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The paper presents a numerical solution to the heat wave motion problem for a degenerate second-order nonlinear parabolic equation with a source term. The nonlinearity is conditioned by the power dependence of the heat conduction coefficient on temperature. The problem for the case of two spatial variables is considered with the boundary condition specifying the heat wave motion law. A new solution algorithm based on an expansion in radial basis functions and the boundary element method is proposed. The solution is constructed stepwise in time with finite difference time approximation. At each time step, a boundary value problem for the Poisson equation corresponding to the original equation at a fixed time is solved. The solution to this problem is constructed iteratively as the sum of a particular solution to the nonhomogeneous equation and a solution to the corresponding homogeneous equation satisfying the boundary conditions. The homogeneous equation is solved by the boundary element method. The particular solution is sought by the collocation method using inhomogeneity expansion in radial basis functions. The calculation algorithm is optimized by parallelizing the computations. The algorithm is implemented as a program written in the C++ language. The parallel computations are organized by using the OpenCL standard, and this allows one to run the same parallel code either on multi-core CPUs or on graphic CPUs. Test cases are solved to evaluate the effectiveness of the proposed solution method and the correctness of the developed computational technique. The calculation results are compared with known exact solutions, as well as with the results we obtained earlier. The accuracy of the solutions and the calculation time are estimated. The effectiveness of using various systems of radial basis functions to solve the problems under study is analyzed. The most suitable system of functions is selected. The implemented complex computational experiment shows higher calculation accuracy of the proposed new algorithm than that of the previously developed one.

A direct algorithm for solving a linear programming problem (LP), given in canonical form, is considered. The algorithm consists of two successive stages, in which the following LP problems are solved by a direct method: a non-degenerate auxiliary problem at the first stage and some problem equivalent to the original one at the second. The construction of the auxiliary problem is based on a multiplicative version of the Gaussian exclusion method, in the very structure of which there are possibilities: identification of incompatibility and linear dependence of constraints; identification of variables whose optimal values are obviously zero; the actual exclusion of direct variables and the reduction of the dimension of the space in which the solution of the original problem is determined. In the process of actual exclusion of variables, the algorithm generates a sequence of multipliers, the main rows of which form a matrix of constraints of the auxiliary problem, and the possibility of minimizing the filling of the main rows of multipliers is inherent in the very structure of direct methods. At the same time, there is no need to transfer information (basis, plan and optimal value of the objective function) to the second stage of the algorithm and apply one of the ways to eliminate looping to guarantee final convergence.

Two variants of the algorithm for solving the auxiliary problem in conjugate canonical form are presented. The first one is based on its solution by a direct algorithm in terms of the simplex method, and the second one is based on solving a problem dual to it by the simplex method. It is shown that both variants of the algorithm for the same initial data (inputs) generate the same sequence of points: the basic solution and the current dual solution of the vector of row estimates. Hence, it is concluded that the direct algorithm is an algorithm of the simplex method type. It is also shown that the comparison of numerical schemes leads to the conclusion that the direct algorithm allows to reduce, according to the cubic law, the number of arithmetic operations necessary to solve the auxiliary problem, compared with the simplex method. An estimate of the number of iterations is given.

This paper presents algorithm for modeling set of trajectories of non-stationary time series, based on a numerical scheme for approximating the sample density of the distribution function in a problem with fixed ends, when the initial distribution for a given number of steps transforms into a certain final distribution, so that at each step the semigroup property of solving the Liouville equation is satisfied. The model makes it possible to numerically construct evolving densities of distribution functions during random switching of states of the system generating the original time series.

The main problem is related to the fact that with the numerical implementation of the left-hand differential derivative in time, the solution becomes unstable, but such approach corresponds to the modeling of evolution. An integrative approach is used while choosing implicit stable schemes with “going into the future”, this does not match the semigroup property at each step. If, on the other hand, some real process is being modeled, in which goal-setting presumably takes place, then it is desirable to use schemes that generate a model of the transition process. Such model is used in the future in order to build a predictor of the disorder, which will allow you to determine exactly what state the process under study is going into, before the process really went into it. The model described in the article can be used as a tool for modeling real non-stationary time series.

Steps of the modeling scheme are described further. Fragments corresponding to certain states are selected from a given time series, for example, trends with specified slope angles and variances. Reference distributions of states are compiled from these fragments. Then the empirical distributions of the duration of the system’s stay in the specified states and the duration of the transition time from state to state are determined. In accordance with these empirical distributions, a probabilistic model of the disorder is constructed and the corresponding trajectories of the time series are modeled.

