Webrelaunch 2020

Inverse Probleme (Winter Semester 2023/24)

Schedule
Lecture: Tuesday 14:00-15:30 20.30 0.19
Friday 8:00-9:30 20.30 SR 3.61
Problem class: Thursday 15:45-17:15 20.30 SR 3.68
Lecturers
Lecturer Prof. Dr. Roland Griesmaier
Office hours: Tuesday, 2:00-3:00 PM
Room 1.040 Kollegiengebäude Mathematik (20.30)
Email: roland.griesmaier@kit.edu
Problem classes Dr. Marvin Knöller
Office hours: Friday 10:30-12:00am
Room 1.038 Kollegiengebäude Mathematik (20.30)
Email: marvin.knoeller@kit.edu
Problem classes M. Sc. Raphael Schurr
Office hours: Wednesday, 3:00-4:00 p.m.
Room 1.039 Kollegiengebäude Mathematik (20.30)
Email: raphael.schurr@kit.edu

Course description:
Following Hadamard, a problem whose solution does not depend continuously on the given data is called ill-posed. Prominent examples are the mathematical problems behind tomographic tools like X-ray tomography, ultrasound tomography or electrical impedance tomography as well as seismic imaging, radar imaging, or inverse scattering. The mathematical models used in this context are typically formulated in terms of integral transforms or differential equations. However the aim is not to evaluate the transform or to solve the differential equation but to invert the transform or to reconstruct parameters of this equation given (part of) its solution, respectively. Therefore these problems are called inverse problems.

Standard methods from numerical mathematics typically fail when they are applied to ill-posed problems -- the problem has to be regularized. The course gives an introduction to the functional analytic background of regularization methods for linear ill-posed problems. Results from functional analysis that are required will be provided in the course of the lecture.

Prerequesites:
Linear Algebra 1-2, Analysis 1-3.

Literature:

  • W. Cheney, Analysis for Applied Mathematics, Springer-Verlag, New York, 2001.
  • H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers Group, Dordrecht, 1996.
  • M. Hanke, A Taste of Inverse Problems. Basic Theory and Examples, SIAM, Philadelphia, 2017.
  • A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 3rd ed., Springer-Verlag, New York, 2021.
  • R. Kress, Linear Integral Equations, 3rd ed., Springer-Verlag, New York, 2014.
  • A. Rieder, Keine Probleme mit inversen Problemen, Friedr. Vieweg & Sohn, Braunschweig, 2003.