A convergent overlapping domain decomposition method for total variation minimization

  • Authors:

  • Massimo Fornasier;Andreas Langer;Carola-Bibiane Schönlieb

  • Affiliations:

  • Austrian Academy of Sciences, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Altenbergerstrasse 69, 4040, Linz, Austria;Austrian Academy of Sciences, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Altenbergerstrasse 69, 4040, Linz, Austria;University of Göttingen, Institute for Numerical and Applied Mathematics, Lotzestr. 16-18, 37083, Göttingen, Germany

  • Venue:
  • Numerische Mathematik
  • Year:
  • 2010

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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.