Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data

Anne Gelb; Eitan Tadmor

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 36, Issue: 2, page 155-175
  • ISSN: 0764-583X

Abstract

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This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical for all methods. Here we formulate a new localized reconstruction method adapted from the method developed in Gottlieb and Tadmor (1985) and recently revisited in Tadmor and Tanner (in press). Our procedure incorporates the detection of edges into the reconstruction technique. The method is robust and highly accurate, yielding spectral accuracy up to a small neighborhood of the jump discontinuities. Results are shown in one and two dimensions.

How to cite

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Gelb, Anne, and Tadmor, Eitan. "Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data." ESAIM: Mathematical Modelling and Numerical Analysis 36.2 (2010): 155-175. <http://eudml.org/doc/194099>.

@article{Gelb2010,
abstract = { This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical for all methods. Here we formulate a new localized reconstruction method adapted from the method developed in Gottlieb and Tadmor (1985) and recently revisited in Tadmor and Tanner (in press). Our procedure incorporates the detection of edges into the reconstruction technique. The method is robust and highly accurate, yielding spectral accuracy up to a small neighborhood of the jump discontinuities. Results are shown in one and two dimensions. },
author = {Gelb, Anne, Tadmor, Eitan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Edge detection; nonlinear enhancement; concentration method; piecewise smoothness; localized reconstruction.; edge detection; piecewise smoothness; localized reconstruction.},
language = {eng},
month = {3},
number = {2},
pages = {155-175},
publisher = {EDP Sciences},
title = {Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data},
url = {http://eudml.org/doc/194099},
volume = {36},
year = {2010},
}

TY - JOUR
AU - Gelb, Anne
AU - Tadmor, Eitan
TI - Spectral Reconstruction of Piecewise Smooth Functions from Their Discrete Data
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 36
IS - 2
SP - 155
EP - 175
AB - This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical for all methods. Here we formulate a new localized reconstruction method adapted from the method developed in Gottlieb and Tadmor (1985) and recently revisited in Tadmor and Tanner (in press). Our procedure incorporates the detection of edges into the reconstruction technique. The method is robust and highly accurate, yielding spectral accuracy up to a small neighborhood of the jump discontinuities. Results are shown in one and two dimensions.
LA - eng
KW - Edge detection; nonlinear enhancement; concentration method; piecewise smoothness; localized reconstruction.; edge detection; piecewise smoothness; localized reconstruction.
UR - http://eudml.org/doc/194099
ER -

References

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  8. B.I. Golubov, Determination of the jump of a function of bounded p-variation by its Fourier series. Math. Notes12 (1972) 444-449.  
  9. D. Gottlieb and C.-W. Shu, On the Gibbs phenomenon and its resolution. SIAM Rev. (1997).  
  10. D. Gottlieb and E. Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, in Progress and Supercomputing in Computational Fluid Dynamics, Proceedings of a 1984 U.S.-Israel Workshop, Progress in Scientific Computing, Vol. 6, E.M. Murman and S.S. Abarbanel Eds., Birkhauser, Boston (1985) 357-375.  
  11. G. Kvernadze, Determination of the jump of a bounded function by its Fourier series. J. Approx. Theory92 (1998) 167-190.  
  12. E. Tadmor and J. Tanner, Adaptive mollifiers for high resolution recovery of piecewise smooth data from its spectral information, Foundations of Comput. Math. Online publication DOI: 10.1007/s002080010019 (2001), in press.  
  13. A. Zygmund, Trigonometric Series. Cambridge University Press (1959).  

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