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Mathematical simulation of unsteady natural convection and thermal surface radiation within a rotating square enclosure was performed. The considered domain of interest had two isothermal opposite walls subjected to constant low and high temperatures, while other walls are adiabatic. The walls were diffuse and gray. The considered cavity rotated with constant angular velocity relative to the axis that was perpendicular to the cavity and crossed the cavity in the center. Mathematical model, formulated in dimensionless transformed variables “stream function – vorticity” using the Boussinesq approximation and diathermic approach for the medium, was performed numerically using the finite difference method. The vorticity dispersion equation and energy equation were solved using locally one-dimensional Samarskii scheme. The diffusive terms were approximated by central differences, while the convective terms were approximated using monotonic Samarskii scheme. The difference equations were solved by the Thomas algorithm. The approximated Poisson equation for the stream function was solved by successive over-relaxation method. Optimal value of the relaxation parameter was found on the basis of computational experiments. Radiative heat transfer was analyzed using the net-radiation method in Poljak approach. The developed computational code was tested using the grid independence analysis and experimental and numerical results for the model problem.

Numerical analysis of unsteady natural convection and thermal surface radiation within the rotating enclosure was performed for the following parameters: Ra = 10³–10⁶, Ta = 0–10⁵, Pr = 0.7, ε = 0–0.9. All distributions were obtained for the twentieth complete revolution when one can find the periodic behavior of flow and heat transfer. As a result we revealed that at low angular velocity the convective flow can intensify but the following growth of angular velocity leads to suppression of the convective flow. The radiative Nusselt number changes weakly with the Taylor number.

In this paper, a comparative analysis of the exact and heuristic algorithms presented by the method of branches and boundaries, genetic and ant algorithms, respectively, is carried out to find the optimal solution to the traveling salesman problem using the example of a courier robot. The purpose of the work is to determine the running time, the length of the obtained route and the amount of memory required for the program to work, using the method of branches and boundaries and evolutionary heuristic algorithms. Also, the most appropriate of the listed methods for use in the specified conditions is determined. This article uses the materials of the conducted research, implemented in the format of a computer program, the program code for which is implemented in Python. In the course of the study, a number of criteria for the applicability of algorithms were selected (the time of the program, the length of the constructed route and the amount of memory necessary for the program to work), the results of the algorithms were obtained under specified conditions and conclusions were drawn about the degree of expediency of using one or another algorithm in various specified conditions of the courier robot. During the study, it turned out that for a small number of points  $\leqslant10$, the method of branches and boundaries is the most preferable, since it finds the optimal solution faster. However, when calculating the route by this method, provided that the points increase by more than 10, the operating time increases exponentially. In this case, more effective results are obtained by a heuristic approach using a genetic and ant algorithm. At the same time, the ant algorithm is distinguished by solutions that are closest to the reference ones and with an increase of more than 16 points. Its relative disadvantage is the greatest resource intensity among the considered algorithms. The genetic algorithm gives similar results, but after increasing the points more than 16, the length of the found route increases relative to the reference one. The advantage of the genetic algorithm is its lower resource intensity compared to other algorithms.

The practical significance of this article lies in the potential possibility of using the results obtained for the optimal solution of logistics problems by an automated system in various fields: warehouse logistics, transport logistics, «last mile» logistics, etc.

The article considers minimization of the expectation of convex function. Problems of this type often arise in machine learning and a variety of other applications. In practice, stochastic gradient descent (SGD) and similar procedures are usually used to solve such problems. We propose to use the ellipsoid method with mini-batching, which converges linearly and can be more efficient than SGD for a class of problems. This is verified by our experiments, which are publicly available. The algorithm does not require neither smoothness nor strong convexity of the objective to achieve linear convergence. Thus, its complexity does not depend on the conditional number of the problem. We prove that the method arrives at an approximate solution with given probability when using mini-batches of size proportional to the desired accuracy to the power −2. This enables efficient parallel execution of the algorithm, whereas possibilities for batch parallelization of SGD are rather limited. Despite fast convergence, ellipsoid method can result in a greater total number of calls to oracle than SGD, which works decently with small batches. Complexity is quadratic in dimension of the problem, hence the method is suitable for relatively small dimensionalities.

