Inverse problems and imaging

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.

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.

Lectures and exercises with theory and computations.