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## 02906 Discrete Inverse Problems

### Danish title:

Diskrete inverse problemer

5

### Course type:

 Ph.D.- Mathematics, Physics and Informatics

### Schedule:

Spring
Weekly meeting as agreed with the teachers. Offered in even years.

### Scope and form:

Seminars and small reports on the computer projects

13 weeks

### General course objectives:

The aim of the course is to give a practical introduction to the numerical treatment of inverse problems (also known as ill-posed problems). Such problems arise, e.g., in connection with the solution of first-kind Fredholm integral equations in medical tomography, geoscience, etc. The course gives the background for inverse problems and the mathematics underlying their solution. The theory is illustrated by Matlab exercises, in such a way that the student gets hands-on experience with some common techniques and paradigms.

### Learning objectives:

A student who has met the objectives of the course will be able to:
• Formulate and identify simple models in the technical sciences, in the form of inverse problems
• Understand the inherent difficulties of inverse problems
• Discretize and solve some first-kind Fredholm integral equations
• Understand the mechanisms of regularization, for stabilization of the solution to an inverse problem
• Implement and use numerical "tools" for the analysis and solution of inverse problems by means of regularization
• Use different methods for choosing the regularization parameter (i.e., the amount of regularization)
• Implement and use iterative methods for large-scale problems, e.g., for tomographic image reconstruction
• Understand, analyze and solve selected inverse problems in differential euqations
• Formulate inverse problems in medical imaging and solve such problems
• Identify nonlinear inverse problems and solve them

### Content:

Integral equations of the first kind. The singular value expansion and the Picard condition. Discretization methods. The singular value decomposition.
Regularization methods (TSVD and Tikhonov). Parameter-choice methods. Iterative regularization methods. Tomographical reconstruction methods.
Inverse problems for partial differential equations. The inverse heat conduction problem. Medical imaging, including Computed Tomography and Electrical Impedance tomogprahy. Nonlinear inverse problems and linearization.

### Remarks:

The course is given only if agreed with the teachers.

### Responsible:

Per Christian Hansen, 303B, 010, (+45) 4525 3097,

### Department:

02 Department of Informatics and Mathematical Modeling

### Department involved:

01 Department of Mathematics