Bouligand–Newton type methods for non-smooth ill-posed problems

We consider Newton-type methods for solving nonlinear ill-posed inverse problems in Hilbert spaces where the forward operators are not necessarily Gâteaux differentiable. Modifications are proposed with the non-existing Fréchet derivatives replaced by a family of bounded linear operators satisfying suitable properties. These bounded linear operators can be constructed by the Bouligand subderivatives which are defined as limits of Fréchet derivatives of the forward operator in differentiable points. The Bouligand subderivative mapping in general is not continuous unless the forward operator is Gâteaux differentiable which introduces challenges for convergence analysis of the corresponding Bouligand–Newton type methods. In this paper we will show that, under the discrepancy principle, these Bouligand–Newton type methods are iterative regularization methods of optimal order. Numerical results for an inverse problem arising from a non-smooth semi-linear elliptic equation are presented to test the performance of the methods.