Single-Course English 5 ECTS

# Inverse problems and Imaging

## Overall Course Objectives

The aim of the course is to give a systematic, mathematical introduction to inverse problems in science and engineering. Such problems arise, e.g., in connection with medical imaging, tomography, seismic exploration, and deblurring og images. The course gives the mathematical background for inverse problems and their solution. The theory is illustrated by computational exercises that gives hands-on experience with some common techniques and paradigms.

Learning Objectives

• 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 first-kind Fredholm integral equations
• Understand the mechanisms of regularization for the solution to an inverse problem
• Implement and use computational tools for the analysis and solution of inverse problems by means of regularization
• Use different methods for choosing the regularization parameter
• Implement and use iterative methods for large-scale problems
• Understand, analyze and solve selected inverse problems in differential euqations
• Formulate inverse problems in imaging, computerized tomography and electrical impedance tomography
• Identify nonlinear inverse problems and solve them
• Explain and utilize the Bayesian approach to inverse problems

Course Content

Integral equations of the first kind. The singular value expansion for compact operators in Hilbert spaces and the Picard condition. Discretization methods. Regularization methods (TSVD and Tikhonov). Parameter-choice methods. Iterative regularization methods. Tomographic 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. Bayesian inverse problems.

Recommended prerequisites

01035/02601, Experience with advanced engineering mathematics (e.g. 01035) and numerical computations (e.g., 02601)

Teaching Method

Lectures and exercises with theory and computations. Project work towards the end of the course.

Faculty

Limited number of seats

Minimum: 10.

Please be aware that this course will only be held if the required minimum number of participants is met. You will be informed 8 days before the start of the course, whether the course will be held.

See course in the course database.

## Registration

Language English 13 weeks Compute DTU Lyngby Campus 02624 Candidate Week 35 Week 48 Wed 13-17
Price 7.500,00 DKK