Home – Home – Jornals – Numerical Analysis and Applications 2018 number 4

October 12, 2018 marks the 90th anniversary of the birth of Academician Anatoly Semenovich Alekseev. A.S. Alekseev was a world-famous scientist, a leading expert in the field of theoretical and computational geophysics, mathematical modeling of geophysical phenomena and geoinformatics. In the world scientific community and domestic industry the name is A.S. Alekseeva marks the development and widespread introduction of new mathematical methods for solving fundamental scientific problems of a planetary scale, starting with a study of the mechanisms and prediction of earthquakes and tsunamis and ending with the study of the consequences of large celestial bodies falling to Earth (asteroids, meteorites, fragments of comets, etc.).

Numerical-statistical estimates of correlation characteristics and averaged angle near distributions of the radiation intensity field, passing throw a random medium are obtained. Comparative investigations were performed for an elementary Poisson field and for the «realistic» field of the medium optical density. The obtained estimates confirm the hypothesis about a strong dependence of investigated values on the correlation scale and the one-dimensional distribution of the medium density field.

It is known that the solution of the semilinear matrix equation X - A\overline X B = C can be reduced to solving the classical Stein equation. The normal case means that the coefficients on the left-hand side of the resulting equation are normal matrices. We propose a method for solving the original semilinear equation in the normal case that permits to almost halve the execution time for equations of order n = 3000 compared to the library function dlyap, which solves Stein equations in Matlab.

Grid-characteristic method for the numerical simulation of wave processes in continuum mechanics was initially proposed and successfully applied to periodic hexagonal computational grids. Later, it was proposed to adapt this method to non-periodic triangle and tetrahedral grids, and a broad computational experience has been gained by now. However, this approach encounters some challenges with the calculation of border and contact points when applied to various grid configurations in the areas with complex geometries. In this paper, the method limitations, which cause the problems is considered, and some improvements to overcome them are proposed.

In the Hilbert space, we consider a class of conditionally well-posed inverse problems, for which the Hölder type estimate of conditional stability on a closed convex bounded subset holds. We investigate the Ivanov quasisolution method and its finite dimensional version associated with the minimizing a multi-extremal discrepancy functional over a conditional stability set and over the finite dimensional section of this set, respectively. For these optimization problems, we prove that each their stationary point that is located not too far from the desired solution of the original inverse problem, in reality belongs to a small neighborhood of the solution. Estimates for the diameter of this neighborhood in terms of error levels in input data are also given.

This paper presents the computational algorithms, which make it possible to overcome some complexities with the numerical solution of the boundary-value problems of thermal conductivity when the domain of the solution has a complex form or boundary conditions differ from standard ones. Boundary contours are assumed to be broken lines (the flat case) or triangles (a 3D case). Boundary conditions and calculation results are presented as discrete functions whose values or their averaged values are given at geometric centers of boundary elements. Boundary conditions can be defined on the heat flows through boundary elements as well as on temperature, a linear temperature combination and heat flow intensity both at the boundary of the solution domain and inside it. The solution to the boundary value problem is presented in the form of a linear combination of the fundamental solutions of the Laplace equation and their partial derivatives and, also, any solutions of these equations that are regular in the solution domain, the values of functions for which can be calculated at the points of the boundary of the solution domain and at its internal points. If the solution, which participates in the linear combination, is singular, then its average value according to this boundary element is considered.

The aim of this article is to investigate the local convergence analysis of the multi-step Homeier-like approach in order to approximate the solution of nonlinear equations in Banach spaces, which fulfilled the Lipschitz as well as Hölder continuity condition. The Hölder condition is more relaxer than Lipschitz condition. Also, the existence and uniqueness theorem has been derived and found their error bounds. Numerical examples are available to appear the importance of theoretical discussions.

We propose a method for solving the three-dimensional boundary value problems for the Laplace equation in an unbounded domain. It is based on the non-overlapping decomposition of the exterior domain to the two subdomains such that the initial problem is reduced to the two subproblems, namely, the exterior and the interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To cross-link the solutions at the interface of subdomains (a sphere), we introduce a special operator equation that is approximated by the system of linear algebraic equations. Such a system is solved by iterative methods in the Krylov subspaces. The method is illustrated by solving the model problems confirming its operability.

The approximation neural network algorithm for solving the inverse geoelectrics problems in the class of grid (block) media models is presented. The algorithm is based on constructing an approximate inverse operator using neural networks and makes it possible to formally obtain solutions of the inverse geoelectrics problem with the total number of desired parameters of the medium ~ n 10³. The correctness of the problem of constructing the neural network inverse operators is considered. A posteriori estimates of the degree of ambiguity of the inverse problem solutions are calculated. The operation of the algorithm is illustrated by examples of the 2D, the 3D inversions of synthesized and field geoelectric data, obtained by the MTS method.