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Vorlesung: 03-M-SP-1 Inverse Problems - Details

Vorlesung: 03-M-SP-1 Inverse Problems - Details

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Veranstaltungsname Vorlesung: 03-M-SP-1 Inverse Problems
Untertitel
Veranstaltungsnummer 03-M-SP-1
Semester SoSe 2025
Aktuelle Anzahl der Teilnehmenden 22
Heimat-Einrichtung Mathematik
Veranstaltungstyp Vorlesung in der Kategorie Lehre
Erster Termin Donnerstag, 10.04.2025 12:00 - 14:00, Ort: MZH 2340
Art/Form Lecture including Exercises
Englischsprachige Veranstaltung Ja
ECTS-Punkte 9

Räume und Zeiten

MZH 4140
Donnerstag: 12:00 - 14:00, wöchentlich (9x)
Freitag: 12:00 - 14:00, wöchentlich (12x)
Freitag: 14:00 - 16:00, wöchentlich (11x)
MZH 2340
Donnerstag: 12:00 - 14:00, wöchentlich (3x)
Freitag: 12:00 - 14:00, wöchentlich (1x)
Freitag: 14:00 - 16:00, wöchentlich (2x)

Modulzuordnungen

Kommentar/Beschreibung

Inverse problems are problems where one would like to find an unknown cause for which one can only measure observed effects. This situation occurs, for example, if one can only make indirect measurements of the quantity of interest. Two simple examples: \begin{itemize} \item We measure the position of an object, but would like to know the speed. \item In tomography we measure several projections (X-ray images) of an object, but would like to know the absorption spectrum of said object. \end{itemize} Inverse problems usually suffer from ill-posedness: Solutions may not be unique, they may not exist (for example due to measurement noise), and, most drastically, their solution is unstable in the sense that it does not depend continuously on the data. We will analyze the phenomenon on ill-posedness for linear inverse problems (modeled as linear and continuous maps between Hilbert spaces) to understand the reason for instability. A central goal of the course is to establish the notion of regularization of ill-posed problems (which roughly means the approximate solution by stable methods) and to derive and analyze regularization methods such as Tikhonov regularization, or the Landweber method with early stopping. We will also treat the numerical solution of inverse problems in the lecture and the exercises.

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