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The review is based on some of the authors ’early works of particular scientific, methodological and practical interest and the greatest attention is paid to recent works, where quite detailed numerical studies of not only single, but also double and multiple explosions in a wide range of heights and environmental conditions have been performed . Since the shock wave of a powerful explosion is one of the main damaging factors in the lower atmosphere, the review focuses on both the physical analysis of their propagation and their interaction. Using the three-dimensional algorithms developed by the authors, the effects of interference and diffraction of several shock waves, which are interesting from a physical point of view, in the absence and presence of an underlying surface of various structures are considered. Quantitative characteristics are determined in the region of their maximum values, which is of known practical interest. For explosions in a dense atmosphere, some new analytical solutions based on the small perturbation method have been found that are convenient for approximate calculations. For a number of conditions, the possibility of using the self-similar properties of equations of the first and second kind to solve problems on the development of an explosion has been shown.

Based on numerical analysis, a fundamental change in the structure of the development of the perturbed region with a change in the height of the explosion in the range of 100–120 km is shown. At altitudes of more than 120 km, the geomagnetic field begins to influence the development of the explosion; therefore, even for a single explosion, the picture of the plasma flow after a few seconds becomes substantially three-dimensional. For the calculation of explosions at altitudes of 120–1000 km under the guidance of academician A. Kholodov. A special three-dimensional numerical algorithm based on the MHD approximation was developed. Numerous calculations were performed and for the first time a quite detailed picture of the three-dimensional flow of the explosion plasma was obtained with the formation of an upward jet in 5–10 s directed in the meridional plane approximately along the geomagnetic field. After some modification, this algorithm was used to calculate double explosions in the ionosphere, spaced a certain distance. The interaction between them was carried out both by plasma flows and through a geomagnetic field. Some results are given in this review and are described in detail in the original articles.

The paper presents a mathematical model that describes the process of obtaining biogas from livestock waste. This model describes the processes occurring in a biogas plant for mesophilic and thermophilic media, as well as for continuous and periodic modes of substrate inflow. The values of the coefficients of this model found earlier for the periodic mode, obtained by solving the problem of model identification from experimental data using a genetic algorithm, are given.

For the model of methanogenesis, an optimal control problem is formulated in the form of a Lagrange problem, whose criterial functionality is the output of biogas over a certain period of time. The controlling parameter of the task is the rate of substrate entry into the biogas plant. An algorithm for solving this problem is proposed, based on the numerical implementation of the Pontryagin maximum principle. In this case, a hybrid genetic algorithm with an additional search in the vicinity of the best solution using the method of conjugate gradients was used as an optimization method. This numerical method for solving an optimal control problem is universal and applicable to a wide class of mathematical models.

In the course of the study, various modes of submission of the substrate to the digesters, temperature environments and types of raw materials were analyzed. It is shown that the rate of biogas production in the continuous feed mode is 1.4–1.9 times higher in the mesophilic medium (1.9–3.2 in the thermophilic medium) than in the periodic mode over the period of complete fermentation, which is associated with a higher feed rate of the substrate and a greater concentration of nutrients in the substrate. However, the yield of biogas during the period of complete fermentation with a periodic mode is twice as high as the output over the period of a complete change of the substrate in the methane tank at a continuous mode, which means incomplete processing of the substrate in the second case. The rate of biogas formation for a thermophilic medium in continuous mode and the optimal rate of supply of raw materials is three times higher than for a mesophilic medium. Comparison of biogas output for various types of raw materials shows that the highest biogas output is observed for waste poultry farms, the least — for cattle farms waste, which is associated with the nutrient content in a unit of substrate of each type.

