A convergent overlapping domain decomposition method for total variation minimization

Massimo Fornasier, Andreas Langer, Carola-Bibiane Schönlieb

Research output: Contribution to journalArticlepeer-review

34 Scopus citations

Abstract

In this paper we are concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to the data and a total variation constraint. To our knowledge, this is the first successful attempt of addressing such a strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization. We provide several numerical experiments, showing the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems, respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles. © 2010 Springer-Verlag.
Original languageEnglish (US)
Pages (from-to)645-685
Number of pages41
JournalNumerische Mathematik
Volume116
Issue number4
DOIs
StatePublished - Jun 22 2010
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: Massimo Fornasier and Andreas Langer acknowledge the financial support provided by the FWF project Y 432-N15 START-Preis Sparse Approximation and Optimization in High Dimensions. Carola-B. Schonlieb acknowledges the financial support provided by the DFG Graduiertenkolleg 1023 Identification in Mathematical Models: Synergy of Stochastic and Numerical Methods, the Wissenschaftskolleg (Graduiertenkolleg, Ph.D. program) of the Faculty for Mathematics at the University of Vienna (funded by the Austrian Science Fund FWF) and the FFG project no. 813610 Erarbeitung neuer Algorithmen zum Image Inpainting. Further, this publication is based on work supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). The results of the paper also contribute to the project WWTF Five senses-Call 2006, Mathematical Methods for Image Analysis and Processing in the Visual Arts.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.

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