Advanced Inverse Problems: Nonlinearity and Banach Spaces (Winter Semester 2015/16)
- Lecturer: Prof. Dr. Andreas Rieder
- Classes: Lecture (0123000), Problem class (0123100)
- Weekly hours: 2+2
The main topic of the course is to solve nonlinear ill-posed problems
where operates between the Banach spaces
and
with domain of definition
. This kind of inverse problem gained a lot of interest over the last years because several applications and constraints are formulated quite naturally in a Banach space framework: sparsity, uniform and impulsive noise, preserving discontinuities (edges) etc. In parameter identification tasks, for instance, the searched-for parameter often appears in the governing partial differential equations as an
-coefficient, e.g., electrical impedance tomography.
Knowledge of Functional Analysis and of some basics in Inverse Problems are advantageous.
We have created an ILIAS domain which you can join by following this link. There you will find the tutorial assignments as well as supplemental material.
| Schedule | |||
|---|---|---|---|
| Lecture: | Thursday 8:00-9:30 | SR 3.61 | Begin: 22.10.2015 |
| Problem class: | Monday 9:45-11:15 | SR 3.61 | |
| Lecturers | ||
|---|---|---|
| Lecturer | Prof. Dr. Andreas Rieder | |
| Office hours: Until further notice only on appointment. | ||
| Room 3.040 Kollegiengebäude Mathematik (20.30) | ||
| Email: andreas.rieder(at)kit.edu | Problem classes | Dr. Robert Winkler |
| Office hours: | ||
| Room Kollegiengebäude Mathematik (20.30) | ||
| Email: robert (punkt) winkler (bei) posteo (punkt) de | ||
The following topics are intended to be covered:
Hilbert spaces
- Inexact Newton solvers for nonlinear ill-posed Problems
Banach spaces
- Geometry of Banach spaces
- Gradient-like iterations for linear Problems
- Inexact Newton solvers
- Tikhonov-Phillips Regularization
- The method of Approximate Inverse
References
- L. Beilina, M. Klibanov: Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, 2012
- S. Kabanikhin: Inverse and Ill-Posed Problems, de Gruyter, 2012
- B. Kaltenbacher, A. Neubauer, O. Scherzer: Iterative regularization methods for nonlinear ill-posed problems, de Gruyter, 2008
- F. Margotti: On inexact Newton methods for inverse problems in Banach spaces, PhD thesis, Department of Mathematics, KIT, 2015
- O. Scherzer (ed.): Handbook of Mathematical Methods in Imaging, Springer, 2011
- O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, F. Lenzen: Variational Methods in Imaging, Springer, 2009
- T. Schuster, B. Kaltenbacher, B. Hofmann, K. S. Kazimierski: Regularization methods in Banach spaces, de Gruyter, 2012
The book by Beilina and the two books by Scherzer et al. can be downloaded from the publishers's site via KIT internet access (vpn client). The PhD thesis of Margotti is freely available. Just follow the links from above.