  A key observation underlying this paper is the fact that the range invariance
condition for convergence of regularization methods for nonlinear ill-posed
operator equations -- such as coefficient identification in partial
differential equiations (PDE)s from boundary observations -- can often be
achieved by extending the seached for parameter in the sense of allowing it to
depend on additional variables. This clearly counteracts unique identifiability
of the parameter, though. The second key idea of this paper is now to restore
the original restricted dependency of the parameter by penalization. This is
shown to lead to convergence of variational (Tikhonov type) and iterative
(Newton type) regularization methods. We concretize the abstract convergence
analysis in a framework typical of parameter identification in PDEs in a
reduced and an all-at-once setting. This is further illustrated by three
examples of coefficient identification from boundary observations in elliptic
and time-dependent PDEs.
