A GCV based method for nonlinear ill-posed problems

This paper discusses the inversion of nonlinear ill-posed problems. Such problems are solved through regularization and iteration and a major computational problem arises because the regularization parameter is not known a priori. In this paper we show that the regularization should be made up of two parts. A global regularization parameter is required to deal with the measurement noise, and a local regularization is needed to deal with the nonlinearity. We suggest the generalized cross validation (GCV) as a method to estimate the global regularization parameter and the damped Gauss-Newton to impose local regularization. Our algorithm is tested on the magnetotelluric problem. In the second part of this paper we develop a methodology to implement our algorithm on large-scale problems. We show that hybrid regularization methods can successfully estimate the global regularization parameter. Our algorithm is tested on a large gravimetric problem.