Mathematik - SoSe 2025

Course description:

In many applications, we are interested in studying an object of interest x. However, its properties are often not directly measurable. Instead, we only observe y = F(x), where the function F is known. To infer about x from the observation y, we seek an (approximate) inverse mapping x ≈ F^-1(y). This process is known as solving the inverse problem.

In this course, we will cover three topics. The first is classical regularization theory, which highlights the difficulties of solving inverse problems related to noisy data. In the second part, we move to the variational formulation of inverse problems and discuss optimization strategies. Finally, the last part focuses on explainable data-driven and network-based approaches for inverse problems.  

Planning:

Credits: 9 ECTS

Modules:
MSc Math: WP35   
MSc FiMa: WP12 or WP22
MSc Stat&DS: WP 31 or WP 52
MSc DS: WP 3

Enrolment Key: N*OLC,1s+EGI96rqDea-

Target Audience: MSc Math students 

Lectures: Tuesdays 16:00-18:00 in room B006 and Fridays 10:00-12:00 in room A027 in Theresienstr. 39
Tutorials: Mondays 12:00-14:00 or 16:00-18:00 in room 504 in Akademiestr. 7 (5th floor).

Dienstag 10-12 Uhr in B041 und Mittwoch 8.30-10 Uhr in B041;
Übung Mittwoch 14-16 Uhr in B046

Wir starten am ersten Vorlesungstag, Mittwoch 23.4.2025 um 8.30 Uhr..

Einschreibeschlüssel: fock

Einschreibeschlüssel: Grundlagen2S25

Einschreibeschlüssel: LA2SS25

Einschreibeschlüssel: An2S25

Kursbeschreibung

In dieser Vorlesung werden bedingte Erwartungswerte mithilfe des Satzes von Radon-Nikodým eingeführt und ihre wesentlichen Eigenschaften hergeleitet. Zudem werden die Grundlagen der Theorie stochastischer Prozesse in diskreter Zeit, mit einem starken Fokus auf Martingale, behandelt. Dazu werden wir uns mit relevanten Methoden aus der Maß- und Integrationstheorie befassen und insbesondere das starke Gesetz der großen Zahlen, das Null-Eins-Gesetz von Kolmogorov und die Lemmata von Borel-Cantelli beweisen.

Vorgesehen für Studierende folgender Studiengänge

Course Description

In this seminar, we will study one-dimensional stochastic Volterra integral equations with a particular focus on fractional kernels. For this purpose, fundamental concepts in stochastic analysis, such as semimartingales, stochastic integrals and fractional Brownian motion, will be discussed. The aim of this course is to derive unique strong solutions under Lipschitz conditions on the drift and diffusion coefficients of the stochastic Volterra equation.

Target Participants

Blockkurs in der vorlesungsfreien Zeit, in dem die Grundlagen des Textsatzsystems LaTeX und speziell die Erstellung mathematischer Texte behandelt werden. Nähere Informationen zu Ablauf und Anmeldung auf der Webseite des Kurses.

Enrolment key: aml25

Credits: 6 ECTS (2SWS lecture + 2SWS exercise)

Modules: 

• MSc FiMa: WP23 „Advanced Topics in Computer and Data Science B” 
  
• MSc Math: WP42 „Überblick über ein aktuelles Forschungsgebiet B” 

Description:

Real-world applications of machine learning require not only a strong theoretical foundation but also a solid knowledge of the methodologies, tools, and heuristics essential for implementing machine learning algorithms. However, the practical aspects of machine learning are often overlooked in mathematics programs. This course bridges that gap by providing students with hands-on experience in implementation and empirical analysis of machine learning algorithms — critical skills for those pursuing careers in data analysis or machine learning.

Content:

A key component of the course is extensive programming in Python, using libraries such as NumPy, Matplotlib, Pandas, and scikit-learn. We will work with real datasets, including MNIST handwritten digits, the Boston Housing dataset, Wine dataset, etc.