Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

Error estimates of gaussian-type quadrature formulae for analytic functions on ellipses-a survey of recent results

This paper presents a survey of recent results on error estimates of Gaussian-type quadrature formulas for analytic functions on confocal ellipses

Solution of second kind Fredholm integral equations by means of Gauss and anti-Gauss quadrature rules

This paper is concerned with the numerical approximation of Fredholm integral equa- tions of the second kind. A Nyström method based on the anti-Gauss quadrature formula is developed and investigated in terms of stability and convergence in appro- priate weighted spaces. The Nyström interpolants corresponding to the Gauss and the anti-Gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation, under suitable assumptions which are easily verified for a particular weight function. Hence, an error estimate is available, and the accuracy of the solution can be improved by approximating it by an averaged Nyström interpolant. The effectiveness of the proposed approach is illustrated through different numerical tests

Averaged Nyström interpolants for the solution of Fredholm integral equations of the second kind

Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. This paper discusses Nyström methods based on Gauss quadrature rules for the solution of such integral equations. It is important to be able to estimate the error in the computed solution, because this allows the choice of an appropriate number of nodes in the Gauss quadrature rule used. This paper explores the application of averaged and weighted averaged Gauss quadrature rules for this purpose and introduces new stability properties of them

On a spectral theorem in paraorthogonality theory

Motivated by the works of Delsarte and Genin (1988, 1991), who studied paraorthogonal polynomials associated with positive definite Hermitian linear functionals and their corresponding recurrence relations, we provide paraorthogonality theory, in the context of quasidefinite Hermitian linear functionals, with a recurrence relation and the analogous result to the classical Favard's theorem or spectral theorem. As an application of our results, we prove that for any two monic polynomials whose zeros are simple and strictly interlacing on the unit circle, with the possible exception of one of them which could be common, there exists a sequence of paraorthogonal polynomials such that these polynomials belong to it. Furthermore, an application to the computation of Szegő quadrature formulas is also discussed.The authors thank the referee for her/his valuable suggestions and comments which have contributed to improve the final form of this paper. The research of the first author is supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/101139/2014 and partially supported by the Brazilian Government through the CNPq under the project 470019/2013-1 and the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2012–36732–C03–01. The work of the second and third authors is partially supported by Dirección General de Programas y Transferencia de Conocimiento, Ministerio de Ciencia e Innovación of Spain under the project MTM2011–28781

Analysis of the tilted flat punch in couple-stress elasticity

In the present paper we explore the response of a half-plane indented by a tilted flat punch with sharp corners in the context of couple-stress elasticity theory. Contact conditions arise in a number of modern engineering applications ranging from structural and geotechnical engineering to micro and nanotechnology. As the contact scales reduce progressively the effects of the microstructure upon the macroscopic material response cannot be ignored. The generalized continuum theory of couple-stress elasticity introduces characteristic material lengths in order to describe the pertinent scale effects that emerge from the underlying material microstructure. The problem under investigation is interesting for three reasons: Firstly, the indentor's geometry is simple so that benchmark results may be extracted. Secondly, important deterioration of the macroscopic results may emerge in the case that a tilting moment is applied on the indentor inadvertently or in the case that the flat punch itself is not self-aligning so that asymmetrical contact pressure distributions arise on the contact faces. Thirdly, the voluntary application of a tilting moment on the flat punch during an experiment gives rise to potential capabilities of the flat punch for the determination of the material microstructural characteristic lengths. The solution methodology is based on singular integral equations which have resulted from a treatment of the mixed boundary value problem via integral transforms and generalized functions. The results show significant departure from the predictions of classical elasticity revealing that valuable information may be deducted from the indentation of a tilted punch of a microstructured solid

