A Finite Element Approach for the Dual Rudin--Osher--Fatemi Model and Its Nonoverlapping Domain Decomposition Methods
For researchers in image processing and numerical analysis, this work provides a more adequate discretization for designing domain decomposition methods for total variation regularization.
The paper develops a finite element discretization for the dual Rudin-Osher-Fatemi model using Raviart-Thomas basis, enabling efficient domain decomposition methods. The primal method achieves O(1/n^2) convergence, and the primal-dual method solves local problems with linear convergence.
We consider a finite element discretization for the dual Rudin--Osher--Fatemi model using a Raviart--Thomas basis for $H_0 (\mathrm{div};Ω)$. 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.