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Inverse Probleme/ Inverse Problems (Winter Semester 2011/12)

Schedule
Lecture: Tuesday 15:45-17:15 Z 2 Begin: 18.10.2011
Wednesday 11:30-13:00 Z 1
Problem class: Friday 9:45-11:15 Z 2 Begin: 28.10.2011
Lecturers
Lecturer Prof. Dr. Roland Griesmaier
Office hours: by phone or video call; please make an appointment by email
Room 1.040 Kollegiengebäude Mathematik (20.30)
Email: roland.griesmaier@kit.edu
Problem classes Dr. Tim Kreutzmann
Office hours:
Room Kollegiengebäude Mathematik (20.30)
Email:

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 first part of the course gives an introduction to the functional analytic background of regularization methods for linear ill-posed problems. All results from functional analysis that are needed will be provided during the lecture. In the second part we consider the problem of transmission computerized tomography (X-ray tomography) in detail.

Course administration and mailing list

You can register online for the problem classes using the course administration website. Thereby, you subscribe to the mailing list which you can use to ask question of general interest and that is used to announce organizational issues.

Prerequesites

Linear Algebra 1-2, Analysis 1-3.

Selected Solutions

Solution of exercise 3 of problem set 2
Solution of exercise 3 of problem set 4
Solution of exercise 3 of problem set 5
Solution of exercise 3 of problem set 9
Solution of exercise 3 of problem set 11
Solution of exercise 3 of problem set 14

References

  • 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.
  • A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 1996.
  • R. Kress, Linear Integral Equations, 2nd ed., Springer-Verlag, New York, 1999.
  • F. Natterer, The Mathematics of Computerized Tomography, SIAM, Philadelphia, 2001.
  • A. Rieder, Keine Probleme mit inversen Problemen, Friedr. Vieweg & Sohn, Braunschweig, 2003.