Approximation of smooth functions on compact two-point homogeneous spaces

Estimates of Kolmogorov $n$-widths $d_n(B_p^r, L^q)$ and linear $n$-widths \da_n(B_p^r, L^q), ($1\leq q\leq \infty$) of Sobolev's classes $B_p^r$, ($r>0$, $1\leq p\leq \infty$) on compact two-point homogeneous spaces (CTPHS) are established. For part of $(p, q)\in[1,\infty]\times[1,\infty]$, sharp orders of $d_n(B_p^r, L^q)$ or \da_n (B_p^r, L^q) were obtained by Bordin, Kushpel, Levesley and Tozoni in a recent paper `` J. Funct. Anal. 202 (2) (2003), 307--326''. In this paper, we obtain the sharp orders of $d_n(B_p^r, L^q)$ and \da_n (B_p^r, L^q) for all the remaining $(p,q)$. Our proof is based on positive cubature formulas and Marcinkiewicz-Zygmund type inequalities on CTPHS

B-Spline based uncertainty quantification for stochastic analysis

The consideration of uncertainties has become inevitable in state-of-the-art science and technology. Research in the field of uncertainty quantification has gained much importance in the last decades. The main focus of scientists is the identification of uncertain sources, the determination and hierarchization of uncertainties, and the investigation of their influences on system responses. Polynomial chaos expansion, among others, is suitable for this purpose, and has asserted itself as a versatile and powerful tool in various applications. In the last years, its combination with any kind of dimension reduction methods has been intensively pursued, providing support for the processing of high-dimensional input variables up to now. Indeed, this is also referred to as the curse of dimensionality and its abolishment would be considered as a milestone in uncertainty quantification. At this point, the present thesis starts and investigates spline spaces, as a natural extension of polynomials, in the field of uncertainty quantification. The newly developed method 'spline chaos', aims to employ the more complex, but thereby more flexible, structure of splines to counter harder real-world applications where polynomial chaos fails. Ordinarily, the bases of polynomial chaos expansions are orthogonal polynomials, which are replaced by B-spline basis functions in this work. Convergence of the new method is proved and emphasized by numerical examples, which are extended to an accuracy analysis with multi-dimensional input. Moreover, by solving several stochastic differential equations, it is shown that the spline chaos is a generalization of multi-element Legendre chaos and superior to it. Finally, the spline chaos accounts for solving partial differential equations and results in a stochastic Galerkin isogeometric analysis that contributes to the efficient uncertainty quantification of elliptic partial differential equations. A general framework in combination with an a priori error estimation of the expected solution is provided

Nonlinear Quantizer Design Based on Clenshaw-Curtis Quadrature

Trabalho de Conclusão de Curso (graduação)—Universidade de Brasília, Faculdade de Tecnologia, Departamento de Engenharia Elétrica, 2019.Esta tese visa propor um novo método para projeto de quantizadores não lineares conservadores de momentos estatísticos, baseado na quadratura de Clenshaw-Curtis. Os conceitos básicos de Conversores Analógico Digital são definidos para contextualização do problema discutido e para servir de base para o entendimento dos parâmetros de quantizadores. Então, uma definição formal da Transformada da Incerteza - Unscented Transform (UT) - é proposta para o contexto deste trabalho, e os conceitos básicos de quadratura são aplicados como uma ferramenta matemática para cálculo da UT. Finalmente, a metodologia de projeto do quantizador é detalhada, apresentando a relação entre os nós e pesos de uma quadratura com os parâmetros de quantizadores. O projeto é então aplicado a uma simulação de estudo de caso para verificação dos cálculos teóricos.This thesis aims to provide a novel method for designing nonlinear moment preserving quantizers based on the Clenshaw-Curtis quadrature. The basic concepts of Analog-to-Digital Converters (ADCs) are defined for contextualization of the discussed problem and to serve as a basis for understanding quantizers parameters. Then, a formal definition of the Unscented Transform (UT) is proposed for this work’s context, and the key concepts of quadrature are applied to it as a mathematical tool for UT calculation. Finally, the design method is detailed, presenting the relationship between quadrature’s nodes and weights and the quantizers parameters. This design is applied to a case study simulation, for validation of theoretical calculations