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.