Optimal Solvers for PDE-Constrained Optimization

Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, particularly in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large-dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems, which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other PDEs.