Variational regularisation for inverse problems with imperfect forward operators and general noise models

TL;DR

This paper develops a variational regularisation framework for inverse problems with imperfect forward models and general noise, providing convergence analysis and applicability to diverse fidelity measures.

Contribution

It introduces a comprehensive analysis of variational regularisation with imperfect operators and broad noise models, including convergence rates for various data fidelity terms.

Findings

Convergence rates established for regularized solutions.

Applicability to Wasserstein distances, f-divergences, and norms.

Analysis covers both a-priori and a-posteriori parameter choices.

Abstract

We study variational regularisation methods for inverse problems with imperfect forward operators whose errors can be modelled by order intervals in a partial order of a Banach lattice. We carry out analysis with respect to existence and convex duality for general data fidelity terms and regularisation functionals. Both for a-priori and a-posteriori parameter choice rules, we obtain convergence rates of the regularized solutions in terms of Bregman distances. Our results apply to fidelity terms such as Wasserstein distances, f-divergences, norms, as well as sums and infimal convolutions of those.