**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 needed will be provided during 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*, Springer-Verlag, New York, 1996. - R. Kress,
*Linear Integral Equations*, 2nd ed., Springer-Verlag, New York, 1999. - A. Rieder,
*Keine Probleme mit inversen Problemen*, Friedr. Vieweg & Sohn, Braunschweig, 2003.