The present paper is dedicated to a variational method for the construction of optimal quadrature formulas in the sense of Sard in the Hilbert space W˜2(m,m−1) of complex-valued and periodic ...functions. In this, the coefficients of the optimal quadrature formula are found separately in the case ωh is integer and non-integer cases. In addition, using the constructed optimal quadrature formula, the numerical results of exponentially weighted integrals of certain functions in the case m=2 is presented. The numerical results show that the order of convergence of the optimal quadrature formula is O1N+|ω|2 in the space W˜2(2,1).
In this paper we obtain new results on positive quadrature formulas with prescribed nodes for the approximation of integrals with respect to a positive measure supported on the unit circle.
We revise ...Szegő–Lobatto rules and we present a characterization of their existence. In particular, when the measure on the unit circle is symmetric, this characterization can be used to recover, in a more elementary way, a recent characterization result for the existence of positive quasi Gauss, quasi Radau and quasi Lobatto rules (quasi Gauss-type), due to B. Beckermann et. al. Some illustrative numerical examples are finally carried out in order to show the powerfulness of our results.
In this article, a composite optimal quadrature formula is constructed for an approximate analytical solution of the generalized integral Abel equation in the Sobolev functional space. The optimal ...coefficients of this quadrature formula have been found. In addition, using the constructed composite optimal quadrature formula, numerical results of examples of the generalized Abel integral equation were obtained and compared with the exact solution.
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. Keywords: function approximation, Fourier series, Fourier series coefficients, quadrature formulas, approximation error.
In this article, we study bivariate polynomial interpolation on the node points of degenerate Lissajous figures. These node points form Chebyshev lattices of rank 1 and are generalizations of the ...well-known Padua points. We show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the Lagrange basis polynomials. Further, we prove mean and uniform convergence of the interpolating schemes. For the uniform convergence the growth of the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature.
The computation of matrix functions using quadrature formulas and rational approximations of very large structured matrices using tensor trains (TT), and quantized tensor trains (QTT) is considered ...here. The focus is on matrices with a small TT/QTT rank. Some analysis of the error produced by the use of the TT/QTT representation and the underlying approximation formula used is also provided. Promising experiments on exponential, power, Mittag-Leffler and logarithm function of multilevel Toeplitz matrices, that are among those which generate a low TT/QTT rank representation, are also provided, confirming that the proposed approach is feasible.
In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural ...Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the importance of the inf-sup condition. Our results suggest, somehow counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of suitably high precision.
In this paper, we study the construction of quadrature rules for the approximation of hypersingular integrals that occur when 2D Neumann or mixed Laplace problems are numerically solved using ...Boundary Element Methods. In particular the Galerkin discretization is considered within the Isogeometric Analysis setting and spline quasi-interpolation is applied to approximate integrand factors, then integrals are evaluated via recurrence relations. Convergence results of the proposed quadrature rules are given, with respect to both smooth and non smooth integrands. Numerical tests confirm the behavior predicted by the analysis. Finally, several numerical experiments related to the application of the quadrature rules to both exterior and interior differential problems are presented.
•We introduce new quadrature rules based on quasi interpolation for hypersingular integrals.•We give proofs of the convergence properties of the quadrature with different smoothness hypotheses on the functions.•We perform numerical tests for quadrature itself and for the quadrature applied to IgA-BEM discretization of differential problems.•All results reveal that the new quadratures are accurate and reliable.
The development of effective methods of approximate calculation of integrals using optimal cubature formulas and optimal quadrature formulas with trigonometric weights for defined functions on a ...sphere, the creation of new algorithms for approximate calculation of trigonometric weighted integrals in different classes of functions, as well as the assessment of their errors, is the goal of this work.
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