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Advanced Inverse Problems: Nonlinearity and Banach Spaces (Winter Semester 2015/16)

The main topic of the course is to solve nonlinear ill-posed problems

$F(x)=y$

where F\colon D(F)\subset X\rightarrow Y operates between the Banach spaces X and Y with domain of definition D(F). 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 L^\infty-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: monday, 14:00-15:00, and 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

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.