The article describes the basic approaches of the calculation procedure of payload swim-out (objects of different function with own propulsor) from the underwater carrier a method of a self-exit using modern CFD technologies. It contains the description of swim-out by a self-exit method, its advantages and disadvantages. Also it contains results of research of convergence on a grid of a final-volume model with accuracy-time criterion, and results of comparison of calculation with experiment (validation of models). Validation of models was carried out using the available data of experimental definition of traction characteristics of water-jet propulsor of the natural sample in the development pool. Calculations of traction characteristics of water-jet propulsor were carried out via software package FlowVision ver. 3.10. On the basis of comparison of results of calculations for conditions of carrying out of experiments the error of water-jet propulsor calculated model which has made no more than 5% in a range of advance coefficient water-jet propulsor, realised in the process of swim-out by a selfexit method has been defined. The received value of an error of calculation of traction characteristics is used for definition of limiting settlement values of speed of branch of object from the carrier (the minimum and maximum values). The considered problem is significant from the scientific point of view thanks to features of the approach to modelling hydrojet moving system together with movement of separated object, and also from the practical point of view, thanks to possibility of reception with high degree of reliability of parametres swim-out of objects from sea bed vehicles a method of the self-exit which working conditions are assumed by movement in the closed volumes, already on a design stage.

The article deals with the problem of a distributed system operation reliability. The system core is an open integration platform that provides interaction of varied software for modeling gas transportation. Some of them provide an access through thin clients on the cloud technology “software as a service”. Mathematical models of operation, transmission and computing are to ensure the operation of an automated dispatching system for oil and gas transportation. The paper presents a system solution based on the theory of Markov random processes and considers the stable operation stage. The stationary operation mode of the Markov chain with continuous time and discrete states is described by a system of Chapman–Kolmogorov equations with respect to the average numbers (mathematical expectations) of the objects in certain states. The objects of research are both system elements that are present in a large number – thin clients and computing modules, and individual ones – a server, a network manager (message broker). Together, they are interacting Markov random processes. The interaction is determined by the fact that the transition probabilities in one group of elements depend on the average numbers of other elements groups.

The authors propose a multi-criteria dispersion model of risk assessment for such systems (both in the broad and narrow sense, in accordance with the IEC standard). The risk is the standard deviation of estimated object parameter from its average value. The dispersion risk model makes possible to define optimality criteria and whole system functioning risks. In particular, for a thin client, the following is calculated: the loss profit risk, the total risk of losses due to non-productive element states, and the total risk of all system states losses.

Finally the paper proposes compromise schemes for solving the multi-criteria problem of choosing the optimal operation strategy based on the selected set of compromise criteria.

We describe an engineering-psychological experiment that continues the study of ways to adapt a person to the increasing complexity of logical problems by presenting a series of problems of increasing complexity, which is determined by the volume of initial data. Tasks require calculations in an associative or non-associative system of operations. By the nature of the change in the time of solving the problem, depending on the number of necessary operations, we can conclude that a purely sequential method of solving problems or connecting additional brain resources to the solution in parallel mode. In a previously published experimental work, a person in the process of solving an associative problem recognized color images with meaningful images. In the new study, a similar problem is solved for abstract monochrome geometric shapes. Analysis of the result showed that for the second case, the probability of the subject switching to a parallel method of processing visual information is significantly reduced. The research method is based on presenting a person with two types of tasks. One type of problem contains associative calculations and allows a parallel solution algorithm. Another type of problem is the control one, which contains problems in which calculations are not associative and parallel algorithms are ineffective. The task of recognizing and searching for a given object is associative. A parallel strategy significantly speeds up the solution with relatively small additional resources. As a control series of problems (to separate parallel work from the acceleration of a sequential algorithm), we use, as in the previous experiment, a non-associative comparison problem in cyclic arithmetic, presented in the visual form of the game “rock, paper, scissors”. In this problem, the parallel algorithm requires a large number of processors with a small efficiency coefficient. Therefore, the transition of a person to a parallel algorithm for solving this problem is almost impossible, and the acceleration of processing input information is possible only by increasing the speed. Comparing the dependence of the solution time on the volume of source data for two types of problems allows us to identify four types of strategies for adapting to the increasing complexity of the problem: uniform sequential, accelerated sequential, parallel computing (where possible), or undefined (for this method) strategy. The Reducing of the number of subjects, who switch to a parallel strategy when encoding input information with formal images, shows the effectiveness of codes that cause subject associations. They increase the speed of human perception and processing of information. The article contains a preliminary mathematical model that explains this phenomenon. It is based on the appearance of a second set of initial data, which occurs in a person as a result of recognizing the depicted objects.

