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.
| Original language | English |
|---|---|
| Article number | 024006 |
| Journal | Inverse Problems |
| Volume | 35 |
| Issue number | 2 |
| DOIs | |
| Publication status | Published - 3 Jan 2019 |
Bibliographical note
Funding Information:MB acknowledges the support of ERC via Grant EU FP 7—ERC Consolidator Grant 615216 LifeInverse. A significant portion of the work presented in this paper was done while YK was a Humboldt Fellow at the University of Münster. YK acknowledges the support of the Humboldt Foundation in that period. Currently YK holds a Newton International Fellowship sponsored by the Royal Society, whose support he also acknowledges.
Publisher Copyright:
© 2019 IOP Publishing Ltd.
Keywords
- Bregman distances
- debiasing
- error estimation
- extended support
- operator uncertainty
- source condition
- total variation
ASJC Scopus subject areas
- Theoretical Computer Science
- Signal Processing
- Mathematical Physics
- Computer Science Applications
- Applied Mathematics