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Introduction. The problem of approximation can be considered as the basis of computational methods, namely, the approximation of individual functions or classes of functions by functions that are in some sense simpler than the functions being approximated. Most often, the role of an approximant is played by a set of algebraic polynomials or (in the case of a periodic function) a set of trigonometric polynomials of a given order. The ideas and methods of approximation theory are used in various fields of science, especially applied areas, since tasks related to the need to replace one object with another, close in one sense or another to the first, but easier to study, arise very often. The purpose of the paper is consider the problems of approximation of a function, which is given by its values in a certain set of nodal points on a certain interval and belongs to a certain class of functions by trigonometric Fourier series, using the quadrature formulas for calculating integrals of fast oscillating functions on this class of functions, which are optimal in accuracy and close to them. The main attention is paid to the study of the sources of error of the proposed approach to function approximation. Results. Effective approximation algorithms from classes of differentiable functions with the help of Fourier series are proposed, using the Fourier coefficients optimal in accuracy and close to them on the given classes of quadrature formulas for calculating integrals of fast-oscillating functions to determine the Fourier coefficients. The error estimates of the proposed approximation algorithms using the quadrature formulas for calculating the Fourier coefficients of the optimal accuracy and close to them for calculating integrals of fast-oscillating functions from classes of differential functions with given values at the nodes of a fixed grid are presented. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of these classes are given. Conclusions. Efficient by precision algorithms for approximating functions from classes of differentiable functions by means of Fourier series are constructed using the optimal accuracy and close to them quadrature formulas for calculating integrals of fast-oscillating functions from the above classes of functions to calculate the Fourier coefficients. A comprehensive analysis of the quality of the constructed algorithms for approximating functions by finite sums of the Fourier series is carried out.

Introduction. Function approximation (approximation or restoration) is widely used in data analysis, model building, and forecasting. The goal of function approximation is to find the function that best approximates the original function. This can be useful when the original function is too complex to analyze or when a model needs to be simplified for more efficient computation or interpretation. Function approximation is an important tool in science, engineering, economics, and other fields where data analysis and modeling are required. It allows you to simplify complex functions, identify patterns in the behavior of the object of study, and predict the value of a function beyond the available data. The purpose of the paper is consider the problems of approximation of a function, which on some interval is given by its values in some set of nodal points and belongs to some class of functions by trigonometric Fourier series with a given accuracy and at fulfillment of given constraints on its execution time. The main attention is paid to obtaining estimates of computational complexity (implementation time) and solving the problem of function approximation by Fourier series with a given or maximum possible accuracy using efficient algorithms for solving optimization problems. Results. The general formulation of the problem of approximation of functions by Fourier series in accordance with the technology of solving problems of computational and applied mathematics with specified values of quality characteristics is presented. Estimates of the error of the proposed approximation algorithms using for the computation of Fourier coefficients the optimal in accuracy and close to them quadrature formulas for the computation of integrals from rapidly oscillating functions of the classes of Helder and Lipschitz with given fixed values in the nodes of a fixed grid are given. The corresponding quadrature formulas and constructive estimates of the error of the method of approximation of functions of the specified classes are given. Estimates of computational complexity of the given algorithms are obtained, which allow us to set real constraints on the time of algorithm implementation with a given or maximum possible accuracy. Conclusions. A comprehensive analysis of the quality of the considered algorithms for the approximation of functions by Fourier series using the accuracy-optimal (or close to them) quadrature formulas for the computation of Fourier coefficients for the computation of integrals from rapidly oscillating functions is presented. The estimates of their main characteristics – accuracy and computational complexity – are obtained.

As one of the most effective privacy protection mechanisms,differential privacy has been widely used in many fields.The existing histogram publishing methods for either static data set or dynamic data set mainly protect the privacy of sliding windows in data streams by adding unified noise.This leads to low data availability,high time complexity and weak privacy protection in their practical applications.In this paper,we tackle this problem by integrating the approximate counting techniques into the differential privacy and proposing an adaptive histogram publishing method for sliding window(APS).Firstly,the proposed APS predicates the distributional information of the sliding windows in the data stream by using an approximate counting method.Secondly,it computes an appropriate value suitable for publishing by checking the difference between estimated values and actual values.Finally,it reduces statistical errors within each interval by clustering.Theoretical analysis shows that the APS algorithm can effectively improve data availability and reduce running time while reducing the privacy budget.Experimental results on two different real data sets also verify the superiority of APS algorithm over existing grouping-based histogram publishing algorithms in terms of data availability and running time.

