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

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.

Convergent Non-overlapping Domain Decomposition Methods for Variational Image Segmentation

“…2b) are employed. A different stopping condition E k ≤ TOL e is used as [51,53], which is defined to guarantee the quality of segmentation visually. We set TOL e = −4.292900 × 10 5 , −2.204990 × 10 5 , −3.374890 × 10 6 , which are the numerical energy of the iterative solutions by the primal-dual algorithm without the DDMs for the three images, respectively.…”

“…Chang et al [44] extended the DDMs to the nonlocal total variation (NLTV) based image restoration, where a unified framework was proposed for the NLTV and ROF model. Chang et al [45,46] introduced a stable "unity decomposition" and derived an overlapping DDMs for the dual formulation, where the convergence rates were first deduced for the DDMs of total variation minimization problems. Some variants of the classical DDMs have been studied in [47][48][49][50].…”

“…Comparing with the overlapping DDMs, we prefer to use the non-overlapping DDMs for two reasons: (1) the ease of handling the constraints, where we do not need to introduce any sophisticated technique as [53]; (2) less communication cost and the convergence guarantee. Therefore, we mainly focus on the non-overlapping DDMs for the variational image segmentation tasks.…”

Domain Decomposition Methods in Science and Engineering XXIII

“…For example, see Chang et al (2015), Fornasier (2007), Fornasier et al (2010), Fornasier andSchönlieb (2009), Hintermüller andLanger (2014), and Xu et al (2010). For example, see Chang et al (2015), Fornasier (2007), Fornasier et al (2010), Fornasier andSchönlieb (2009), Hintermüller andLanger (2014), and Xu et al (2010).…”

Block Decomposition Methods for Total Variation by Primal–Dual Stitching

“…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. Although the theoretical global convergence is not known, the block decomposition method is highly useful due to the simple structure of the method.…”