NDLI: Convergence Rate of Overlapping Domain Decomposition Methods for the Rudin--Osher--Fatemi Model Based on a Dual Formulation

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Tai, Xue-Cheng Wang, Li-Lian Yang, Danping Chang, Huibin
Copyright Year	2015
Abstract	This paper is concerned with overlapping domain decomposition methods (DDMs), based on successive subspace correction (SSC) and parallel subspace correction (PSC), for the Rudin--Osher--Fatemi (ROF) model in image restoration. In contrast to recent attempts, we work with a dual formulation of the ROF model, where one significant difficulty resides in the decomposition of the global constraint of the dual variable. We introduce a stable unity decomposition using a set of partition of unity functions, which naturally leads to overlapping DDMs based on the dual formulation. The main objective of this paper is to rigorously analyze the convergence of the SSC and PSC algorithms and derive the rate of convergence $O(n^{-1/2})$, where $n$ is the number of iterations. Moreover, we characterize the explicit dependence of the convergence rate on the subdomain overlapping size and other important parameters. To the best of our knowledge, such a convergence rate has not yet been claimed for domain decomposition related algorithms for the ROF model.
Starting Page	564
Ending Page	591
Page Count	28
File Format	PDF
DOI	10.1137/140965016
e-ISSN	19364954
Journal	SIAM Journal on Imaging Sciences (SJISBI)
Issue Number	1
Volume Number	8
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2015-03-10
Access Restriction	Subscribed
Subject Keyword	successive subspace correction parallel subspace correction Image processing overlapping domain decomposition method Rudin--Osher--Fatemi model convergence rate Multigrid methods; domain decomposition dual formulation Finite difference methods
Content Type	Text
Resource Type	Article

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