On Primal-dual Method For PDE Optimization Problems

The primal-dual algorithm is an efficient algorithm which obtains the global solution by iterating the primal and dual variables alternatively. The primal-dual algorithm has wide applications in different areas. In this paper, we consider solving two PDE optimization problems by using the primal-dual method. The one is the image restoration problem. We focus on the total variation image restoration model to remove the Gaussian white noise and blur in the observed image. The existing methods can not handle the nondifferentiability of the total variation term effectively. But we can solve the total variation model exactly by using primal-dual algorithm and the result of the image restoration is well. The other one is the PDE-constrained optimization problem, we propose an algorithm based on primal-dual method for it and we also give theoretical analysis. After discretizing the optimization problem and combining Lagrange multiplier, the original problem is converted into a saddle-point problem. We apply the primal-dual method to the resulting problems and the solution of the original problems thus is obtained. Both theoretical analysis and numerical experiments show that the proposed method is very efficient for solving the PDE-constrained optimization problems.