Home – Home – Jornals – Numerical Analysis and Applications 2024 number 2

This paper reviews recent publications that describe mathematical models with stochastic differential equations (SDEs) and applications in various fields. The purpose of this paper is to briefly describe Rosenbrock-type methods for approximate solution of SDEs. It shows how the performance of the numerical methods can be improved and the accuracy of calculations can be increased without increasing the implementation complexity too much. The paper also proposes a new Rosenbrock-type method for SDEs with multiplicative non-commutative noise. Its testing is carried out by modeling rotational diffusion.

In this paper we formulate the requirements for choosing approximation bases in constructing cost-effective optimized computational (numerical) functional algorithms for approximating probability densities for a given sample, with special attention to stability and approximation of the bases. It is shown that to meet the requirements and construct efficient approaches to conditional optimization of the numerical schemes, the best choice is a multi-linear approximation and a corresponding special case for both kernel and projection computational algorithms for nonparametric density estimation, which is a multidimensional analogue of the frequency polygon.

The main goal of the work is to simulate heat transfer in structural elements of an aircraft under random temperature changes on its outer surface due to rapid changes in environmental parameters. In this case, to model the heat transfer a one-dimensional boundary value problem of the third kind is taken for the heat conduction equation. Random disturbances are specified at the boundary corresponding to the outer surface. The numerical solution is based on an application of the Galerkin method. Modeling the random disturbances of the external environment is carried out using a Wiener integral in a system of differential equations written in integral form. Calculations for a problem with a known exact solution show that when moving away from the boundary with random disturbances, the numerical solution of the boundary value problem with disturbances converges to the known exact solution of the unperturbed boundary value problem. Based on an expansion of the solution to the boundary value problem in trigonometric functions, theoretical estimates are obtained for the influence of a disturbance on the outer surface as a function of the wall thickness and the disturbance magnitude.

The paper deals with numerical models related to radiation transfer in ice clouds. A mathematical model of crystal particles of irregular shape and an algorithm for modeling such particles based on constructing a convex hull of a set of random points are considered. Two approaches to simulating radiation transfer in optically anisotropic clouds are studied. One approach uses pre-calculated scattering phase functions for crystals of various shapes and orientations. In the other approach, no knowledge of the phase functions is required; the radiation scattering angle is modeled directly in the interaction of a photon with crystal faces. This approach makes it possible to simply adjust the input parameters of the problem to changing microphysical characteristics of the environment, including shape, orientation, transparency of particles and roughness of their boundaries, and does not require time-consuming preliminary calculations. The impact of flutter on radiation transfer by a cloud layer and angular distributions of reflected and transmitted radiation are studied.

The paper presents efficiently realized approximations of random functions, which are developed by the authors and numerically modeled for the study of stochastic processes of particle transport, including criticality fluctuations of processes in random media with multiplication. Efficient correlation randomized algorithms for approximating an ensemble of particle trajectories using the correlation function or only the correlation scale of a medium are constructed. A simple grid model of an isotropic random field is formulated reproducing a given average correlation length, which ensures high accuracy in solving stochastic transfer problems for a small correlation scale. The algorithms are tested by solving a test problem of photon transfer and a problem of estimating the overexponential average particle flux in a random medium with multiplication.

The paper presents an approximate algorithm for modeling a stationary discrete random process with marginal and bivariate distributions of its consecutive components in the form of a mixture of two Gaussian distributions. The algorithm is based on a combination of the conditional distribution method and the rejection method. An example of application of the proposed algorithm for simulating time series of daily maximum air temperatures is given.

A continuous-discrete stochastic model is constructed to describe the evolution of a spatially heterogeneous population. The population structure is defined in terms of a graph with two vertices and two unidirectional edges. The graph describes the presence of individuals in the population at the vertices and their transitions between the vertices along the edges. Individuals enter the population from an external source at each of the vertices of the graph. The duration of movement of individuals along the edges of the graph is constant. Individuals may die or turn into individuals of other populations not considered in the model. The assumptions of the model are formulated, the probabilistic formalization of the model and the numerical simulation algorithm based on the Monte Carlo method are given. Distribution patterns of the population are studied. The results of a computational experiment are presented.

The paper deals with Monte Carlo modeling of spatiotemporal signals of wide-angle lidars for probing atmospheric clouds. Using computational experiments, we study the features of lidar signals for monostatic and bistatic sensing schemes which make it possible to analyze the optical and microphysical properties of the cloud environment. When probing thin cloud layers, the lidar signal looks like an expanding and attenuating light ring. It is shown that for a bistatic scheme a second ring, which appears for a short time inside the main one, is characteristic of the lidar signal.

In this paper, a new stochastic algorithm for solving the system of Lame equations based on the Slobodianskii representation is proposed, in which the recovery of boundary conditions for the harmonic functions involved is carried out implicitly using the method of fundamental solutions, while the unknown coefficients in this method are calculated using a stochastic projection method. Results of numerical experiments for several examples of two- and three-dimensional boundary value problems are presented, which demonstrate the high efficiency of the proposed method.