Iterative Methods For PDE-Constrained Optimization

In this paper, we consider the preconditioned iterative methods for the system of linear equation. arising from the discrete PDE-constrained optimization. As we know that the Krylov subspace iteration method may converge very slow when the coefficient has large condition number and bad spectrum distribution. Moreover, as the coefficient matrix is indefinite, the iterative method may fail to converge. Therefore, we need to construct good preconditioners which can be used to reduce the condition number and get better spectrum distribution and, hence, make the iterative method converge faster.We present a class of preconditioners based on the Schilder factorization. The properties of this kind of preconditioner are studied. Moreover, by making use of the special structure of the coefficient matrix, we reduce the original problem to a saddle point problem which has lower dimension and result in less computations. Precon-ditioners for this saddle problem are considered in detail and results concerning the eigenvalues of the preconditioned matrix are given. Finally, we discuss the approxima-tion issues in implementation. A few numerical experiments are used to illustrate the effectiveness of our preconditioners.