  This paper is 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 data and a total
variation constraint. To our knowledge, this is the first successful attempt of
addressing such 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.
