Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T12:06:56.218Z Has data issue: false hasContentIssue false
  • English
  • Français

A Review of David Gottlieb’s Work on the Resolution of the Gibbs Phenomenon

Published online by Cambridge University Press:  20 August 2015

Sigal Gottlieb ,
Jae-Hun Jung  and
Saeja Kim
Sigal Gottlieb*
Affiliation:
Mathematics Department, University of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA
Jae-Hun Jung*
Affiliation:
Mathematics Department, Suny Buffalo, Buffalo, New York 14260-2900, USA
Saeja Kim*
Affiliation:
Mathematics Department, University of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

Given a piecewise smooth function, it is possible to construct a global expansion in some complete orthogonal basis, such as the Fourier basis. However, the local discontinuities of the function will destroy the convergence of global approximations, even in regions for which the underlying function is analytic. The global expansions are contaminated by the presence of a local discontinuity, and the result is that the partial sums are oscillatory and feature non-uniform convergence. This characteristic behavior is called the Gibbs phenomenon. However, David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing. In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

Keywords

42A1041A1041A25Gibbs phenomenonpost-processingGalerkin approximationcollocation approximationspectral methodsexponential accuracy
Type
Review Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 497 - 519
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Archibald, R., Gelb, A. and Yoon, J., Determining the locations of the discontinuities in the derivatives of functions, Appl. Numer. Math., 58(5) (2008), 577–592.Google Scholar
[2]Archibald, R. and Gelb, A., A method to reduce the Gibbs ringing artifact in MRI scans while keeping tissue boundary integrity, IEEE Trans. Med. Imag., 21 (2002), 305–319.CrossRefGoogle ScholarPubMed
[3]Archibald, R. and Gelb, A., Reducing the effects of noise in image reconstruction, J. Sci. Comput., 17 (2002), 167–180.Google Scholar
[4]Archibald, R., Chen, K., Gelb, A. and Renaut, R., Improving tissue segmentation of human brain MRI through pre-processing by the Gegenbauer reconstruction method, Neuro. Image., 20(1) (2003), 489–502.Google Scholar
[5]Bateman, H., Higher Transcendental Functions, Vol. 2, (edited by Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G.), McGraw-Hill, New York, 1953.Google Scholar
[6]Bôcher, M., Introduction to the theory of Fourier series, Ann. Math., 7 (1906), 81–152.Google Scholar
[7]Boyd, J. P., Trouble with Gegenbauer reconstruction for defeating Gibbs phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations, J. Comput. Phys., 204 (2005), 253–264.Google Scholar
[8]The Encyclopedia Britannica, 11th edition, Vol 4, 1910.Google Scholar
[9]Cai, W., Gottlieb, D. and Shu, C.-W., On one-sided filters for spectral Fourier approximations of discontinuous functions, SIAM J. Numer. Anal., 29 (1992), 905–916.Google Scholar
[10]Canuto, C. and Quarteroni, A., Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comput., 38 (1982), 67–86.Google Scholar
[11]Cates, D. and Gelb, A., Determining the locations of the discontinuities in derivatives of functions using spectral methods, Numer. Algorithms., 46 (2007), 59–84.Google Scholar
[12]Cook, R., Gibb’s phenomenon in Fourier-Bessel series and integrals, Proc. London. Math. Soc., 27 (1928), 171–192.Google Scholar
[13]Cook, R., Gibb’s phenomenon in Schlömlich series, J. London. Mathe. Soc., 4 (1928), 18–21.Google Scholar
[14]Du Bois-Reymond, P., Über die sprungweisen Werthänderungen analytischer functionen, Math. Ann., 7 (1874), 241–261.Google Scholar
[15]Funaro, D., Polynomial Approximation of Differential Equations, Springer-Verlag, 1992.Google Scholar
[16]Gelb, A. and Gottlieb, D., The resolution of the Gibbs phenomenon for spliced functions in one and two dimensions, Comput. Math. Appl., 33 (1997), 35–58.Google Scholar
[17]Gelb, A., Parameter optimization and reduction of round off error for the Gegenbauer reconstruction method, J. Sci. Comput., 20(3) (2004), 433–459.Google Scholar
[18] A.Gelb and Jackiewicz, Z., Determining analyticity for parameter optimization of the Gegenbauer reconstruction method, SIAM J. Sci. Comput., 27(3) (2005), 1014–1031.Google Scholar
