estimating the functionals of stochastic differential equations (SDE) solutions by means of statistical modelling and comparing the functional algorithms for global solution of boundary-value problems for the diffusion equation. The research has been made within the framework of combined approach to constructing Monte Carlo algorithms. As a result, new estimates have been constructed as well as new optimization methods and effective distribution schemes for calculations suggested. The research outcomes are presented in three parts.

In the first part, we suggest a new modification of the Euler method. It is based on simultaneous estimation of the functional with two different time-steps and extrapolation. This modification is numerically compared with two well-known techniques of improving the deterministic error order.

In the second part, we consider the problem of numerical solution of the system of forward–backward SDEs and its Cauchy problem for a quasilinear parabolic equation. We propose a layer method of solving the Cauchy problem with simultaneous estimation of the solution gradient, which is based on the probabilistic representation of the solution.

The third part is dedicated to comparison of the three functional Monte Carlo algorithms for solving the Dirichlet problem for the Helmholtz equation. This comparison is made within the framework of C-approach, where the metric of C-space with convergence in probability is considered as an error measure of the numerical method.

One of the major advantages of the Monte Carlo methods is a simple distribution of calculations into several independent processors. All the weight algorithms of the statistical modelling suggested in this work have been tested with the system of parallel statistical simulation MONC. The first part of this system is the program that distributes calculations between LAN-computers, gathers and averages out the results. The second part of the system is the 128-bit random number generator with astronomically large period.

Note. Abstracts are published in author's edition

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