A renewed interest in phase frame analysis of distribution systems has surfaced in recent literature due to rapid expansion of grid-integrated distributed generation and non-linear loads, whose grid imbalance-increasing effects require more detailed analysis and modeling of distribution systems for their proper assessment. In this paper, under motivation of such recent usage of phase frame components, we propose analytical equations for N-phase neutral-equipped line admittance matrix under transposition assumption. By using the proposed equations, we evaluate precision losses caused by using the transposition assumption for modeling three-phase four-wire distribution lines. Moreover, differently from previous works, we validate the transposition assumption and the admittance matrix values obtained for various combinations of cable type and line geometry. A relevant reason for analyzing transposition assumption in more detail is the fact that it may be useful in distribution system computations, due to its advantageous decoupling of symmetrical components. In this sense, the present work provides additional discussion and illustrative evaluations of the line transposition assumption, which may be useful for analyzing its applicability under different circumstances.

This paper presents a new nonlinear projection based model reduction using convolutional Variational AutoEncoders (VAEs). This framework is applied on transient incompressible flows. The accuracy is obtained thanks to the expression of the velocity and pressure fields in a nonlinear manifold maximising the likelihood on pre-computed data in the offline stage. A confidence interval is obtained for each time instant thanks to the definition of the reduced dynamic coefficients as independent random variables for which the posterior probability given the offline data is known. The parameters of the nonlinear manifold are optimized as the ones of the decoder layers of an autoencoder. The parameters of the conditional posterior probability of the reduced coefficients are the ones of the encoder layers of the same autoencoder. The optimization of both sets of the encoder and the decoder parameters is obtained thanks to the application of a variational Bayesian method, leading to variational autoencoders. This Reduced Order Model (ROM) is not a regression model over the offline pre-computed data. The numerical resolution of the ROM is based on the Chorin projection method. We apply this new nonlinear projection-based Reduced Order Modeling (ROM) for a 2D Karman Vortex street flow and a 3D incompressible and unsteady flow in an aeronautical injection system.

Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and the Standard Leapfrog scheme with 1/3 weight coefficient. The weight coefficients are obtained as a result of solving the problem of minimizing the approximation error. Numerical experiments show the advantage of the developed scheme in comparison with other modifications of the Upwind Leapfrog scheme in the case when the convective transport prevails over the diffusion one. The proposed difference scheme solves transport problems more effectively than classical difference schemes in the case when the Péclet number falls in the range from 2 to 20. It follows that the considered difference scheme allows hydrodynamic problems to be solved in regions of complex shape effectively.

Although differential evolution (DE) algorithms perform well on a large variety of complicated optimization problems, only a few theoretical studies are focused on the working principle of DE algorithms. To make the first attempt to reveal the function of binomial crossover, this paper aims to answer whether it can reduce the approximation error of evolutionary algorithms. By investigating the expected approximation error and the probability of not finding the optimum, we conduct a case study comparing two evolutionary algorithms with and without binomial crossover on two classical benchmark problems: OneMax and Deceptive. It is proven that using binomial crossover leads to the dominance of transition matrices. As a result, the algorithm with binomial crossover asymptotically outperforms that without crossover on both OneMax and Deceptive, and outperforms on OneMax, however, not on Deceptive. Furthermore, an adaptive parameter strategy is proposed which can strengthen the superiority of binomial crossover on Deceptive.

When the multi-receiver synthetic aperture sonar (SAS) works with a wide-bandwidth signal, the performance of the range-Doppler (R-D) algorithm is seriously affected by two approximation errors, i.e., point target reference spectrum (PTRS) error and residual quadratic coupling error. The former is generated by approximating the PTRS with the second-order term in terms of the instantaneous frequency. The latter is caused by neglecting the cross-track variance of secondary range compression (SRC). In order to improve the imaging performance in the case of wide-bandwidth signals, an improved R-D algorithm is proposed in this paper. With our method, the multi-receiver SAS data is first preprocessed based on the phase center approximation (PCA) method, and the monostatic equivalent data are obtained. Then several sub-blocks are generated in the cross-track dimension. Within each sub-block, the PTRS error and residual quadratic coupling error based on the center range of each sub-block are compensated. After this operation, all sub-blocks are coerced into a new signal, which is free of both approximation errors. Consequently, this new data is used as the input of the traditional R-D algorithm. The processing results of simulated data and real data show that the traditional R-D algorithm is just suitable for an SAS system with a narrow-bandwidth signal. The imaging performance would be seriously distorted when it is applied to an SAS system with a wide-bandwidth signal. Based on the presented method, the SAS data in both cases can be well processed. The imaging performance of the presented method is nearly identical to that of the back-projection (BP) algorithm.

In this paper, a direct adaptive fuzzy control using a supervisory compensator is designed for the robust tracking of a MEMS gyroscope sensor. The parameters of the membership functions are adjusted according to the designed adaptive law for the purpose of tracking a reference trajectory. A fuzzy controller that can approximate the unknown nonlinear function and compensate the system