Spectral reconstruction of piecewise smooth functions from their discrete data

Anne Gelb, Eitan Tadmor

Research output: Contribution to journalArticle

22 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)155-175
Number of pages21
JournalMathematical Modelling and Numerical Analysis
Volume36
Issue number2
DOIs
StatePublished - 2002

Fingerprint

Piecewise Smooth Functions
Discrete Data
Discontinuity
Recovery
Jump
Interpolants
One Dimension
Two Dimensions
High Resolution
Coefficient

Keywords

  • Concentration method
  • Edge detection
  • Localized reconstruction
  • Nonlinear enhancement
  • Piecewise smoothness

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics
  • Modeling and Simulation

Cite this

Spectral reconstruction of piecewise smooth functions from their discrete data. / Gelb, Anne; Tadmor, Eitan.

In: Mathematical Modelling and Numerical Analysis, Vol. 36, No. 2, 2002, p. 155-175.

Research output: Contribution to journalArticle

@article{f0f0c4eee837426daf92d4033daacb90,
title = "Spectral reconstruction of piecewise smooth functions from their discrete data",
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.",
keywords = "Concentration method, Edge detection, Localized reconstruction, Nonlinear enhancement, Piecewise smoothness",
author = "Anne Gelb and Eitan Tadmor",
year = "2002",
doi = "10.1051/m2an:2002008",
language = "English (US)",
volume = "36",
pages = "155--175",
journal = "ESAIM: Mathematical Modelling and Numerical Analysis",
issn = "0764-583X",
publisher = "EDP Sciences",
number = "2",

}

TY - JOUR

T1 - Spectral reconstruction of piecewise smooth functions from their discrete data

AU - Gelb, Anne

AU - Tadmor, Eitan

PY - 2002

Y1 - 2002

N2 - 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.

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.

KW - Concentration method

KW - Edge detection

KW - Localized reconstruction

KW - Nonlinear enhancement

KW - Piecewise smoothness

UR - http://www.scopus.com/inward/record.url?scp=0035999253&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035999253&partnerID=8YFLogxK

U2 - 10.1051/m2an:2002008

DO - 10.1051/m2an:2002008

M3 - Article

AN - SCOPUS:0035999253

VL - 36

SP - 155

EP - 175

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 2

ER -