APPROXIMATION OF EVOLUTIONARY PROCESSES WITH DISTRIBUTED PARAMETERS ON A NETWORK
The paper considers the approximation of mathematical models of network-like evolutionary transport processes as applied to differential systems with distributed parameters on a network (graph). An approach is indicated that uses the application of the theory of classical computational methods, which consists in reducing the investigated problem to systems of algebraic equations (auxiliary finite-dimensional problems) in which the values of the grid functions at the points of partition of the edges of the graph are unknown. At the same time, there is a fairly wide opportunity for choosing different types of convergent difference schemes that are significantly different from each other: explicit difference schemes, implicit difference schemes, analogs of Krank-Nicholson difference schemes (below, in order not to load the study with technical difficulties, explicit difference schemes are used). It should be noted a characteristic feature of the studied mathematical models inherited by the rheological structure of the graph — the presence of singular points of the graph at which the differential equation is not determined (nodes or vertices of the graph) and is replaced by generalized Kirchhoff conditions. The formalisms of the latter describe the laws of continuum transfer at these points and require a separate approach to approximation issues (in the work, for the sake of simplicity, classical difference relations are used). It should also be noted that the use of an implicit difference scheme or the Crank-Nicholson scheme for approximation requires additional analysis of auxiliary finite-dimensional problems (solvability, uniform boundedness of approximations to the solution of the original problem), but it significantly increases the accuracy of calculating approximations. The use of an explicit difference scheme is freed from studying some of these issues, however (and this, when analyzing some applied problems, can be a significant obstacle to use) gives a rather large error in determining the solution to the original problem. The given particular examples of applied character illustrate the ways of numerical analysis of differential systems with carriers on an arbitrary network (graph). The results obtained are quite simply transferred to the study of wave processes and oscillation phenomena in transport processes by numerical methods.
Keywords:evolutionary transport processes on networks, approximation, difference scheme, features at network nodes, numerical methods.
PIECEWISE NEURAL MODEL BASED ON SPLIT SIGNALS FOR BERNOULLI MEMRISTORS
UDC 519.65; 621.3.01
E.B. Solovyeva, H.A. Harchuk
Actuality of the investigation theme is specified by complexity of mathematical modeling of nonlinear dynamic devices, since the analytical solutions of the nonlinear differential equation systems of high size are not always obtained, and numerical solutions are often accompanied by the problem of poor conditionality. In this situation, behavioral modeling is effective, herewith the object of investigation is represented as a “black or gray box”, and its mathematical model is constructed using the sets of the input and output signals. Behavioral modeling is important in conditions of restricted information of new elements and technologies, as well as under the complexity and variety of models built at the component level. The behavioral modeling of memristive devices actively developed using nanotechnology for energy-saving equipment is represented. A method of behavioral modeling of the transfer characteristics of memristive devices by means of piecewise neural models based on split signals is proposed. To reduce the dimension on approximating nonlinear operators and, therefore, to simplify mathematical models, are applied the following: neural networks, the signal splitting method that enables to adapt the model to the type of the input signals, and a piecewise approximation method for operators of nonlinear dynamic systems. On the basis of the proposed method, a piecewise neural model is constructed. This model includes five three-layer neural networks of simple structure (3x2x1, 100 parameters) and provides a significantly higher accuracy of modeling the transfer characteristic of memristors, the current dynamics of which are described by the Bernoulli differential equation, in comparison with the two-layer piecewise neural and piecewise polynomial models. The described results are of practical value for the behavioral modeling of memristors and various memristive devices, as well as of other nonlinear dynamic systems, since they develop a universal approach for approximating nonlinear operators based on neural networks.
Keywords:nonlinear dynamic system, mathematical modeling, nonlinear operator, nonlinear model, approximation, neural network, memristor.