This work contains a description of the results of modeling the process of maintaining the readiness of a section of the road network under strikes of with specified parameters. A one-dimensional section of road up to 40 km long with a total number of strikes up to 100 during the work of the brigade is considered. A simulation model has been developed for carrying out work to maintain it in working condition by several groups (engineering teams) that are part of the engineering and road division. A multicopter-type unmanned aerial vehicle is used to search for the points of appearance of obstacles. Life cycle schemes of the main participants of the tactical scene have been developed and an event-driven model of the tactical scene has been built. The format of the event log generated as a result of simulation modeling of the process of maintaining a road section is proposed. To visualize the process of maintaining the readiness of a road section, it is proposed to use visualization in the cyclogram format.

An XSL style has been developed for building a cyclogram based on an event log. As an algorithm for making a decision on the assignment of barriers to brigades, the simplest algorithm has been adopted, prescribing choosing the nearest barrier. A criterion describing the effectiveness of maintenance work on the site based on the assessment of the average speed of vehicles on the road section is proposed. Graphs of the dependence of the criterion value and the root-meansquare error depending on the length of the maintained section are plotted and an estimate is obtained for the maximum length of the road section maintained in a state of readiness with specified values for the selected quality indicator with specified characteristics of striking and performance of repair crews. The expediency of carrying out work to maintain readiness by several brigades that are part of the engineering and road division operating autonomously is shown.

The influence of the speed of the unmanned aerial vehicle on the ability to maintain the readiness of the road section is analyzed. The speed range for from 10 to 70 km/h is considered, which corresponds to the technical capabilities of multicoptertype reconnaissance unmanned aerial vehicles. The simulation results can be used as part of a complex simulation model of an army offensive or defensive operation and for solving the problem of optimizing the assignment of tasks to maintain the readiness of road sections to engineering and road brigades. The proposed approach may be of interest for the development of military-oriented strategy games.

The paper deals with a heat wave motion problem for a degenerate second-order nonlinear parabolic equation with power nonlinearity. The considered boundary condition specifies in a plane the motion equation of the circular zero front of the heat wave. A new numerical-analytical algorithm for solving the problem is proposed. A solution is constructed stepby- step in time using difference time discretization. At each time step, a boundary value problem for the Poisson equation corresponding to the original equation at a fixed time is considered. This problem is, in fact, an inverse Cauchy problem in the domain whose initial boundary is free of boundary conditions and two boundary conditions (Neumann and Dirichlet) are specified on a current boundary (heat wave). A solution of this problem is constructed as the sum of a particular solution to the nonhomogeneous Poisson equation and a solution to the corresponding Laplace equation satisfying the boundary conditions. Since the inhomogeneity depends on the desired function and its derivatives, an iterative solution procedure is used. The particular solution is sought by the collocation method using inhomogeneity expansion in radial basis functions. The inverse Cauchy problem for the Laplace equation is solved by the null field method as applied to a circular domain with a circular hole. This method is used for the first time to solve such problem. The calculation algorithm is optimized by parallelizing the computations. The parallelization of the computations allows us to realize effectively the algorithm on high performance computing servers. The algorithm is implemented as a program, which is parallelized by using the OpenMP standard for the C++ language, suitable for calculations with parallel cycles. The effectiveness of the algorithm and the robustness of the program are tested by the comparison of the calculation results with the known exact solution as well as with the numerical solution obtained earlier by the authors with the use of the boundary element method. The implemented computational experiment shows good convergence of the iteration processes and higher calculation accuracy of the proposed new algorithm than of the previously developed one. The solution analysis allows us to select the radial basis functions which are most suitable for the proposed algorithm.

