Alireza Ataei

Optimal control problems Constrained by a partial differential equation (PDE) arise in various important applications, Such as in engineering and natural sciences. In this thesis, we first introduce a distributed control problem and deseribe the discretization of this problem and the problem after discretization becomes a system of liner equations, which we use the GMRES iterative method to solve it. Since the convergence rate of this method is too slow to solve the system of liner equations, we use preconditioning to accelerate the converegence rate. Then we introduce many of existing preconditioning and how to employment them. Next, we intoduce a new iteration method and a new preconditioning technique for an elliptic. PDE- constrained optimal control problem. Finally, we see the number of iteration steps and the convergence speed to the answer in sloving this problem with GMRES method.