A Finite Element Approach for the Dual Rudin--Osher--Fatemi Model and Its Nonoverlapping Domain Decomposition Methods

TL;DR

This paper introduces a finite element discretization for the dual Rudin--Osher--Fatemi model that enables effective domain decomposition methods with proven convergence rates, improving computational efficiency for image processing tasks.

Contribution

It proposes a novel finite element discretization with a splitting property and develops primal and primal-dual domain decomposition methods with linear and quadratic convergence rates.

Findings

Achieves $O(1/n^2)$ convergence for the primal domain decomposition method.

Local problems in the primal-dual method converge linearly.

Numerical results demonstrate the effectiveness of the proposed methods.

Abstract

We consider a finite element discretization for the dual Rudin--Osher--Fatemi model using a Raviart--Thomas basis for $H_0 (\mathrm{div};\Omega)$. Since the proposed discretization has splitting property for the energy functional, which is not satisfied for existing finite difference based discretizations, it is more adequate for designing domain decomposition methods. In this paper, a primal domain decomposition method is proposed, which resembles the classical Schur complement method for the second order elliptic problems, and it achieves $O(1/n^2)$ convergence. A primal-dual domain decomposition method based on the method of Lagrange multipliers on the subdomain interfaces is also considered. Local problems of the proposed primal-dual domain decomposition method can be solved in linear convergence rate. Numerical results for the proposed methods are provided.