Convergence rates and structure of solutions of inverse problems with imperfect forward models. (3rd January 2019)

• Record Type: Journal Article
• Authors: Burger, Martin; Korolev, Yury; Rasch, Julian
• Is Part Of: Inverse problems. Volume 35:Number 2(2019)
• Publication Date: 2019-01-03
• Subjects: operator uncertainty -- total variation -- extended support -- Bregman distances -- source condition -- error estimation -- debiasing; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aaf6f5
• Languages: English
• ISSNs: 0266-5611

Abstract

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 -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 -based regularisation.