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A total variation diminishing interpolation operator and applications


Authors: Sören Bartels, Ricardo H. Nochetto and Abner J. Salgado
Journal: Math. Comp. 84 (2015), 2569-2587
MSC (2010): Primary 65D05, 49M25, 65K15, 65M60, 65N15, 49J40
DOI: https://doi.org/10.1090/mcom/2942
Published electronically: March 30, 2015
MathSciNet review: 3378839
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Abstract: We construct an interpolation operator that does not increase the total variation and is defined on continuous first degree finite elements over Cartesian meshes for any dimension $ d$ and right triangular meshes for $ d = 2$. The operator is stable and exhibits second order approximation properties in any $ L^p$, $ 1\leq p \leq \infty $. With the help of it we provide improved error estimates for discrete minimizers of the total variation denoising problem and for total variation flows. We also explore computationally the limitations of the total variation diminishing property over non-Cartesian meshes.


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Sören Bartels
Affiliation: Abteilung für Angewandte Mathematik, Albert-Ludwigs-Universität Freiburg Hermann-Herder-Str. 1079104 Freiburg i.Br., Germany.
Email: bartels@mathematik.uni-freiburg.de

Ricardo H. Nochetto
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
Email: rhn@math.umd.edu

Abner J. Salgado
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Address at time of publication: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: asalgad1@utk.edu

DOI: https://doi.org/10.1090/mcom/2942
Keywords: Total variation, interpolation, approximation, finite elements
Received by editor(s): November 5, 2012
Received by editor(s) in revised form: July 10, 2013, and February 12, 2014
Published electronically: March 30, 2015
Additional Notes: This work was partially supported by NSF grants DMS-0807811 and DMS-1109325. The third author was also partially supported by an AMS-Simons grant.
Article copyright: © Copyright 2015 American Mathematical Society

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