  Many inverse problems can be described by a PDE model with unknown parameters
that need to be calibrated based on measurements related to its solution. This
can be seen as a constrained minimization problem where one wishes to minimize
the mismatch between the observed data and the model predictions, including an
extra regularization term, and use the PDE as a constraint. Often, a suitable
regularization parameter is determined by solving the problem for a whole range
of parameters -- e.g. using the L-curve -- which is computationally very
expensive. In this paper we derive two methods that simultaneously solve the
inverse problem and determine a suitable value for the regularization
parameter. The first one is a direct generalization of the Generalized Arnoldi
Tikhonov method for linear inverse problems. The second method is a novel
method based on similar ideas, but with a number of advantages for nonlinear
problems.
