Domain Decomposition Methods for Total Variation Minimization

Abstract

In this paper, overlapping domain decomposition methods (DDMs) are used for solving the Rudin-Osher-Fatemi (ROF) model in image restoration. It is known that this problem is nonlinear and the minimization functional is non-strictly convex and non-differentiable. Therefore, it is difficult to analyze the convergence rate for this problem. In this work, we use the dual formulation of the ROF model in connection with proper subspace correction. With this approach, we overcome the problems caused by the non-strict-convexity and non-differentiability of the ROF model. However, the dual problem has a global constraint for the dual variable which is difficult to handle for subspace correction methods. We propose a stable unit decomposition, which allows us to construct the successive subspace correction method (SSC) and parallel subspace correction method (PSC) based domain decomposition. Numerical experiments are supplied to demonstrate the efficiency of our proposed methods.