Inverse Problems
Winter term 2023/24

Lecturer: Prof. Dr. Benedikt Wirth
Tutorial: Marco Mauritz
News: There is a new room for the tutorials: SR 1C

Information on the lecture

Time, location: Mon 14:00 to 16:00, weekly, M4
Fri 14:00 to 16:00, weekly, M4
Start: 09.10.2023
Contents: Inverse problems are problems in which indirect measurements are used to calculate certain quantities in mathematical models. Examples are determination of thermal conductivity (e.g. of metal in blast furnaces) through measuring the temperature at the boundary of an object, the calculation of volatilities of stocks from option prices, or the reconstruction of medical images in computerized tomography. Due to the indirect measurements, inverse problems are often ill posed, i.e. small measurement errors in the observed data may lead to arbitrarily large deviations in the solution. Since in reality measurements are always currupted by noise, this is a serious problem that calls for special mathematical techniques (called regularization) to stabally approximate the solution. Additionally, numerical solutions and large amount of data containing little information are challenging problems in many applications of inverse problems. This lecture gives insights into theory as well as applications of inverse problems and is suited as a preparation for a possible Master's thesis in this area. The first part of the lecture deals with theory and regularization of inverse problems and the second part will discuss methods for optimization and discretization as well as applications (focus will be on image processing).
Prerequisites:  Bachelor-Courses in Applied Math.
Exam: This lecture may be recognized as specializing modules 'applied mathematics' and 'scientific computing' (probably also as a preparation for a master's thesis).
type 1: oral examination (30 min) at the end of term + some type of homework assessment to be announced
type 2: oral examination (20 min) at the end of term
Material: notes
Literature:
  • Engl, Hanke, Neubauer: Regularization of Inverse Problems, Springer 2000.
  • Louis: Inverse und schlecht gestellte Probleme, Teubner 1989.
  • Natterer, Wübbeling: Mathematical methods in image reconstruction, SIAM 2001.
  • Natterer: The mathematics of computerized tomography, SIAM 2001.
  • Helgason: The Radon transform, Birkhäuser 1999.
  • Hochstadt: Integral equations, Wiley 1973.

Information on the tutorial

Time, location: Wed 10:00 to 12:00, weekly
Room: SR 1C
Start: 18.10.2023