Preconditioned iterative methods for PDE-constrained optimization problems with pointwise state constraints

Abstract: Optimization problems constrained by partial differential equations (PDEs) arise in a variety of fields when one wants to optimize a system governed by a PDE. The goal is to compute a control variable such that a state variable is as close as possible to some desired state when control and state are coupled by some PDE. The control and state may have additional conditions acting on them, such as, the so-called box constraints which define upper and lower bounds on these variables. Here we study the optimal control of the Poison equation with pointwise inequality constraints on the state variable with Moreau-Yosida regularization. The state constraints make the optimality system nonlinear and a primal-dual active set method is used to solve it. In each nonlinear step a large saddle point system has to be solved. This system is generally ill-conditioned and preconditioning is required to efficiently solve the system with an iterative solution method. The preconditioner should also be efficiently realizable. The convergence rate is also dependent on model and discretization parameters, for this reason, a preconditioning technique needs to be applicable to a wide range of parameters. Three preconditioners are tested for the problem and compared in terms of iteration counts and execution time for wide range of problem and discretization parameters.