A convergent overlapping domain decomposition method for total variation minimization

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. Mathematics Subject Classification (2000)

Wavelet Decomposition Method for $L_2/$/TV-Image Deblurring

“…Our approach. Domain decomposition and subspace correction methods for functionals of the form (1.7) were already proposed in [24,25]. There some of the authors of this paper mainly focused on the splitting of the physical domain Ω into smaller subdomains Ω = i Ω i and studied an alternating minimization algorithm on each subspace.…”

“…In this paper, we show additional properties of the limit of a sequence produced by the subspace correction algorithm proposed by Fornasier and Schönlieb [25] for L2/T V -minimization problems. An important but missing property of such a limiting sequence in [25] is the convergence to a minimizer of the original minimization problem, which was obtained in [24] with an additional condition of overlapping subdomains. We can now determine when the limit is indeed a minimizer of the original problem.…”

Block Decomposition Methods for Total Variation by Primal–Dual Stitching

“…Note that, if the non-differentiable function is separable, e.g., 1 -norm minimization problem, then we can establish a global convergence of block decomposition methods [4,11] and a block coordinate method [23]. Due to the non-differentiability and non-separability of TV, theoretical global convergence is only available for overlapped domain decompositions [12,17]. Note that recently, dual based approaches [8,18,19] are also introduced to overcome the difficulties of TV in domain decomposition applications.…”

Domain Decomposition Methods for Total Variation Minimization

“…Chang et al [6] extended the DDMs for the nonlocal total variation(NLTV) based image restoration, where the authors also pointed out that their proposed DDMs for NLTV can be adopted to solve the ROF model directly. In addition to these works, some variants of the classical DDMs have been proposed in [10,11,12,17]. In [10,11,12], the authors introduced the surrogate functional to form an approximation(or iterative proximity-map) of the subproblems.…”

“…In addition to these works, some variants of the classical DDMs have been proposed in [10,11,12,17]. In [10,11,12], the authors introduced the surrogate functional to form an approximation(or iterative proximity-map) of the subproblems. Then the subproblems were solved by oblique thresholding.…”