COMPARISON OF THE ACCURACY OF EXPERIMENTAL DATA APPROXIMATION USING THE LEAST RELATIVE SQUARES METHOD WITH THE LEAST SQUARES METHOD
A.B. Golovanchikov, M.K Doan, A.B. Petrukhin, N.A. Merentsov
The results of comparing the accuracy of approximation of experimental or tabular data obtained using the standard method of least squares (LSM) and the proposed method of least relative squares (LRSM), for example, a given table dependence of the viscosity of a water-glycerine solution on the mass concentration of glycerol. The advantage of the latter is shown as the sum and average values of the local relative deviation of calculated data of viscosity of the desired solution obtained by LRSM, with similar data obtained by standard LSM and maximum values of these relative deviations. So, calculated using LSM average relative deviations of theoretical viscosity of an aqueous solution of glycerin from the specified table, in absolute value equal to 12.9%; LRSM of 5.8%, i.e., below 2 times. Accordingly the largest relative deviations in the LSM are 17.9%, and LRSM – 10.6 %, that is, reduced by 68%. It is proposed to determine the conditional values of parallel experiments based on the experimental data of the main experiment. To do this, the calculation of conditional numerical values of the i-th parallel experience is determined by the method of piecewise linear approximation of i-1 and i+1 numerical values of the main experience or table data. A correlation analysis is performed to determine the correlation coefficients, reproducibility, adequacy, and significance of the coefficients of the resulting regression equation.
Keywords:linearization, approximation, absolute and relative deviations, LSM and LRSM, correlation coefficient, reproducibility, adequacy, significance.
APPROXIMATION OF EVOLUTIONARY DIFFERENTIAL SYSTEMS WITH DISTRIBUTED PARAMETERS ON THE NETWORK AND MOMENT METHODS
The paper considers evolutionary problems underlying the mathematical description of oscillatory and hydrodynamic processes in network-like objects (waveguides, hydraulic networks, etc.). The main attention is paid to the analysis of the properties of the elliptic operator (the one-dimensional Laplace operator) with distributed parameters on the network, establishing the spectral completeness of the system of eigenfunctions in the class of square-integrable functions. Conditions are obtained that guarantee Neumann stability (spectral stability) of difference schemes for evolutionary problems; a solution to the moment method control problem is presented. The methods for studying evolutionary problems are based on the properties of a positive definite elliptic operator: a system of eigenfunctions forms a basis in the space of functions summable with a square; series in the system of eigenfunctions admit a priori estimates of the solutions of the evolutionary problem; approximation of an elliptic operator reduces it to a finite-dimensional operator in a finite-dimensional space of grid functions with a natural Euclidean norm, which (a finite-dimensional operator) approximates the original with any predetermined accuracy in the sense of the norm of the space of functions summable squared. For evolutionary problems, an explicit first-order approximation scheme on the graph grid (parabolic system) and an explicit second-order approximation scheme (hyperbolic system) are used. The oscillatory properties of the obtained operators are established, similar to the classical oscillatory properties. For difference schemes of parabolic and hyperbolic systems of equations, conditions are obtained that guarantee countable spectral stability (stability in the sense of Neumann) and, therefore, the possibility of obtaining analogues of A.F. Filippova on the convergence of difference schemes in terms of approximation steps of a graph grid. To illustrate the applicability of the approach used, the control problem is considered – the translation of evolutionary systems of parabolic and hyperbolic types from given initial to given final states; conditions are obtained that guarantee the controllability of the systems under study.
Keywords: laplace operator on a graph, evolution problems, approximation, difference schemes, stability, convergence, method of moments.
MULTIOBJECTIVE OPTIMIZATION OF BAND-PASS FILTERS CHARACTERISTICS USING HEURISTIC ALGORITHM
The problem of multiobjective optimization of band pass filter (BPF) is investigated. There were three objectives considered: a nonuniformity of attenuation in the pass band, a minimum attenuation in the stop band and a nonuniformity of delay time in the pass band. The first method of solution was to derive Pareto-optimal approximations for BPF from such for low pass filter (LPF) by means of reactance frequency transform. The second method was to search optimal approximation for BPF directly. In the both methods the heuristic algorithm was applied. This algorithm searches out the set of Pareto optimal solutions and uses multistart for searching of global extreme for each point of Pareto front. The numerical experiments showed that for cases with wide pass band of BPF and comparatively poor-quality indexes of gain frequency response the second method can give lower nonuniformity of delay time than the first one when other objectives are the same. This result means that Pareto-optimality does not persist in the frequency transform in such cases. At once in the cases with narrow pass band of BPF or comparatively good quality indexes of gain frequency response the direct search provides no preferences.
