The lectures will give an introduction to linear inverse problems and their (classical) theory. This will by done by means of some simple linear problems, in the form of deconvolution problems. The lectures will start with some "toy problems" from geophysics and signal/image processing, and then use all the tool-of-the-trade from numerical linear algebra, primarily the SVD, to explain the inherent difficulties. With this background, you will be ready for the parameter-estimation problems discussed later in the week. The lectures will be based on the surver paper on deconvolution methods and Toeplitz matrices. However, the Toeplitz (computational) aspects will be toned down and the focus will be on the above-mentioned aspects.

The lectures will focus on parameter identification problems for PDEs. Also these lectures will start with basic material, typical challenges, problem formulation (output least squares, etc), and thereafter move on to more complex methods and solution strategies (Lagrange multipliers, level set, software aspects, etc). Examples will be taken from material testing and geophysics, and the inverse conductivity problem will be mentioned briefly.

The lectures will discuss parameter identification by level set and related methods. We start with the phase filed model where one need to find interfaces for motion by mean curvature. Then some other interface problem like image segmentation and optimal shape design problems will be presented. After this, a relation between the phase field model and the level set method will be explained. In the end, some new piecewise constant level set methods will be presented and their applications, strengthens and weakness will be discussed. Some details shall be given for these new piecewise constant level set methods.

The lectures will focus on various inverse problems arising in connection with ECG (electrocardiogram) recordings. More specifically, the three lectures can roughly be described as follows:

Emphasis will be put on making the lectures as self-contained as possible, and we assume no specialized knowledge, except for a general knowledge of PDEs, numerical methods and analysis.

Published October 20, 2010

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