Convergence Analysis of Primal-dual Based Methods for Total Variation Minimization with Finite Element Approximation

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 primaldual 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-dualbased prediction-correction scheme.