Large Noise in Variational Regularization

TL;DR

This paper develops a Banach space framework for variational regularization of inverse problems affected by large, unbounded noise, providing new existence results and error estimates, including for stochastic noise like white noise.

Contribution

It introduces a general Banach space setting for inverse problems with large noise, extending classical source conditions to approximate source conditions, and derives error estimates for various noise models.

Findings

Established existence of solutions under broad noise conditions.

Derived error estimates in Bregman distances for regularization.

Applied results to common regularizers like total variation and Besov norms.

Abstract

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.