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## IAM664 - Inverse Problems

**Credit: **
3(3-0); **ECTS: **
8.0

**Instructor(s): **
Gerhard-Wilhelm Weber

**Prerequisites: **
Consent of instructor (Basic knowledge in numerical and statistical methods and, if possible, in probability theory)

#### Course Catalogue Description

Classification of inverse problems, linear regression, discretizing continuous inverse problems, rank-deficiency, Tikhonov regularization, iterative methods, other regularization techniques, Fourier techniques, nonlinear inverse problems, Bayesian methods. Computer applications and MATLAB exercises are important elements of the course.

#### Course Objectives

- The objective of this course is to promote fundamental understanding of parameter estimation and inverse problems methodology, specifically regarding such issues like uncertainty, ill-posedness, regularization, bias and resolution using examples from various fields of applications, e.g., engineering, financial mathematics, economics, the environmental sector, Operational Research, computational biology and social sciences.

#### Course Learning Outcomes

- At the end of the course, students should have a good overview of modern scientific methods in inverse problems. They should also be able to choose and work them out appropriately in contexts of project applications and of their theses.

#### Tentative (Weekly) Outline

- Introduction
- Linear Regression
- Least Squares Theory
- Discretizing Continuous Inverse Problems
- Rank Deficiency and Ill-Conditioning
- Tikhonov Regularization
- Iterative Methods
- Fourier Techniques
- Other Regularization Techniques
- Nonlinear Inverse Problems
- Nonlinear Regression
- Nonlinear Least Squares
- Bayesian Methods
- Application to Tomography
- Discrete Tomography

#### Course Textbook(s)

- A. Aster, B. Borchers, C. Thurber, Parameter Estimation and Inverse Problems, Academic Press, 2nd edition, 2012

#### Supplementary Materials and Resources

- Books:
- J. Baumeister, Stable Solutions of Inverse Problems, Vieweg, 1987
- H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, 1996
- P.C. Hansen, Rank-Deficient and Ill-Posed Problems, SIAM, 1996
- G.T. Herman, A. Kuba, Discrete Tomography: Foundations, Algorithms and Applications, Birkhaeuser, 1999
- A.N. Tikhonov, V.Y. Arsenin, Solution of Ill-Posed Problems, Wiley, 1977
- Resources:
- Lecture Notes: Furthermore, lecture notes and recent research articles will be provided during the course
- MATLAB Student Version is available to download on MathWorks website, http://www.mathworks.com, or METU FTP Severs (Licenced)

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