[19]Gelb, A. and Tanner, J., Robust reprojection methods for the resolution of Gibbs phenomenon, Appl. Comput. Harmon. A., 20 (2006), 3–25.Google Scholar
[20]Gelb, A. and Tadmor, E., Adaptive edge detectors for piecewise smooth data based on the minmod limiter, J. Sci. Comput., 28 (2006), 279–306.Google Scholar
[21]Gelb, A. and Tadmor, E., Detection of edges in spectral data, Appl. Comput. Harmon. A., 7 (1999), 101–135.Google Scholar
[22]Gelb, A. and Tadmor, E., Detection of edgesin spectral data II: nonlinear enhancement, SIAM J. Numer. Anal., 38(4) (2000), 1389–1408.Google Scholar
[23]Gibbs, J. W., Letter to the Editor, Fourier’s Series, Nature, 59(1522) (1898), 200.Google Scholar
[24]Gibbs, J. W., Letter to the Editor, Fourier’s Series, Nature, 59(1539) (1899), 606.Google Scholar
[25]Gottlieb, D. and Shu, C.-W., A general theory for the resolution of the Gibbs phenomenon, Tricomi’s Ideas and Contemporary Applied Mathematics, Accademia Nazionale Dei Lincy, ATTI Dei Convegni Lincey, 147 (1998), 39–48.Google Scholar
[26]Gottlieb, D., Shu, C.-W., Solomonoff, A. and Vandeven, H., On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math., 43 (1992), 81–92.Google Scholar
[27]Gottlieb, D. and Shu, C.-W., On the Gibbs phenomenon III: recovering exponential accuracy in a sub-interval from the spectral sum of piecewise analytic function, SIAM J. Numer. Anal., 33 (1996), 280–290.Google Scholar
[28]Gottlieb, D. and Shu, C.-W., On the Gibbs phenomenon IV: recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comput., 64 (1995), 1081–1095.Google Scholar
[29]Gottlieb, D. and Shu, C.-W., On the Gibbs phenomenon V: recovering exponential accuracy from collocation point values of a piecewise analytic function, Numer. Math., 71 (1995), 511–526.Google Scholar
[30]Gottlieb, D. and Shu, C.-W., On the Gibbs phenomenon and its resolution, SIAM Rev., 39 (1997), 644–668.Google Scholar
[31]Gottlieb, D., The effect of local features on global expansions, The SIAM 2008 John von Neumann Lecture, SIAM Annual Meeting, (2008), http://tinyurl.com/yce34h9.Google Scholar
[32]Gottlieb, D. and Gottlieb, S., Spectral methods for compressible reactive flows, Comptes Rendus Mcanique: High-order methods for the numerical simulation of vortical and turbulent flows, 333(1) (2005), 3–16.Google Scholar
[33]Gottlieb, S., Gottlieb, D. and Shu, C.-W., Recovering high order accuracy in WENO computations of steady state hyperbolic systems, J. Sci. Comput., 28 (2006), 307–318.Google Scholar
[34]Hesthaven, J. S., Gottlieb, S. and Gottlieb, D., Spectral Methods for Time Dependent Problems, Cambridge Monographs on Applied and Computational Mathematics (No. 21), Cambridge University Press, 2006.Google Scholar
[35]Hewitt, E. and Hewitt, R., The Gibbs-Wilbraham phenomenon: an episode in Fourier analysis, Arch. Hist. Exact. Sci., 21 (1979), 129–160.Google Scholar
[36]Jung, J.-H., Gottlieb, S., Kim, S. O., Bresten, C. L. and Higgs, D., Recovery of high order accuracy in radial basis function approximation for discontinuous problems, J. Sci. Comput., 45 (2010), 359–381.Google Scholar
[37]Love, A. E. H., Letter to the Editor, Fourier’s Series, Nature, 58(1511) (1898), 569–570.Google Scholar
[38]Michelson, A. A., Letter to the Editor, Fourier’s Series, Nature, 58(1510) (1898), 544–545.Google Scholar
[39]Michelson, A. and Stratton, S., A new harmonic analyzer, Philo. Mag., 45 (1898), 85–91.Google Scholar
[40]Sarra, Scott A., Algorithm 899: The Matlab Postprocessing Toolkit, ACM Trans. Math. Software., 2009.Google Scholar
[41]Shu, C.-W. and Wong, P., A note of the accuracy of spectral method applied to nonlinear conservation laws, J. Sci. Comp., 10(3) (1995), 357–369.Google Scholar
[42]Tadmor, E., Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal., 26 (1989), 30–44.Google Scholar
[43]Tadmor, E., Filters, mollifiers and the computation of the Gibbs phenomenon, Acta. Numer., 16 (2007), 305–378.Google Scholar
[44]Trefethen, L. N., Pachn, R., Platte, R. B. and Driscoll, T. A., Chebfun Version 2, Oxford University, 2008, http://www.maths.ox.ac.uk/chebfun/.Google Scholar
[45]Wilbraham, H., On a certain periodic function, Camb. Dublin. Math. J., 3 (1848), 198–201.Google Scholar
[46]Wilton, J., The Gibbs phenomenon in series of Schlömlich type, Messenger. Math., 56 (1926), 175–181.Google Scholar
[47]Wilton, J., The Gibbs phenomenon in Fourier-Bessel series, J. Reine. Angew. Math., 159 (1928), 144–153.Google Scholar