Preconditioning PDE-constrained optimization with L^1-sparsity and control constraints

PDE-constrained optimization aims at finding optimal setups for partial differential
equations so that relevant quantities are minimized. Including nonsmooth L1 sparsity
promoting terms in the formulation of such problems results in more practically relevant
computed controls but adds more challenges to the numerical solution of these problems.
The needed L1-terms as well as additional inclusion of box control constraints require
the use of semismooth Newton methods. We propose robust preconditioners for different
formulations of the Newton equation. With the inclusion of a line-search strategy and an
inexact approach for the solution of the linear systems, the resulting semismooth Newton's
method is reliable for practical problems. Our results are underpinned by a theoretical
analysis of the preconditioned matrix. Numerical experiments illustrate the robustness of
the proposed scheme.