In recent years, the use of neural network models for solving aerodynamics problems has become widespread. These models, trained on a set of previously obtained solutions, predict solutions to new problems. They are, in essence, interpolation algorithms. An alternative approach is to construct a neural network operator. This is a neural network that reproduces a numerical method used to solve a problem. It allows to find the solution in iterations. The paper considers the construction of such an operator using the UNet neural network with a spatial attention mechanism. It solves flow problems on a rectangular uniform grid that is common to a streamlined body and flow field. A correction mechanism is proposed to clarify the obtained solution. The problem of the stability of such an algorithm for solving a stationary problem is analyzed, and a comparison is made with other variants of its construction, including pushforward trick and positional encoding. The issue of selecting a set of iterations for forming a train dataset is considered, and the behavior of the solution is assessed using repeated use of a neural network operator.

A demonstration of the method is provided for the case of flow around a rounded plate with a turbulent flow, with various options for rounding, for fixed parameters of the incoming flow, with Reynolds number $\text{Re} = 10^5$ and Mach number $M = 0.15$. Since flows with these parameters of the incoming flow can be considered incompressible, only velocity components are directly studied. At the same time, the neural network model used to construct the operator has a common decoder for both velocity components. Comparison of flow fields and velocity profiles along the normal and outline of the body, obtained using a neural network operator and numerical methods, is carried out. Analysis is performed both on the plate and rounding. Simulation results confirm that the neural network operator allows finding a solution with high accuracy and stability.

Conversion of numeric data to the spiking form and information losses in this process are serious problems limiting usage of spiking neural networks in applied informational systems. While physical values are represented by numbers, internal representation of information inside spiking neural networks is based on spikes — elementary objects emitted and processed by neurons. This problem is especially hard in the reinforcement learning applications where an agent should learn to behave in the dynamic real world because beside the accuracy of the encoding method, its dynamic characteristics should be considered as well. The encoding algorithm based on the Gaussian receptive fields (GRF) is frequently used. In this method, one numeric variable fed to the network is represented by spike streams emitted by a certain set of network input nodes. The spike frequency in each stream is determined by proximity of the current variable value to the center of the receptive field corresponding to the given input node. In the standard GRF algorithm, the receptive field centers are placed equidistantly. However, it is inefficient in the case of very uneven distribution of the variable encoded. In the present paper, an improved version of this method is proposed which is based on adaptive selection of the Gaussian centers and spike stream frequencies. This improved GRF algorithm is compared with its standard version in terms of amount of information lost in the coding process and of accuracy of classification models built on spike-encoded data. The fraction of information retained in the process of the standard and adaptive GRF encoding is estimated using the direct and reverse encoding procedures applied to a large sample from the triangular probability distribution and counting coinciding bits in the original and restored samples. The comparison based on classification was performed on a task of evaluation of current state in reinforcement learning. For this purpose, the classification models were created by machine learning algorithms of very different nature — nearest neighbors algorithm, random forest and multi-layer perceptron. Superiority of our approach is demonstrated on all these tests.

The paper considers the problem of parameter identification of discrete-time linear stochastic systems in the state space with additive and multiplicative noise. It is assumed that the state and measurements equations of a discrete-time linear stochastic system depend on an unknown parameter to be identified.

A new approach to the construction of gradient parameter identification methods in the class of discrete-time linear stochastic systems with additive and multiplicative noise is presented, based on the application of modified weighted Gram – Schmidt orthogonalization (MWGS) and the discrete-time information-type filtering algorithms.

The main theoretical results of this research include: 1) a new identification criterion in terms of an extended information filter; 2) a new algorithm for calculating derivatives with respect to an uncertainty parameter in a discrete-time linear stochastic system based on an extended information LD filter using the direct procedure of modified weighted Gram – Schmidt orthogonalization; and 3) a new method for calculating the gradient of identification criteria using a “differentiated” extended information LD filter.

The advantages of this approach are that it uses MWGS orthogonalization which is numerically stable against machine roundoff errors, and it forms the basis of all the developed methods and algorithms. The information LD-filter maintains the symmetry and positive definiteness of the information matrices. The algorithms have an array structure that is convenient for computer implementation.

All the developed algorithms were implemented in MATLAB. A series of numerical experiments were carried out. The results obtained demonstrated the operability of the proposed approach, using the example of solving the problem of parameter identification for a mathematical model of a complex mechanical system.