The paper considers the process of flow formation in an outflow of a binary gas mixture through a thin slit into vacuum. An approach to modeling the flows of rarefied gas mixtures in the transient regime is proposed based on the direct solution of the Boltzmann kinetic equation, in which the conservative projection method is used to calculate the collision integrals. Calculation formulas are provided; the calculation procedure is described in detail in relation to the flow of a binary gas mixture. The Lennard–Jones potential is used as an interaction potential of molecules. A software modeling environment has been developed that makes it possible to study the flows of gas mixtures in a transitional regime on systems of cluster architecture. Due to the use of code parallelization technologies, an acceleration of calculations by 50–100 times was obtained. Numerical simulation of a two-dimensional outflow of a binary argon-neon gas mixture from a vessel into vacuum through a thin slit is carried out for various values of the Knudsen number. The graphs of the dependence of gas mixture components output flow on time in the process of establishing the flow are obtained. Non-stationary regions of strong separation of gas mixture components, in which the molecular densities ratio reaches 10 or more, were discovered. The discovered effect can have applications in the problem of gas mixtures separation.

This article reviews both historical achievements and modern results in the field of Markov Decision Process (MDP) and convex optimization. This review is the first attempt to cover the field of reinforcement learning in Russian in the context of convex optimization. The fundamental Bellman equation and the criteria of optimality of policy — strategies based on it, which make decisions based on the known state of the environment at the moment, are considered. The main iterative algorithms of policy optimization based on the solution of the Bellman equations are also considered. An important section of this article was the consideration of an alternative to the $Q$-learning approach — the method of direct maximization of the agent’s average reward for the chosen strategy from interaction with the environment. Thus, the solution of this convex optimization problem can be represented as a linear programming problem. The paper demonstrates how the convex optimization apparatus is used to solve the problem of Reinforcement Learning (RL). In particular, it is shown how the concept of strong duality allows us to naturally modify the formulation of the RL problem, showing the equivalence between maximizing the agent’s reward and finding his optimal strategy. The paper also discusses the complexity of MDP optimization with respect to the number of state–action–reward triples obtained as a result of interaction with the environment. The optimal limits of the MDP solution complexity are presented in the case of an ergodic process with an infinite horizon, as well as in the case of a non-stationary process with a finite horizon, which can be restarted several times in a row or immediately run in parallel in several threads. The review also reviews the latest results on reducing the gap between the lower and upper estimates of the complexity of MDP optimization with average remuneration (Averaged MDP, AMDP). In conclusion, the real-valued parametrization of agent policy and a class of gradient optimization methods through maximizing the $Q$-function of value are considered. In particular, a special class of MDPs with restrictions on the value of policy (Constrained Markov Decision Process, CMDP) is presented, for which a general direct-dual approach to optimization with strong duality is proposed.

The second part presents numerical studies of the parameters of the lower ionosphere at altitudes of 40–90 km when heated by powerful high-frequency radio waves of various frequencies and powers. The problem statement is considered in the first part of the article. The main attention is paid to the interrelation between the energy and kinetic parameters of the disturbed $D$-region of the ionosphere in the processes that determine the absorption and transformation of the radio beam energy flux in space and time. The possibility of a significant difference in the behavior of the parameters of the disturbed region in the daytime and at nighttime, both in magnitude and in space-time distribution, is shown. In the absence of sufficiently reliable values of the rate constants for a number of important kinetic processes, numerical studies were carried out in stages with the gradual addition of individual processes and kinetic blocks corresponding at the same time to a certain physical content. It is shown that the energy thresholds for inelastic collisions of electrons with air molecules are the main ones. This approach made it possible to detect the effect of the emergence of a self-oscillating mode of changing parameters if the main channel for energy losses in inelastic processes is the most energy-intensive process — ionization. This effect may play a role in plasma studies using high-frequency inductive and capacitive discharges. The results of calculations of the ionization and optical parameters of the disturbed $D$-region for daytime conditions are presented. The electron temperature, density, emission coefficients in the visible and infrared ranges of the spectrum are obtained for various values of the power of the radio beam and its frequency in the lower ionosphere. The height-time distribution of the absorbed radiation power is calculated, which is necessary in studies of higher layers of the ionosphere. The influence on the electron temperature and on the general behavior of the parameters of energy losses by electrons on the excitation of vibrational and metastable states of molecules has been studied in detail. It is shown that under nighttime conditions, when the electron concentration begins at altitudes of about 80 km, and the concentration of heavy particles decreases by two orders of magnitude compared to the average $D$-region, large-scale gas-dynamic motion can develop with sufficient radio emission power The algorithm was developed based on the McCormack method and two-dimensional gas-dynamic calculations of the behavior of the parameters of the perturbed region were performed with some simplifications of the kinetics.

