We consider the total variation minimization model with consistent finite element discretization. It has been shown in the literature that this model can be reformulated as a saddle-point problem and be efficiently solved by the primal-dual method. The convergence for this application of the primal-dual method has also been analyzed. In this paper, we focus on a more general primal-dual scheme with a combination factor for the model and derive its convergence. We also establish the worst-case convergence rate measured by the iteration complexity for this general primal-dual scheme. Furthermore, we propose a prediction-correction scheme based on the general primal-dual scheme, which can significantly relax the step size for the discretization in the time direction. Its convergence and the worst-case convergence rate are established. We present preliminary numerical results to verify the rationale of considering the general primal-dual scheme and the primal-dual-based prediction-correction scheme.
View Convergence Analysis of Primal-Dual Based Methods for Total Variation Minimization with Finite Element Approximation