Inverse problems occur in many real world application areas such as physics, material sciences, medical imaging or geosciences, thus have attracted many mathematicians' interests from the past decades. Normally, these problems are modeled by certain differential equations with multiple parameters, and the inverse problems are focused on the reconstruction of the unknown parameters from the measurements of the solutions. They are also closely connected to hot topics such as machine learning and neural networks.
The inverse problems are always ill-posed, which means the existence, uniqueness and stability of the solutions do not hold. What's worse, these inverse problems are usually non-linear. Thus special measures, such as regularization or linearization, have to be applied in order develop efficient numerical methods. We will go through the regularization technique for both linear inverse problems, and nonlinear ones combined with the linearization technique in this lecture. Despite the universal tools to solve general inverse problems, we will also introduce the Radon transformation and the factorization method, which are particular high efficient methods in the fields of computational tomography (CT) and Electrical impedance tomography (EIT).
The content (planned) of the lecture is listed as follows:
- Introduction: ill-posed problems
- Regularization techniques for linear inverse problems
- Linearization methods for nonlinear inverse problems
- The computational tomography and the Radon transform
- The electrical impedance tomography and the factorization method
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An Introductory Note on Machine Learning
Requirements: Analysis I, II, III, Linear Algebra I, II
Examination: oral exam
Literature:
- A.Kirsch: An introduction to the mathematical theory of inverse problems. Applied Mathematical Sciences 120, Springer, 2021
- A.Rieder: Keine Probleme mit Inversen Problemen. Vieweg, 2003
- M. Hanke, A Taste of Inverse Problems. Basic Theory and Examples, SIAM, Philadelphia, 2017.
- H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers Group, Dordrecht, 1996.
- Trainer/in: Ruming Zhang