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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.

Sidansvarig: karin.johansson@liu.se