Modeling of channel processes in the study of coastal channel deformations requires the calculation of hydrodynamic flow parameters that take into account the existence of secondary transverse currents formed at channel curvature. Three-dimensional modeling of such processes is currently possible only for small model channels; for real river flows, reduced-dimensional models are needed. At the same time, the reduction of the problem from a three-dimensional model of the river flow movement to a two-dimensional flow model in the cross-section assumes that the hydrodynamic flow under consideration is quasi-stationary and the hypotheses about the asymptotic behavior of the flow along the flow coordinate of the cross-section are fulfilled for it. Taking into account these restrictions, a mathematical model of the problem of the a stationary turbulent calm river flow movement in a channel cross-section is formulated. The problem is formulated in a mixed formulation of velocity — “vortex – stream function”. As additional conditions for problem reducing, it is necessary to specify boundary conditions on the flow free surface for the velocity field, determined in the normal and tangential direction to the cross-section axis. It is assumed that the values of these velocities should be determined from the solution of auxiliary problems or obtained from field or experimental measurement data.

To solve the formulated problem, the finite element method in the Petrov – Galerkin formulation is used. Discrete analogue of the problem is obtained and an algorithm for solving it is proposed. Numerical studies have shown that, in general, the results obtained are in good agreement with known experimental data. The authors associate the obtained errors with the need to more accurately determine the circulation velocities field at crosssection of the flow by selecting and calibrating a more appropriate model for calculating turbulent viscosity and boundary conditions at the free boundary of the cross-section.

Considered the application of physics-informed neural networks using multi layer perceptrons to solve Cauchy initial value problems in which the right-hand sides of the equation are continuous monotonically increasing, decreasing or oscillating functions. With the use of the computational experiments the influence of the construction of the approximate neural network solution, neural network structure, optimization algorithm and software implementation means on the learning process and the accuracy of the obtained solution is studied. The analysis of the efficiency of the most frequently used machine learning frameworks in software development with the programming languages Python and C# is carried out. It is shown that the use of C# language allows to reduce the time of neural networks training by 20–40%. The choice of different activation functions affects the learning process and the accuracy of the approximate solution. The most effective functions in the considered problems are sigmoid and hyperbolic tangent. The minimum of the loss function is achieved at the certain number of neurons of the hidden layer of a single-layer neural network for a fixed training time of the neural network model. It’s also mentioned that the complication of the network structure increasing the number of neurons does not improve the training results. At the same time, the size of the grid step between the points of the training sample, providing a minimum of the loss function, is almost the same for the considered Cauchy problems. Training single-layer neural networks, the Adam method and its modifications are the most effective to solve the optimization problems. Additionally, the application of twoand three-layer neural networks is considered. It is shown that in these cases it is reasonable to use the LBFGS algorithm, which, in comparison with the Adam method, in some cases requires much shorter training time achieving the same solution accuracy. The specificity of neural network training for Cauchy problems in which the solution is an oscillating function with monotonically decreasing amplitude is also investigated. For these problems, it is necessary to construct a neural network solution with variable weight coefficient rather than with constant one, which improves the solution in the grid cells located near by the end point of the solution interval.