The results can be used to develop methods for identifying parameters in mathematical models that are represented in state space by discrete-time linear stochastic systems with additive and multiplicative noise.

We developed an algorithm for fast simulation of VLSI CMOS (Very Large Scale Integration with Complementary Metal-Oxide-Semiconductors) with an accuracy control. The algorithm provides an ability of parallel numerical experiments in multiprocessor computational environment. There is computation speed up by means of block-matrix and structural (DCCC) decompositions application. A feature of the approach is both in a choice of moments and ways of parameters synchronization and application of multi-rate integration methods. Due to this fact we have ability to estimate and control error of given characteristics.

The stationary flame spread rate has been calculated using the relationship based on the thermodynamic variational principle. It has been shown that proposed numerical algorithm provides the stable convergence under any initial approximation, which could be noticeably far from the searched solution.

We consider a numerically stable direct multiplicative algorithm of solving linear equations systems, which takes into account the sparseness of matrices presented in a packed form. The advantage of the algorithm is the ability to minimize the filling of the main rows of multipliers without losing the accuracy of the results. Moreover, changes in the position of the next processed row of the matrix are not made, what allows using static data storage formats. Linear system solving by a direct multiplicative algorithm is, like the solving with $LU$-decomposition, just another scheme of the Gaussian elimination method implementation.

In this paper, this algorithm is the basis for solving the following problems:

Problem 1. Setting the descent direction in Newtonian methods of unconditional optimization by integrating one of the known techniques of constructing an essentially positive definite matrix. This approach allows us to weaken or remove additional specific difficulties caused by the need to solve large equation systems with sparse matrices presented in a packed form.

Problem 2. Construction of a new mathematical formulation of the problem of quadratic programming and a new form of specifying necessary and sufficient optimality conditions. They are quite simple and can be used to construct mathematical programming methods, for example, to find the minimum of a quadratic function on a polyhedral set of constraints, based on solving linear equations systems, which dimension is not higher than the number of variables of the objective function.

Problem 3. Construction of a continuous analogue of the problem of minimizing a real quadratic polynomial in Boolean variables and a new form of defining necessary and sufficient conditions of optimality for the development of methods for solving them in polynomial time. As a result, the original problem is reduced to the problem of finding the minimum distance between the origin and the angular point of a convex polyhedron, which is a perturbation of the $n$-dimensional cube and is described by a system of double linear inequalities with an upper triangular matrix of coefficients with units on the main diagonal. Only two faces are subject to investigation, one of which or both contains the vertices closest to the origin. To calculate them, it is sufficient to solve $4n – 4$ linear equations systems and choose among them all the nearest equidistant vertices in polynomial time. The problem of minimizing a quadratic polynomial is $NP$-hard, since an $NP$-hard problem about a vertex covering for an arbitrary graph comes down to it. It follows therefrom that $P = NP$, which is based on the development beyond the limits of integer optimization methods.

A numerically stable direct multiplicative method for solving systems of linear equations that takes into account the sparseness of matrices presented in a packed form is considered. The advantage of the method is the calculation of the Cholesky factors for a positive definite matrix of the system of equations and its solution within the framework of one procedure. And also in the possibility of minimizing the filling of the main rows of multipliers without losing the accuracy of the results, and no changes are made to the position of the next processed row of the matrix, which allows using static data storage formats. The solution of the system of linear equations by a direct multiplicative algorithm is, like the solution with LU-decomposition, just another scheme for implementing the Gaussian elimination method.

The calculation of the Cholesky factors for a positive definite matrix of the system and its solution underlies the construction of a new mathematical formulation of the unconditional problem of quadratic programming and a new form of specifying necessary and sufficient conditions for optimality that are quite simple and are used in this paper to construct a new mathematical formulation for the problem of quadratic programming on a polyhedral set of constraints, which is the problem of finding the minimum distance between the origin ordinate and polyhedral boundary by means of a set of constraints and linear algebra dimensional geometry.

To determine the distance, it is proposed to apply the known exact method based on solving systems of linear equations whose dimension is not higher than the number of variables of the objective function. The distances are determined by the construction of perpendiculars to the faces of a polyhedron of different dimensions. To reduce the number of faces examined, the proposed method involves a special order of sorting the faces. Only the faces containing the vertex closest to the point of the unconditional extremum and visible from this point are subject to investigation. In the case of the presence of several nearest equidistant vertices, we investigate a face containing all these vertices and faces of smaller dimension that have at least two common nearest vertices with the first face.