## Two problems of the Monte Carlo method theory

In this paper two nonsolved problems of the Monte Carlo theory are presented. The first of them concernes the uniform boundedness of the "walk on spheres" estimates for the Helmgoltz equation. Another problem is the important example from the minimax Monte Carlo theory for evaluating of many integrals.

## "Walk on spheres" algorithms for solving Helmholtz equation in the n-dimensional Euclidean space

In this paper the algorithms of Monte Carlo methods for solving the *n*-dimensional Helmholtz equation are investigated. The dependence of the computational efficiency of the algorithms on *n* is studied.

## Weak convergence of randomized spectral models of Gaussian random vector fields

Convergence of randomized spectral models of homogeneous vector fields is studied in the sense of convergence in distribution in a uniform metric of the Banach space of continuous functions. Under quite moderate restrictions on the parameters of the spectral model, weak convergence to a Gaussian field is shown if the...

## Calculation of time constant of particle breeding by Monte Carlo method using parametric derivatives

The paper contains the results of time constant calculations for the process of particle breeding. The calculations are based on the estimates of parametric derivatives of the particle flux. The transfer process of radiation is assumed to be stationary.

**Attention!** Please, also see this article.

## Special models of non-stationary random processes and non-homogeneous fields

In this paper some methods of statistical simulation of non-Gaussian non-stationary scalar and vectorial processes and non-homogeneous spatial fields with continuous argument on the basis of synthesis of discrete models and models on point fluxes are considered.

## Spectral models of vector-valued random fields

Numerical models of vector-valued random fields are extensively used in solving applied problems. These models have become the subject of many investigations. The paper deals with methods of numerical modeling of homogeneous vector-valued random fields based on the spectral decomposition. General relations for spectral models are obtained and particular algorithms...

## A probabilistic representation for systems of elliptic equations

New probabilistic representations for systems of elliptic equations are constructed in the form of expectations over the Markov chains. It is shown that this approach gives the effective Monte Carlo algorithms even in the cases, where the classical probabilistic representation based on the Wiener and diffusion processes fails. As an...

## Comparison of two procedures for global stochastic estimation of functions

Numerical stochastic procedures for estimating integrals depending on parameter are considered. The discrete mesh on the domain of definition of parameter is introduced, and the Monte Carlo algorithms for estimating integral in mesh points are used. The independent Monte Carlo estimates and the "depended tests" method are compared. It is...

## Solution of two-dimensional Prandtl equations by Monte Carlo method

New numerical method for approximating two-dimensional flow field for the viscous incompressible fluid in the vicinity of the flat boundary is introduced. Using the vorticity formulation of the Prandtl equation we come to the heat equation with nonlinear right-hand side. We consider various boundary value problems for this equation and...

## A new Monte Carlo method for calculation of covariance function of solution of biharmonic equation

The article is devoted to the new Monte Carlo method for the calculation of covariance function of the solution of biharmonic equation when its right-hand side is a random field. The comparison of this method with the randomization algorithm of the Monte Carlo method is presented. The numerical results of...