Regularizing Newton-Kaczmarz Methods for Nonlinear Ill-Posed Problems

We introduce a class of stabilizing Newton-Kaczmarz methods for nonlinear ill-posed problems and analyze their convergence and regularization behaviour. As usual for iterative methods for solving nonlinear ill-posed problems, conditions on the nonlinearity (or the derivatives) have to be imposed in order to obtain convergence. As we shall discuss in general and in some specific examples, the nonlinearity conditions obtained for the Newton-Kaczmarz methods are less restrictive than those for previously existing iteration methods and can be verified for several practical applications. We also discuss the discretization and efficient numerical solution of the linear problems arising in each step of a Newton-Kacmarz method, and carry out numerical experiments for a model problem.