Preconditioning Newton methods for Optimal Control Problems with Sparsity Constraints

We address the solution of PDE-constrained optimal control problems via semismooth Newton methods. Specifically, we consider problems with control constraints and with nonsmooth costs that are known to promote sparse optimal controls, i.e. controls which are identically zero on large parts of the control domain [HSW]. A typical example is the L$^1$ cost that has been used, e.g., for the optimal placement of control devices [S]. Following a discretize-then-optimize approach, we analyze the convergence properties of the Newton method applied to the discretization of optimal control problems with nonsmooth regularization terms. Moreover, we present the study of the impact of the control sparsity on the structure of the arising linear systems and propose preconditioners which exploit this information. Numerical experiments on 3D problems are presented.