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Inverse Problems - Einzelansicht

Grunddaten
Veranstaltungsart Vorlesung Langtext
Veranstaltungsnummer 100410 Kurztext
Semester WiSe 2022/23 SWS 4
Erwartete Teilnehmer/-innen Studienjahr
Max. Teilnehmer/-innen
Credits Belegung Belegpflicht
Hyperlink
Sprache englisch
Termine Gruppe: [unbenannt] iCalendar Export für Outlook
  Tag Zeit Rhythmus Dauer Raum Raum-
plan
Lehrperson Status Bemerkung fällt aus am Max. Teilnehmer/-innen
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Mo. 12:00 bis 14:00 woch 10.10.2022 bis 30.01.2023  Orléans-Ring 12 - SRZ 204        
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Do. 12:00 bis 14:00 woch 13.10.2022 bis 02.02.2023  Orléans-Ring 12 - SRZ 204        
Gruppe [unbenannt]:
 


Zugeordnete Person
Zugeordnete Person Zuständigkeit
Wübbeling, Frank, Dr. verantwort
Studiengänge
Abschluss - Studiengang Sem ECTS Bereich Teilgebiet
Master - Mathematics (88 F23 20) -
Master - Mathematik (88 105 13) -
Master - Mathematik (88 105 10) -
Prüfungen / Module
Prüfungsnummer Modul
18004 Vorlesung 2 (mit Studienleistung) - Master Mathematik Version 2013
17004 Vorlesung 2 (mit Studienleistung) - Master Mathematik Version 2013
17001 Vorlesung 1 - Master Mathematik Version 2013
18001 Vorlesung 1 - Master Mathematik Version 2013
20001 Lecture 1 - Master Mathematics Version 2020
19001 Lecture 1 - Master Mathematics Version 2020
19003 Lecture 2 - Master Mathematics Version 2020
20003 Lecture 2 - Master Mathematics Version 2020
11007 Vorlesung zur angewandten Mathematik 1 - Master Mathematik Version 2013
11009 Vorlesung zur angewandten Mathematik 2 - Master Mathematik Version 2013
11012 Vorlesung zur angewandten Mathematik 3 - Master Mathematik Version 2013
11007 Lecture 1 (Applied Mathematics) - Master Mathematics Version 2020
11009 Lecture 2 (Applied Mathematics) - Master Mathematics Version 2020
11012 Lecture 3 (Applied Mathematics) - Master Mathematics Version 2020
Zuordnung zu Einrichtungen
Fachbereich 10 Mathematik und Informatik
Inhalt
Kommentar

Simulations are a typical object of study in applied mathematics. Some examples are:

  • Predicting the weather of tomorrow measuring the weather of today
  • Given a sound source (a singer) and the shape of the auditorium, compute what is heard by the listeners
  • Assume an X-Ray is attenuated by a bone. How much of the ray intensity is left after the bone?

These problems, which can also be solved by experiments, are direct problems. Often, their modeling involves Differential Equations. Typically, the solution to the problem exists, is unique, and is stable in the sense that small measurement errors will lead to small errors in the result. According to Hadamard, problems with these properties are well-posed.

However, many problems do not fall into this category. Some examples are:

  • Given what a listener hears in an auditorium, can you identify the sound source? (Inverse Scattering Problem, in: Kac 1966, Can one hear the shape of a drum?)
  • Given the amount of ray intensity that is lost when an X-ray travels through the body, can you identify the bone shape inside the body? (computerized tomography)
  • Assume you measure resistance on the surface of the body. Can you identify the conductivity inside the body? (impedance tomography)
  • Assume you measure an EKG, which more or less measures the electric current that the heart sends out. Can you identify the shape of the heart? (inverse electrocardiography)
  • Measuring the weather of today, can you identify the weather of yesterday? (time-reversal)

These problems can not be solved with experiments, but only mathematically. Their name is inverse problems. Typically, they lose all the nice properties above: there is no unique solution, and if it exists, it does not depend on the data in a stable way. Problems with these properties are ill-posed.

We will start off the lecture by examining inverse problems in an analytical way. In the second part, we will use this to look at various problems and compute solution strategies.

While we always have an application in mind, of course, no physical knowledge is needed. However, it will be part of the homeworks to implement and analyze the algorithms we develop, so some programming experience is useful. The examples in the lecture will be given in Python, but all programming languages are acceptable.

Inverse Problems are a field that borrow stuff from many fields. While everything is covered in the lecture, it helps to have a basic understanding of Functional Analysis, PDE and/or stochastics.

I ask that all participants join the Learnweb course even before the lecture starts, so that I can pass on some information, if needed. The course already contains some videos of 2020 with an informal introduction to two inverse problems. The password is RadonTrafo. 

Literatur

Specified in the Learnweb. A script for this lecture will be provided, so there is no compulsory literature you should buy.

Voraussetzungen

Bachelor-Courses in Applied Math.

Leistungsnachweis

Solving the exercises and an oral exam.


Strukturbaum
Keine Einordnung ins Vorlesungsverzeichnis vorhanden. Veranstaltung ist aus dem Semester WiSe 2022/23 , Aktuelles Semester: SoSe 2023