Alternating Direction Method Of Multipliers For Solving Optimal Control Problem Governed By Stokes Equation

In this thesis, we consider an alternating direction method of multipliers(ADMM) for solving the optimal control problem governed by the Stokes equation. The standard continuous problems are discretized by the finite element method to generate the discrete algebraic systems,and then are solved by the ADMM. The standard ADMM can not work well directly due to the construction of three separable variables. We extend the standard ADMM to three-variables problem to construct extended ADMM, which possess a global O(1/k) convergence rate as the standard ADMM. Based on the extended ADMM with the strongly convex assumption of the objective function, we construct an acceleration scheme to improve the efficiency of the algorithm and prove its convergence theoretically. We also adopt the ”restart” technique to modify the acceleration scheme in order to reduce the assumption, and then the modified algorithm can be applied to our model problems. The error estimates of our algorithms are presented for several different type optimization problems. Finally, numerical experiments are shown to demonstrate the efficiency of the algorithms.