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.

Keywords: edge detection, nonlinear enhancement, concentration method, piecewise smoothness, localized reconstruction

@article{M2AN_2002__36_2_155_0, author = {Gelb, Anne and Tadmor, Eitan}, title = {Spectral reconstruction of piecewise smooth functions from their discrete data}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, pages = {155--175}, publisher = {EDP-Sciences}, volume = {36}, number = {2}, year = {2002}, doi = {10.1051/m2an:2002008}, zbl = {1056.42001}, mrnumber = {1906813}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2002008/} }

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 - Modélisation Mathématique et Analyse Numérique PY - 2002 SP - 155 EP - 175 VL - 36 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2002008/ UR - https://zbmath.org/?q=an%3A1056.42001 UR - https://www.ams.org/mathscinet-getitem?mr=1906813 UR - https://doi.org/10.1051/m2an:2002008 DO - 10.1051/m2an:2002008 LA - en ID - M2AN_2002__36_2_155_0 ER -

%0 Journal Article %A Gelb, Anne %A Tadmor, Eitan %T Spectral reconstruction of piecewise smooth functions from their discrete data %J ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique %D 2002 %P 155-175 %V 36 %N 2 %I EDP-Sciences %U https://doi.org/10.1051/m2an:2002008 %R 10.1051/m2an:2002008 %G en %F M2AN_2002__36_2_155_0

Gelb, Anne; Tadmor, Eitan. Spectral reconstruction of piecewise smooth functions from their discrete data. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 36 (2002) no. 2, pp. 155-175. doi : 10.1051/m2an:2002008. http://www.numdam.org/articles/10.1051/m2an:2002008/

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