Newton and quasi-Newton methods of absolute optimization based on Cholesky factorization with adaptive step and finite difference approximation of the first and the second derivatives. In order to raise effectiveness of the quasi-Newton methods a modified version of Cholesky decomposition of quasi-Newton matrix is suggested. It solves the problem of step scaling while descending, allows approximation by non-quadratic functions, and integration with confidential neighborhood method. An approach to raise Newton methods effectiveness with finite difference approximation of the first and second derivatives is offered. The results of numerical research of algorithm effectiveness are shown.

In this paper, a time-dependent natural convective heat transfer in a closed square cavity filled with non- Newtonian fluid was considered in the presence of an isothermal energy source located on the lower wall of the region under consideration. The vertical boundaries were kept at constant low temperature, while the horizontal walls were completely insulated. The behavior of a non-Newtonian fluid was described by the Ostwald de Ville power law. The process under study was described by transient partial differential equations using dimensionless non-primitive variables “stream function – vorticity – temperature”. This method allows excluding the pressure field from the number of unknown parameters, while the non-dimensionalization allows generalizing the obtained results to a variety of physical formulations. The considered mathematical model with the corresponding boundary conditions was solved on the basis of the finite difference method. The algebraic equation for the stream function was solved by the method of successive lower relaxation. Discrete analogs of the vorticity equation and energy equation were solved by the Thomas algorithm. The developed numerical algorithm was tested in detail on a class of model problems and good agreement with other authors was achieved. Also during the study, the mesh sensitivity analysis was performed that allows choosing the optimal mesh.

As a result of numerical simulation of unsteady natural convection of a non-Newtonian power-law fluid in a closed square cavity with a local isothermal energy source, the influence of governing parameters was analyzed including the impact of the Rayleigh number in the range 104–106, power-law index $n = 0.6–1.4$, and also the position of the heating element on the flow structure and heat transfer performance inside the cavity. The analysis was carried out on the basis of the obtained distributions of streamlines and isotherms in the cavity, as well as on the basis of the dependences of the average Nusselt number. As a result, it was established that pseudoplastic fluids $(n < 1)$ intensify heat removal from the heater surface. The increase in the Rayleigh number and the central location of the heating element also correspond to the effective cooling of the heat source.

This article studies a method of constructing a predictive neural network model of a time series based on determining the composition of input variables, constructing a training sample and training itself using the back propagation method. Traditional methods of constructing predictive models of the time series are: the autoregressive model, the moving average model or the autoregressive model — the moving average allows us to approximate the time series by a linear dependence of the current value of the output variable on a number of its previous values. Such a limitation as linearity of dependence leads to significant errors in forecasting.

Mining Technologies using neural network modeling make it possible to approximate the time series by a nonlinear dependence. Moreover, the process of constructing of a neural network model (determining the composition of input variables, the number of layers and the number of neurons in the layers, choosing the activation functions of neurons, determining the optimal values of the neuron link weights) allows us to obtain a predictive model in the form of an analytical nonlinear dependence.

The determination of the composition of input variables of neural network models is one of the key points in the construction of neural network models in various application areas that affect its adequacy. The composition of the input variables is traditionally selected from some physical considerations or by the selection method. In this work it is proposed to use the behavior of the autocorrelation and private autocorrelation functions for the task of determining the composition of the input variables of the predictive neural network model of the time series.

In this work is proposed a method for determining the composition of input variables of neural network models for stationary and non-stationary time series, based on the construction and analysis of autocorrelation functions. Based on the proposed method in the Python programming environment are developed an algorithm and a program, determining the composition of the input variables of the predictive neural network model — the perceptron, as well as building the model itself. The proposed method was experimentally tested using the example of constructing a predictive neural network model of a time series that reflects energy consumption in different regions of the United States, openly published by PJM Interconnection LLC (PJM) — a regional network organization in the United States. This time series is non-stationary and is characterized by the presence of both a trend and seasonality. Prediction of the next values of the time series based on previous values and the constructed neural network model showed high approximation accuracy, which proves the effectiveness of the proposed method.

