On Preconditioned Subspace Method For Solving State Constrained Stokes Optimal Control Problems

This thesis devotes to the numerical methods for solving the state constrained optimal control problem. The problem is transformed into the optimal control of the Stokes equations by the Moreau-Yosida regularization techniques. Then by the Q2-Q1 mixed finite element discretization, this class of problems become a semi-smooth optimization problem. We consider an inexact semi-smooth Newton method for the KKT conditions, the subproblem of which is a large saddle point problem.We focus on the efficient preconditioned Krylov methods for this problem. We investigate a class of splitting preconditioners based on the different splittings of the coefficient matrix, and three preconditioners Pi, P2 and P3 are proposed. Their spectral properties are analyzed in theory. Meanwhile, to avoid the inverse of the Schur complement, we apply an inexact proximal parallel splitting method. Final-ly, several numerical experiments are presented to illustrate the efficiency of our preconditioners.The contributions of this thesis include:(1)Based on the modified generalized shift-splitting preconditioner PMGSS in [38], three splitting preconditioners are given to solve the optimal control problem constrained by the Stokes equations. Theory analysis and numerical experiments are given.(2)An inexact proximal parallel splitting method(PPSM) is applied the sub-problems of the optimization problem, which avoids the inverse of the Schur complement.