The article considers splitting schemes in geometric directions that approximate the initial-boundary value problem for p-dimensional (p≥3 ) hyperbolic equation by chain of two-dimensional-one-dimensional problems. Two ways of constructing splitting schemes are considered with an operator factorized on the upper layer, algebraically equivalent to the alternating direction scheme, and additive schemes of total approximation. For the first scheme, the restrictions on the shape of the region G при p=3 can be weakened in comparison with schemes of alternating directions, which are a chain of three-point problems on the upper time layer, the region G can be a connected union of cylindrical regions with generators parallel to the axis OX₃. In the second case, for a three-dimensional equation of hyperbolic type, an additive scheme is constructed, which is a chain «two-dimensional problem – one-dimensional problem» and approximates the original problem in a summary sense (at integer time steps). The stability and convergence of the constructed schemes are proved: with the factorized rate O(ǁhǁ²+τ²), and with the additive rate O(ǁhǁ²+τ) , where ǁhǁ is the norm of the step of the spatial grid,, τ is the time step, under the appropriate restrictions on the smoothness of the functions included in the statement of the initial-boundary value problem. For the numerical implementation of the constructed schemes – the numerical solution of two-dimensional elliptic problems – one can use fast direct methods based on the Fourier algorithm, cyclic reduction methods for three-point vector equations, combinations of these methods, and other methods. The proposed two-dimensional splitting schemes in a number of cases turn out to be more economical in terms of total time expenditures, including the time for performing computations and exchanges of information between processors, compared to traditional splitting schemes based on the use of three-point difference problems for multiprocessor computing systems, with different structures of connections between processors type «ruler», «matrix», «cube», with universal switching.

The paper studies the model of «Power-Society» system with two clans and bipolar reaction of the society. The «Power-Society» model describes the dynamics of distribution of power in hierarchy. This dynamics is influenced by society. Continuous-time «Power-Society» model has the form of parabolic equation in the case of continuous hierarchy, and the form of system of ordinary differential equations in the case of discrete hierarchy. The discrete-time model considered in this paper has the form of five dynamical equations. Bipolar reaction of the society refers to the situation with two stable distributions of power. In other words, for each government official two values are possible for the volume of power. Each of these values is considered by society as desirable. If each official holds the greater volume, we say that there is the «strong hand» distribution, if they all hold the smaller volume, this is the participatory distribution. Bureaucratic clans are an association of bureaucrats united by common interests and pursuing common goals, generally speaking, different from those of society as a whole. The paper considers a simple hierarchy of five officials, of which one is the head and four others form two competing clans. The system is studied numerically. It is shown, in particular, that in this system, the clan's lust for power significantly affects how quickly it manages to increase its power, however, the achieved amount of power itself almost does not depend on the lust for power, but is determined by the reaction of society.

The paper presents a discrete optimal control problem with constraints on which a method for calculation of migration flows, where qualified and unskilled workers are distinguished, is discussed. At the same time, the optimality criterion in the problem is associated with the achievement of the maximum output with the minimization of the total number of migrants. Numerical calculations are provided that illustrate the sustainable growth scenario over a 10 year period. The work objectives included the development of an approach for calculating the necessary size of working-age population migration and its components to achieve optimal output growth. A macromodel is proposed, which is a discrete optimal control problem. An algorithm for the control synthesis is pro-posed. Numerical modeling is carried out. The obtained results can be used in migration flows planning and management processes.

This paper covers the creation and numerical realization of proposed mathematical model of hydrodynamical processes in shallow water based on contemporary information technology and new computational methods that allow improve the prediction accuracy of the environmental situation using the example of the Taganrog Bay in the Azov Sea basin. The proposed mathematical hydrodynamics model takes into account surges, dynamically reconstructed geometry, elevation of the level and coastline, wind currents and friction against the bottom, Coriolis force, turbulent exchange, evaporation, river flow, deviation of the pressure field value from the hydrostatic approximation, the salinity and temperature impact. The discretization of the mathematical model of hydrodynamics was performed using the splitting schemes for physical processes. The constructed discrete analogs possess the properties of conservatism, stability, and convergence. Numerical algorithms are also proposed for solving the arising SLAEs that improve the accuracy of predictive modeling. The practical significance of this research is software implementation of the developed model and the determination of limits and prospects of its application. The experimental software development was based on a graphics accelerator for mathematical simulation the possible scenarios of shallow water ecosystems in consideration the environmental factors influence. The decomposition methods taking into account the CUDA architecture specifications were used at parallel implementation for computationally labors diffusion-convection problems.

The paper proposes an algorithm for the formation of a small training set, which ensures a reasonable quality of a surrogate machine learning model trained using this set. The algorithm uses multilayer perceptron to estimate heuristics and select the best next sample for the inclusion in a set. The paper tests the algorithm proposed applying it to the problem of deformation and breaking of a thin thread under the action of a transverse load pulse on it. The possibility to generalize the approach and apply it to building surrogate machine learning models for other physical problems is discussed.