In this paper, we introduce a new version of Gauss–Newton method for solving a system of nonlinear equations based on ideas of the residual upper bound for a system of nonlinear equations and a quadratic regularization term. The introduced Gauss–Newton method in practice virtually forms the whole parameterized family of the methods solving systems of nonlinear equations and regression problems. The developed family of Gauss–Newton methods completely consists of iterative methods with generalization for cases of non-euclidean normed spaces, including special forms of Levenberg–Marquardt algorithms. The developed methods use the local model based on a parameterized proximal mapping allowing us to use an inexact oracle of «black–box» form with restrictions for the computational precision and computational complexity. We perform an efficiency analysis including global and local convergence for the developed family of methods with an arbitrary oracle in terms of iteration complexity, precision and complexity of both local model and oracle, problem dimensionality. We present global sublinear convergence rates for methods of the proposed family for solving a system of nonlinear equations, consisting of Lipschitz smooth functions. We prove local superlinear convergence under extra natural non-degeneracy assumptions for system of nonlinear functions. We prove both local and global linear convergence for a system of nonlinear equations under Polyak–Lojasiewicz condition for proposed Gauss– Newton methods. Besides theoretical justifications of methods we also consider practical implementation issues. In particular, for conducted experiments we present effective computational schemes for the exact oracle regarding to the dimensionality of a problem. The proposed family of methods unites several existing and frequent in practice Gauss–Newton method modifications, allowing us to construct a flexible and convenient method implementable using standard convex optimization and computational linear algebra techniques.

Modern methods of complex planning the execution of task packages in multistage systems are characterized by the presence of restrictions on the dimension of the problem being solved, the impossibility of guaranteed obtaining effective solutions for various values of its input parameters, as well as the impossibility of registration the conditions for the formation of sets from the result and the restriction on the interval duration of time of the system operating. The decomposition of the generalized function of the system into a set of hierarchically interconnected subfunctions is implemented to solve the problem of scheduling the execution of task packages with generating sets of results and the restriction on the interval duration of time for the functioning of the system. The use of decomposition made it possible to employ the hierarchical approach for planning the execution of task packages in multistage systems, which provides the determination of decisions by the composition of task groups at the first level of the hierarchy decisions by the composition of task packages groups executed during time intervals of limited duration at the second level and schedules for executing packages at the third level the hierarchy. In order to evaluate decisions on the composition of packages, the results of their execution, obtained during the specified time intervals, are distributed among the packages. The apparatus of the theory of hierarchical games is used to determine complex solutions. A model of a hierarchical game for making decisions by the compositions of packages, groups of packages and schedules of executing packages is built, which is a system of hierarchically interconnected criteria for optimizing decisions. The model registers the condition for the formation of sets from the results of the execution of task packages and restriction on duration of time intervals of its operating. The problem of determining the compositions of task packages and groups of task packages is NP-hard; therefore, its solution requires the use of approximate optimization methods. In order to optimize groups of task packages, the construction of a method for formulating initial solutions by their compositions has been implemented, which are further optimized. Moreover, a algorithm for distributing the results of executing task packages obtained during time intervals of limited duration by sets is formulated. The method of local solutions optimization by composition of packages groups, in accordance with which packages are excluded from groups, the results of which are not included in sets, and packages, that aren’t included in any group, is proposed. The software implementation of the considered method of complex optimization of the compositions of task packages, groups of task packages, and schedules for executing task packages from groups (including the implementation of the method for optimizing the compositions of groups of task packages) has been performed. With its use, studies of the features of the considered planning task are carried out. Conclusion are formulated concerning the dependence of the efficiency of scheduling the execution of task packages in multistage system under the introduced conditions from the input parameters of the problem. The use of the method of local optimization of the compositions of groups of task packages allows to increase the number of formed sets from the results of task execution in packages from groups by 60% in comparison with fixed groups (which do not imply optimization).