Large Noise in Variational Regularization.

In this paper we consider variational regularization methods for inverseproblems with large noise that is in general unbounded in the image space ofthe forward operator. We introduce a Banach space setting that allows to definea reasonable notion of solutions for more general noise in a larger spaceprovided one has sufficient mapping properties of the forward operators.

A key observation, which guides us through the subsequent analysis, is thatsuch a general noise model can be understood with the same setting asapproximate source conditions (while a standard model of bounded noise isrelated directly to classical source conditions). Based on this insight weobtain a quite general existence result for regularized variational problemsand derive error estimates in terms of Bregman distances. The latter arespecialized for the particularly important cases of one- and p-homogeneousregularization functionals.

As a natural further step we study stochastic noise models and in particularwhite noise, for which we derive error estimates in terms of the expectation ofthe Bregman distance. The finiteness of certain expectations leads to a novelclass of abstract smoothness conditions on the forward operator, which can beeasily interpreted in the Hilbert space case. We finally exemplify the approachand in particular the conditions for popular examples of regularizationfunctionals given by squared norm, Besov norm and total variation,respectively.