On Block Preconditioners For Distributed Control PDE-constrained Optimization Problems

Recently, there are many methods for solving the PDE-constrained optimization problems. Some proposed the Krylov methods for solving the structured system of linear equations arising from the Galerkin finite-element discretizations of the distributed con-trol problems. Preconditioning technique as an efficient tool has been widely applied. In this thesis, we first give the convergence analyses for the preconditioned GMRES meth-ods. By applying the special structures and properties of the eigenvector matrices of the preconditioned matrices, we derive upper bounds for the2-norm condition numbers of the eigenvector matrices and give asymptotic convergence factors of the preconditioned GM-RES methods with the block-counter-diagonal and the block-counter-triangular precon-ditioners. Then,by analyzing the block-counter-diagonal preconditioned linear system resulting from discretization of the distributed control problems, we derive a new equiva-lent linear system and apply the MINRES method coupled with a proposed block-diagonal preconditioner to solve it. At the same time, we also discuss the spectral properties of the new equivalent linear system. Theoretical analyses and experiments show that the the-oretically estimated bounds can indicate the real condition numbers of the eigenvector matrices. And the proposed preconditioned MINRES method is efficient for solving the elliptic PDE-constrained optimization distributed control problems especially when the regularization parameter is suitably small.