A Convergent Overlapping Domain Decomposition Method for Total Variation Minimization

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.Comment: Matlab code and numerical experiments of the methods provided in this paper can be downloaded at the web-page: http://homepage.univie.ac.at/carola.schoenlieb/webpage_tvdode/tv_dode_numerics.ht

Pseudo-linear convergence of an additive Schwarz method for dual total variation minimization. ETNA - Electronic Transactions on Numerical Analysis

In this paper, we propose an overlapping additive Schwarz method for total variation minimization based on a dual formulation. The O(1/n)-energy convergence of the proposed method is proven, where n is the number of iterations. In addition, we introduce an interesting convergence property of the proposed method called pseudo-linear convergence; the energy decreases as fast as for linearly convergent algorithms until it reaches a particular value. It is shown that this particular value depends on the overlapping width δ, and the proposed method becomes as efficient as linearly convergent algorithms if δ is large. As the latest domain decomposition methods for total variation minimization are sublinearly convergent, the proposed method outperforms them in the sense of the energy decay. Numerical experiments which support our theoretical results are provided