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On the resolution power of Fourier extensions for oscillatory functions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 260卷 312-336页

作者： Adcock, Ben Huybrechs, Daan Purdue Univ Dept Math W Lafayette IN 47907 USA Katholieke Univ Leuven Dept Comp Sci Louvain Belgium

functions that are smooth but non-periodic on a certain interval possess Fourier series that lack uniform convergence and suffer from the Gibbs phenomenon. However, they can be represented accurately by a Fourier series that is periodic on a larger interval. This is commonly called a Fourier extension. When constructed in a particular manner, Fourier extensions share many of the same features of a standard Fourier series. In particular, one can compute Fourier extensions which converge spectrally fast whenever the function is smooth, and geometrically fast if the function is analytic, much the same as the Fourier series of a smooth/analytic and periodic function. With this in mind, the purpose of this paper is to describe, analyse and explain the observation that Fourier extensions, much like classical Fourier series, also have excellent resolution properties for representing oscillatory functions. The resolution power, or required number of degrees of freedom per wavelength, depends on a user-controlled parameter and, as we show, it varies between 2 and pi. The former value is optimal and is achieved by classical Fourier series for periodic functions, for example. The latter value is the resolution power of algebraic polynomial approximations. Thus, Fourier extensions with an appropriate choice of parameter are eminently suitable for problems with moderate to high degrees of oscillation. (C) 2013 Elsevier B.V. All rights reserved.

关键词： Fourier series Fourier extension oscillatory functions Function approximation

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Weighted Averages and Double Exponential Algorithms

Weighted Averages and Double Exponential Algorithms

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17th International Symposium on Antennas and Propagation (ISAP)

作者： Mosig, Juan R. Ecole Polytech Fed Lausanne Lab Electromagnetisme & Acoust LEMA CH-1015 Lausanne Switzerland

This paper reviews two simple numerical algorithms particularly useful in Computational ElectroMagnetics (CEM): the Weighted Averages (WA) algorithm and the Double Exponential (DE) quadrature. After a short historical introduction and an elementary description of the mathematical procedures underlying both techniques, they are applied to the evaluation of Sommerfeld integrals, where WA and DE combine together to provide a numerical tool of unprecedented quality. It is also shown that both algorithms have a much wider range of applications. A generalization of the WA algorithm, able to cope with integrands including products of Bessel and similar oscillatory functions, is described. Similarly, the original DE algorithm is adapted with exceptional results to the evaluation of the multidimensional singular integrals arising in the discretization of Integral-Equation based CEM formulations. The new possibilities of WA and DE algorithms are demonstrated through several practical numerical examples.

关键词： weighted averages double exponential Sommerfeld integrals multidimensional singular integrals oscillatory functions

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Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday

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COMPUTER PHYSICS COMMUNICATIONS 2012年第12期183卷 2499-2512页

作者： Paternoster, Beatrice Univ Salerno Dipartmento Matemat I-84084 Fisciano Italy

The standard monograph in this area is the book Exponential fitting by Ixaru and Vanden Berghe (Kluwer. Boston - Dordrecht - London, 2004) but a fresh look on things is necessary because many new contributions have been accumulated in the meantime. With no claim that our investigation is exhaustive we consider various directions of interest, try to integrate the new contributions in a natural, easy to follow way, and also detect some open problems of acute interest. (C) 2012 Elsevier B.V. All rights reserved.

关键词： Exponential fitting oscillatory functions Frequency-dependent coefficients

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Differential inequalities and equations in Banach spaces with a cone

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2008年第1期69卷 245-255页

作者： Herrmann, Leopold Czech Tech Univ Prague Inst Tech Math Prague 12135 2 Czech Republic

A class of second-order abstract dissipative evolution differential operators D with 0 is an element of ker D is shown for which the fact that a non-zero t vertical bar -> u(t) belongs to a cone and - Du to a dual cone may hold only oil time intervals whose length is less than or equal to a defined number. Then oscillatory functions are dealt with in the framework of Banach spaces with a cone and conditions for the existence of a uniform oscillatory time for solutions of the equation Du = 0 are given. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： abstract dissipative evolution differential operators ordered Banach spaces cone in a Banach space conjugate points oscillatory functions uniform oscillatory time

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The construction of cubature rules for multivariate highly oscillatory integrals

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MATHEMATICS OF COMPUTATION 2007年第260期76卷 1955-1980页

作者： Huybrechs, Daan Vandewalle, Stefan Univ Kentucky Dept Comp Sci Louvain Belgium

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule. The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.

