Abstract
For the resolution of the Gibbs phenomenon, the inverse polynomial and the statistical filter methods were proposed independently. In this paper, we show how these two methods are different and similar, both mathematically and numerically. After comparing these methods, we propose a hybrid inverse polynomial and statistical filter method for the resolution of the Gibbs phenomenon.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Abdi, M. Hosseini, An investigation of resolution of 2-variate Gibbs phenomenon, Appl. Math. Comput. 203 (2008), 714–732
J.P. Boyd, Jun Rong Ong, Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions, part I: Single-interval schemes, Commun. Comput. Phys. 5 (2009), 484–497
D. Gottlieb, C.-W Shu, On the Gibbs phenomenon and its resolution, SIAM Rev. 39 (1997), 644–668
D. Gottlieb, C.-W. Shu, A. Solomonoff, H. Vandeven, 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
T. Hrycak, K. Gröchenig, Pseudospectral Fourier reconstruction with the modified inverse polynomial reconstruction method, J. Comput. Phys. 229 (2010), 933–946
J.-H. Jung, W. Stefan, A simple regularization of the polynomial interpolation for the resolution of the Runge phenomenon, J. Sci. Comput. DOI: 10.1007/s10915-010-9397-7
J.-H. Jung, B. D. Shizgal, Generalization of the inverse polynomial reconstruction method in the resolution of the Gibbs phenomena, J. Comput. Appl. Math. 172 (2004), 131–151
J.-H. Jung, B. D. Shizgal, On the numerical convergence with the inverse polynomial reconstruction method for the resolution of the Gibbs phenomenon, J. Comput. Phys. 224l (2007), 477–488
M. Krebs, Reduktion des Gibbs-Phänomens in der Magnetresonanztomographie, Master’s thesis, Technical University of Dortmund, 2007
B. D. Shizgal, J.-H Jung, Towards the resolution of the Gibbs phenomena, J. Comput. Appl. Math. 161 (2003), 41–65
A. Solomonoff, Reconstruction of a discontinuous function from a few Fourier coefficients using bayesian estimation, J. Sci. Comput. 10 (1995), 29–80
Acknowledgements
The author acknowledges the useful discussion with Alex Solomonoff and also thanks Tomasz Hrycak for allowing me to cite his unpublished manuscript, which is now published [5]. This work has been supported by the NSF under Grant No. DMS-0608844. The author would like to dedicate this paper to the late Prof. David Gottlieb, whose inspiration helped to make much of this work.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Berlin Heidelberg
About this paper
Cite this paper
Jung, JH. (2011). A Hybrid Method for the Resolution of the Gibbs Phenomenon. In: Hesthaven, J., Rønquist, E. (eds) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 76. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15337-2_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-15337-2_19
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15336-5
Online ISBN: 978-3-642-15337-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)