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

TL;DR

This paper advances the theoretical understanding of inverse problems with imperfect forward models, providing convergence rates, solution structure insights, and a debiasing approach for TV regularisation.

Contribution

It develops a duality-based framework for convergence analysis, studies the structure of solutions, and introduces a debiasing method for inverse problems with imperfect operators.

Findings

Convergence rates in Bregman distances under source conditions.

Structure of TV-regularised solutions and their level set convergence.

A novel debiasing approach for pointwise error estimation.

Abstract

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