Some Investigations On Preconditioned Iterative Methods For Solving Optimal Control Problems Of Stokes Equations

In the biological, physical, economic and other fields, the control problems gov-erned by the partial differential equations exist almost everywhere. Because of the large scale and the complexity of these problems, scientific computing becomes the important task for the solution to them. This kind of problems usually become sad-dle point problems or generalized saddle point problems by discrete-then-optimize or optimize-then-discrete. But the coefficient matrix of the obtained system of e-quations is often ill-conditioned, so it may converge very slowly if we use Krylov subspace methods directly to solve the problems. At this moment we need to pre-condition the equations, so as to reduce the grade of the minimal polynomial, then we can use preconditioned Krylov subspace methods to improve the rate of conver-gence. As a consequence, choosing appropriate preconditioners is the key to solve the problems efficiently.In this thesis, we study the optimal control problem controlled by the stokes equations. We find that by some permutation, the (1,1) block of the matrix ob-tained from discretizion can have very special structure. Based on this structure, we provide some effective preconditioners for this problem. The proposed precondition-ers are an approximate block counter triangular preconditioner and some constraint preconditioners. Then the distributions of the eigenvalues of the corresponding pre-conditioned matrices are discussed. Finally, some numerical experiments are given to illustrate the efficiency of the preconditioners.The innovation points of this thesis include:(1) By permutating the coefficient matrix, we first give a new approximation of the (1,1) block. Then the block counter triangular preconditioned iterative meth-ods are applied for solving the optimal control problem controlled by the stokes equations.(2) We give some new constraint preconditioners corresponding to GMRES method.