INSTITUTE OF APPLIED MATHEMATICS
Last Updated:
28/08/2017 - 21:10

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

  1. Introduction
  2. Linear Regression
  3. Least Squares Theory
  4. Discretizing Continuous Inverse Problems
  5. Rank Deficiency and Ill-Conditioning
  6. Tikhonov Regularization
  7. Iterative Methods
  8. Fourier Techniques
  9. Other Regularization Techniques
  10. Nonlinear Inverse Problems
  11. Nonlinear Regression
  12. Nonlinear Least Squares
  13. Bayesian Methods
  14. Application to Tomography
  15. 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)

More Info on METU Catalogue

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