  In this work we consider the stable numerical solution of large-scale
ill-posed nonlinear least squares problems with nonzero residual. We propose a
non-stationary Tikhonov method with inexact step computation, specially
designed for large-scale problems. At each iteration the method requires the
solution of an elliptical trust-region subproblem to compute the step. This
task is carried out employing a Lanczos approach, by which an approximated
solution is computed. The trust region radius is chosen to ensure the resulting
Tikhonov regularization parameter to satisfy a prescribed condition on the
model, which is proved to ensure regularizing properties to the method. The
proposed approach is tested on a parameter identification problem and on an
image registration problem, and it is shown to provide important computational
savings with respect to its exact counterpart.
