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The goal of this paper is to further develop an approach to inverse problemswith imperfect forward operators that is based on partially ordered spaces.Studying the dual problem yields useful insights into the convergence of theregularised solutions and allow us to obtain convergence rates in terms ofBregman distances - as usual in inverse problems, under an additionalassumption on the exact solution called the source condition. These results areobtained for general absolutely one-homogeneous functionals. In the specialcase of TV-based regularisation we also study the structure of regularisedsolutions and prove convergence of their level sets to those of an exactsolution. Finally, using the developed theory, we adapt the concept ofdebiasing to inverse problems with imperfect operators and propose an approachto pointwise error estimation in TV-based regularisation.

Convergence rates and structure of solutions of inverse problems with imperfect forward models