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Fallstudien der Mathematischen Modellbildung: Teil 1 (MA 2902) - WS 18/19

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DozentMassimo Fornasiermassimo.fornasierematma.tum.de MI 02.10.058
ÜbungsleitungNada Sissounosissounoematma.tum.de MI 02.10.037
VorlesungstermineTue. 12:15 - 13:4500.06.011, MI Hörsaal 3 (5606.EG.011)
Wed. 17:00 - 18.30CH 21010, Hans-Fischer-Hörsaal (5401.01.101K)
ÜbungsterminMo. 16:00 - 18:0000.06.011, MI Hörsaal 3 (5606.EG.011)

Aktuelles

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Lecture The lecture by Prof. Fornasier will be held - as long as it is not differently recommended - in hybrid format, both in presence and online. The presence teaching will take place on Tue. 12:15 - 13:45 and Wed. 17:00 - 18.30 in HS3. The access to the HS3 will be allowed from 15 minutes before the start of the lecture until 5 minutes after the start of the lecture. During these 20 minutes the compliance with the 3G rule will be checked for ALL students. The lectures will be both streamed simultaneously online and recorded via zoom and the recordings will be posted on this website. For this reason and despite the fact that presence teaching will be offered, given the current critical situation of the pandemics of COVID-19 in Munich, students are warmly invited to consider also the attendance of the lecture online. For those students who would choose to attend the lecture online, the zoom access is the following
Tue.

In case the situation of the pandemics will require it or the authorities will require it, teaching in presence will be suspended and completely delivered online.


Tutorials
The tutorial by Mr. Michael Rauchensteiner will be offered online starting on Monday 29.11. More information about the time

Inhalt der Vorlesung

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Can one of the most important Italian Renaissance frescoes reduced in hundreds of thousand fragments
by a bombing during the Second World War bere-composed after more than 60 years from its damage? Can we
reconstruct the missing parts and can we say something about their original color?
Our lectures on Fallstudien der Modellbildung starts by exemplifying, hopefully effectively by taking advantage of the seduction of art,
how mathematics today can be applied in real-life problems which were considered unsolvable only few years ago,
see reference [1].
We shall continue our lectures digging into the world of digitalization, i.e. how images, sounds, signals, opinions,
emotions, etc. can be made numbers and then elaborated via mathematical algorithms. For that we will need to learn
some of the fundamentals of harmonic analysis, see reference [3], in particular the Fourier theorem in Hilbert spaces
and its concrete application to define Fourier series, transforms, and the algorithm of the Fast Fourier Transform (FFT),
see [Chapter 1, 2] and [3]. This will introduces us to the problem of estimating how good is the approximation in
computing a Fourier Transform of a function from its samples, and the Shannon sampling theory, see [Chapter 2, 2] and [3].
To address the analysis of signals in their time-frequency nature, we shall explore the tools provided by Gabor
analysis, in particular the so-called Gabor transform and its discretization via frames, see [Chapter 3, 2] and [4,5,6].
We conclude the lectures again by returning to the beginning, and by analysing in details the mathematics behind the fresco
restoration problem, as our main inspirational Fallstudium.

In these lectures we follow very closely the Skriptum [2], which collects in short several results from other texts, in particular
[3,4,5,6]. As the Skriptum is currently available only in Italian we shall give at the end of each lecture a synthesis in the form
of Slides (Folien) in English, which will be posted online in PDF. The course and the exercises in this part of Fallstudien will
be held in English.

Slides

Introduction lecture, Oct. 16, 2018
Slides of the lecture 1, Oct. 17, 2018
Slides of the lecture 2, Oct. 23-24, 2018
Slides of the lecture 3, Oct. 30-31, 2018
Slides of the lecture 4, Nov. 6-7, 2018

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Übungsblätter

uploaded in moodle!

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Themen

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Bewertung/Hausarbeiten

Richtlinien zu Hausarbeiten und Bewertung

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Literatur

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1. M. Fornasier, Mathematics enters the picture, Proceedings of the conference Mathknow 2008 [ .pdf ]
2. M. Fornasier Introduzione all'analisi armonica numerica (Italian), Lecture notes, 2007 112 pp. [ .pdf ]
3. D. W. Kammler, A First Course in Fourier Analysis, Prentice Hall, Upper Saddle River, New Jersey 07458, 2000. [ ref ]
4. C. Heil, A Basis Theory Primer, Birkhaeuser, 1998. [ .pdf ]
5. O. Christensen, An Introduction to Frames and Riesz Bases, Birkhaeuser, 2003.
6. H. G. Feichtinger, F. Luef, T. Werther, A Guided Tour from Linear Algebra to the Foundations of Gabor Analysis, Univ. of Vienna, August 2005 [ .pdf ]