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# MAI0099

Regularization Theory/

Regulariseringteori

**Number of credits:** 8 hp

**Examiner:** Natan Kruglyak

**Lecturer:** Natan Kruglyak and Fredrik Berntsson

**Course literature:** Compendium based on the lectures and book: A. Kirsch, * An Introduction to the Mathematical Theory of Inverse Problems.*

**Aim: ** After the course the student will be able to use concepts and methods of regularisation theory to concrete inverse problems in science and tecnology. This mean that a student which has taken the course is expected to be able:

- understand definitions and properties of Hilbert spaces and operators;
- investigate properties of simple concrete operators;
- find the singular value for concrete operators;
- understand the notion of well-posed and ill-posed problem;
- understand the algorithms of considered regularization methods and apply them to concrete problems using MATLAB;
- describe definitions and derive relations between the central concepts of the course and apply these relations to solve concrete problems;
- interpret, communicate and argue using mathematical notions.

**Course contents:** Euclidian and Hilbert spaces. Orthonormal systems. Linear, bouded and compact operators. Adjoint operators. Singular value decomposition for compact operators in Hilbert spaces, Ill-posed problems. Linear regression. Least square method. Cutoff and Tikhonov regularization. The discrepancy principle of Morozov. Landweber iteration method. Galerkin and Collocation methods.

**Organisation:** Lectures and seminars. Compulsory assignments may be given during the course.

**Examination:** Oral presentation of the MATLAB project and written exam on the theory of inverse problems.

**Prerequisites:** Basic Linear Algebra and Basic Fourier analysis.

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Last updated: 2022-11-14