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401-3654-00L 12 Credits DR , MSC D-USYS , D-MAVT , D-MTEC , D-MATH , D-BIOL , D-CHAB

Inverse Problems: Theory and Numerical Treatment

Lecturers & Examiners: Prof. Dr. Ralf Hiptmair
VVZ CR n/a

Last Updated: 2026-02-05 15:19:32

Abstract

This course gives an introduction into the mathematical theory of inverse problems, techniques for their regularization and numerical methods to solve them. It covers(i) linear ill-posed operator equations, (ii) impedance tomography (iii) inverse acoustic scattering.

Objective

Goals of this course are: 1. Familiarity with the notion of an ill-posed problem and related operator theory 2. Knowledge about regularization procedure for linear operator equations 3. Insight into regularization by discretization 4. Iterative regularization of non-linear ill-posed problems 5. Knowledge about theory and numerical methods for impedance tomography 6. Knowledge about theory and numerical treatment of inverse acoustic scattering problems

Content

1. Examples of inverse problems and ill-posedness 2. Singular value decomposition of compact operators 3. Regularization 3.1 Abstract regularization procedures for ill-posed linear operator equations 3.2 Tikhonov regularization for linear problems 3.3 Iterative regularization for linear problems 3.4Regularization by discretization 4. Theory of non-linear ill-posed problems 5. Impedance tomography 5.1 Theory 5.2 Numerical methods 6. Inverse acoustic scattering 6.1 The direct acoustic scattering problem 6.2. Boundary integral equations and the far field 6.3. Theory of inverse acoustic scattering 6.4. Iteration methods for inverse acoustic scattering 6.5. Sampling and probe methods

Resources

Lecture Notes

Lecture notes will not be available

Literature

1. A. Rieder: Keine Probleme mit Inversen Problemen, Vieweg, 2003 (in German) 2. H. W. Engl , M. Hanke and A. Neubauer: Regularization of Inverse Problems, Kluwer, 1996 3. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer, 1996 4. C. W. Groetsch: Inverse Problems in the Mathematical Sciences, Vieweg, Braunschweig 1993 5. Isakov, Victor: Inverse problems for partial differential equations. Second edition. Applied Mathematical Sciences, 127. Springer, New York, 2006. 6. C.R. Vogel: Computational Methods for Inverse Problems, SIAM, 2002 7. R. Potthast: Point sources and multipoles in inverse scattering

General Information

Language
English
Levels
DR , MSC

Examination

Type
session examination
Mode
oral 30 minutes

Course Components

Type Title Time & Place Hours
lecture Inverse Problems: Theory and Numerical Treatment
(Räume HG G26.x wegen Umbau geschlossen)
  • Tue 15:15-17:00 (HG G 19.1)
  • Tue 15:15-17:00 (HG G 26.1)
  • Fri 08:15-10:00 (HG E 33.5)
  • Fri 08:15-10:00 (HG G 26.3)
4 h weekly
exercise Inverse Problems: Theory and Numerical Treatment
  • Tue 17:15-19:00 (HG G 19.1)
  • Fri 13:15-15:00 (HG G 19.1)
2 h weekly

Offered In