关键词： cubature formulas oscillatory functions steepest descent

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On the evaluation of highly oscillatory integrals by analytic continuation

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2006年第3期44卷 1026-1048页

作者： Huybrechs, Daan Vandewalle, Stefan Katholieke Univ Leuven Dept Comp Sci B-3001 Louvain Belgium

We consider the integration of one-dimensional highly oscillatory functions. Based on analytic continuation, rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand. The accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency, and for the case of an increasing number of points and fixed frequency. These results are then used to obtain quadrature rules for more general oscillatory integrals, i.e., for functions that exhibit some smoothness but that are not analytic. The approach described in this paper is related to the steepest descent method, but it does not employ asymptotic expansions. It can be used for small or moderate frequencies as well as for very high frequencies. The approach is compared with the oscillatory integration techniques recently developed by Iserles and Norsett.

关键词： oscillatory functions steepest descent method numerical quadrature

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Numerical evaluation of Hilbert transforms for oscillatory functions: A convergence accelerator approach

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COMPUTER PHYSICS COMMUNICATIONS 2002年第2期145卷 256-266页

作者： King, FW Smethells, GJ Helleloid, GT Pelzl, PJ Univ Wisconsin Dept Chem Eau Claire WI 54702 USA

The numerical evaluation of Hilbert transforms on the real line for functions that exhibit oscillatory behavior is investigated. A fairly robust numerical procedure is developed that is based on the use of convergence accelerator techniques. Several different types of oscillatory behavior are examined that can be successfully treated by the approach given. A few examples of functions whose oscillations are too extreme to deal with are also discussed. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： Hilbert transforms oscillatory functions numerical quadrature convergence acceleration

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Reply to the "Comment on Reliable operations on oscillatory functions" by Dr. Ixaru [Comput. Phys. Commun. 134 (2001)]

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COMPUTER PHYSICS COMMUNICATIONS 2001年第2期134卷 269-272页

作者： Adam, G Adam, S Inst Phys & Nucl Engn Dept Theoret Phys RO-76900 Bucharest Romania

Stimulated by the 'Comment' of L. Ixaru [Comput. Phys. Commun. 134 (2001) previous paper], we investigate the relationships between the exponentially fitted numerical differentiation and Simpson quadrature of oscillatory functions [L. Ixaru, Comput. Phys. Commun. 105 (1997) 1] (LI), and the subsequently reported corresponding formulae [Gh. Adam, S. Adam, Comput. Phys. Commun. 125 (2000) 127] (A&A). We find that the A&A formulae achieve an optimum in the absolute sense, while the Lr formulae satisfy a Rayleigh-Ritz optimum principle. As it concerns their practical interest, the A&A formulae are to be preferred whenever the problems to be solved involve harmonic series (trigonometric polynomials). However, if the oscillatory function is nothing but a black box for the user, then only the exponentially fitted formulae are available. (C) 2001 Elsevier Science B.V. All rights reserved.

关键词： oscillatory functions numerical differentiation numerical integration interpolatory Simpson quadrature exponential fitting discretization errors

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A comment on "Reliable operations on oscillatory functions" [Comput. Phys. Commun. 125 (2000) 127]

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COMPUTER PHYSICS COMMUNICATIONS 2001年第2期134卷 267-268页

作者： Ixaru, LG Inst Phys & Nucl Engn Bucharest Romania

关键词： oscillatory functions numerical differentiation numerical integration exponential fitting

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Reliability conditions in quadrature algorithms

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COMPUTER PHYSICS COMMUNICATIONS 2003年第1期154卷 49-64页

作者： Adam, G Adam, S Plakida, NM Joint Inst Nucl Res Bogolubov Lab Theoret Phys Dubna 141980 Russia Inst Phys & Nucl Engn Dept Theoret Phys R-76900 Bucharest Romania

The detection of insufficiently resolved or ill-conditioned integrand structures is critical for the reliability assessment of the quadrature rule outputs. We discuss a method of analysis of the profile of the integrand at the quadrature knots which allows inferences approaching the theoretical 100% rate of success, under error estimate sharpening. The proposed procedure is of the highest interest for the solution of parametric integrals arising in complex physical models. (C) 2003 Elsevier B.V. All rights reserved.

关键词： numerical integration reliability interpolatory quadrature Gauss-Kronrod quadrature discretization errors oscillatory functions

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