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.

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.

Linear Algebra 1-2, Analysis 1-3.

Problem set 1 of October 21, 2011
Problem set 2 of October 28, 2011
Problem set 3 of November 04, 2011
Problem set 4 of November 11, 2011
Problem set 5 of November 18, 2011
Problem set 6 of November 25, 2011
Problem set 7 of December 02, 2011
Problem set 8 of December 09, 2011
Problem set 9 of December 16, 2011
Problem set 10 of December 23, 2011
Problem set 11 of January 13, 2012
Problem set 12 of January 20, 2012
Problem set 13 of January 27, 2012
Problem set 14 of February 03, 2012

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.