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Contents 1 Course Title 2 Course Code 3 Credit 4 Prerequisites 5 Content 6 Aims 7 Learning Outcomes 8 Suggested Textbooks 9 Outline 10 Resources 11 First Meeting

Course Title

Inverse Problems

Course Code

IAM 664

Credit

(3-0)3

Prerequisites

Basic knowledge in numerical methods, differential equations.

Content

This course introduces into inverse and parameter estimation problems of various, e.g., linear and nonlinear, continuous and discrete, deterministic and stochastic nature. These problems are of strongly growing practical importance. We analyze and numerically-algorithmically treat them. Special attention is paid to computational aspects.

Aims

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, computational biology and social sciences.

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 thesis.

Suggested Textbooks

• A. Aster, B. Borchers, Cliff Thurber, Parameter Estimation and
• Inverse Problems, 2004, Academic Press.
• 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.

Furthermore, a manuscript and recent research articles will be provided during the course.

Outline

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

Resources

MATLAB 6.1

First Meeting

First Meeting (and Lecture / Exercise)

Tuesday, February 17, 9.40-12.30 IAM Building (S-209)