Volume 22, Issue 5

Two methods for analysing stochastic control systems with distributed changes of structures are considered: statistical simulation and the spectral method. Algorithms for solving the problem are presented in the paper. The comparison of the methods and their efficiency are demonstrated on model problems.

A method for estimation of parameters for a parabolic boundary value problem with a Neumann boundary condition is proposed in the paper. This method utilizes probabilistic representations of the solution to the boundary value problem considered and also of its parametric derivatives in the form of functionals of the solution to a system of stochastic differential equations (SDE). Estimates for the solution and its parametric derivatives are obtained by statistical modelling of SDE trajectories by the Euler method. The results of numerical experiments are presented.

Based on solving test problems for nonlinear kinetic equations, the efficiency of different variants of weighted ‘value’ modelling of evolution of multiparticle ensembles has been studied. Approximations of solutions to model problems for the Boltzmann and Smolukhovsky equations are taken as value functions. It is shown that the most essential gain is achieved for the combination of approximate value modelling of the free path length of the ensemble with value modelling of the number of a pair of interacting particles. The technique developed here may be useful for solving actual problems of coagulation and relaxation of a system of particles, for which the considered model solutions are approximations.

A problem on estimation of the parameters of time asymptotics of polarized radiation beams in layers of scattering and absorbent media is considered in the paper. Based on two approaches to solving this problem, i.e., the method of iteration of the resolvent of the corresponding transfer operator and the method of parametric differentiation of a special representation of the solution to a nonstationary equation, concordant estimates for time asymptotics constants are obtained.

A special class of random nonstationary processes on Poisson point flows with piecewise-constant and piecewise-linear trajectories is considered in the paper. Expressions for mean values of these processes are obtained as functions of time.