Semismooth Newton Methods in Function Space: Theory, Numerics and Applications

Motivated by constrained optimal control problems for PDEs we present a generalized differentiability concept in function space which allows to consider generalized (semismooth) versions of Newtons method for solving nonsmooth operator equations. The local rate of convergence turns out to be q-superlinear. The implementation of the method in terms of a primal-dual active set strategy is discussed and a mesh independence result is established. The talk ends by highlighting several application areas ranging from constrained optimal control of PDEs to problems in image restoration.