Identification of the Support of Nonsmoothness

We consider a class of nonlinear equations in function spaces for which the nonlinearity can be split into smooth and nonsmooth parts. Such problems arise in optimal control problems for parabolic partial differential equations with bound constraints on the control. In many situations the nonsmooth part acts on functions having support in a set of small measure. If this small set can be well identified one can exploit this structure and apply Newton-like methods away from the small set.In this paper we extend and simplify earlier work on optimal control problems by developing a simple approach for construction of the splitting into smooth and nonsmooth parts. This paper extends earlier work on the generalization of the classical projected Newton method to infinite dimensional optimal control problems to the more general setting of splitting algorithms for nonlinear equations.