Keywords: :transfer function, band pass filter, approximation, Pareto optimality, heuristic algorithm.
PREDICTION OF EYE-DIAGRAM PARAMETERS FROM TRANSIENT AND GAIN-FREQUENCY CHARACTERISTICS USING NEURAL NETWORK
A capability of prediction of the eye-diagram width and height with using artificial neural network (ANN) was investigated. For this purpose, were simulated more than 750 examples of telecommunication channels with different transfer functions. Eye-diagrams were composed for all examples by means of convolution of random pulse sequence and pulse response and parameters of these eye-diagrams were measured. Some ANN was learned. Their input variables were transient characteristic delay time, raise time, magnitude of voltage peak and oscillation duration as well as a gain value at the half of clock rate. For each of predicted parameters distinct ANN was chosen for different ranges of input variables. Root mean square errors of eye-diagram parameters prediction using these ANN were in the range of 2 – 4%. Correlation coefficient of predicted and known values was more then 0,98. Sufficient decreasing of computational time is achieved compare with estimation of the eye width and height using eye-diagram modeling. This method can be used for optimization of communication channel characteristics when eye-diagram parameters are the components of the goal function.
Keywords: eye-diagram, transient characteristic, gain-frequency characteristic, neural network, approximation.
THE FORECASTING PARAMETERS OF A DISTRIBUTED SYSTEM OF PRODUCTION RESOURCES
In this paper we consider the problem of formation of production resources in a distributed system. As a forecasting tool approximated the real values of the parameters of the eight versions of the tests (baseline values) in each series of tests as means of checking the optimality of the choice of method of approximation, the calculated relative forecast errors (for ninth, tenth and eleventh variants of the tests and test values). By type of location parameter values in the coordinate system with the x – numbered versions of the tests made the assumption that for forecasting of change of parameters in the first and the third task can be used linear approximation, while the second task of the hyperbolic approximation. The relative approximation errors at the forecast test values is equal to not more than 7% for the first task, 0% for the second task, and no more than 2% for the third problem.
Keywords: distributed, online, approximation, prediction.
THE ANALYSIS OF THE COVERAGE IN COMMUNICATION SYSTEMS BASED ON GEOMETRICAL ALGORITHMS
A. P. Preobrazhensky
This paper discusses the issues related to improvement of radio coverage in the communication systems. It is proposed to reduce this problem to geometric to analyze whether a given point to the polygon, where the polygon represents the coverage area of certain base station. We consider several algorithms. In the method of ray tracing produced by a ray of given points in certain directions and counting the number of times the beam is the intersection of edges of the polygon. Also considered trigonometric algorithm when the points hold the rays for all the vertices of the polygons. The analysis rpm which is oriented boundary of the polygon around a point.
Keywords: communication, radio coverage, algorithm, base station, polygon, approximation.
THE STUDY OF THE CHARACTERISTICS OF PROPAGATION OF ELECTROMAGNETIC WAVES IN THE INTERNAL AREAS INDOORS
E.S.Zatsepin, A.G.Sklyar, D.V.Rusanov
In the paper we consider the problem of propagation of electromagnetic waves in the internal area of the premises. Based on the analysis of possible approaches to solving this task the experimental method was chosen. The results obtained for different variants of propagation of electromagnetic waves through different environments were obtained. The structure of subsystem of data processing in the study of the propagation of the radio waves in the room was proposed.
Keywords: : wireless communication, experiment, method, optimization, method, coverage, approximation.