  In this paper we consider variational regularization methods for inverse
problems with large noise that is in general unbounded in the image space of
the forward operator. We introduce a Banach space setting that allows to define
a reasonable notion of solutions for more general noise in a larger space
provided one has sufficient mapping properties of the forward operators.
  A key observation, which guides us through the subsequent analysis, is that
such a general noise model can be understood with the same setting as
approximate source conditions (while a standard model of bounded noise is
related directly to classical source conditions). Based on this insight we
obtain a quite general existence result for regularized variational problems
and derive error estimates in terms of Bregman distances. The latter are
specialized for the particularly important cases of one- and p-homogeneous
regularization functionals.
  As a natural further step we study stochastic noise models and in particular
white noise, for which we derive error estimates in terms of the expectation of
the Bregman distance. The finiteness of certain expectations leads to a novel
class of abstract smoothness conditions on the forward operator, which can be
easily interpreted in the Hilbert space case. We finally exemplify the approach
and in particular the conditions for popular examples of regularization
functionals given by squared norm, Besov norm and total variation,
respectively.
