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

Abstract

Abstract We consider Newton-type methods for solving nonlinear ill-posed inverse problems in Hilbert spaces where the forward operators are not necessarily G\^{a}teaux differentiable. Modifications are proposed with the non-existing Fr'{e}chet 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 Fr'{e}chet derivatives of the forward operator in differentiable points. The Bouligand subderivative mapping in general is not continuous unless the forward operator is G\^{a}teaux 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.