Modul
Bayes'sche inverse Probleme und deren Verbindungen zum maschinellen Lernen [M-MATH-106328]
Leistungspunkte
4Turnus
Jedes SommersemesterDauer
1 SemesterSprache
EnglischLevel
4Version
1Verantwortung
Einrichtung
- KIT-Fakultät für Mathematik
Bestandteil von
Teilleistungen
Identifier | Name | LP |
---|---|---|
T-MATH-112842 | Bayes'sche inverse Probleme und deren Verbindungen zum maschinellen Lernen | 4 |
Erfolgskontrolle(n)
Die Modulprüfung erfolgt in Form einer mündlichen Gesamtprüfung (ca. 30 Minuten).
Qualifikationsziele
After completing the module's classes and the exam, students will be familiar with the theory of inverse problems. They will be able to apply the Bayesian framework to a given inverse problem and assess the
well-posedness of the Bayesian posterior. In addition, students will be able to describe the basics of several solution methods for accessing the Bayesian posterior, including approximation and machine-learning techniques, and their limitations. Finally, they will be able to name and discuss essential theoretical concepts for Bayesian inversion in Banach spaces and describe the suitable sampling-based solution techniques. In particular, the course prepares students to write a thesis in the field of Uncertainty Quantification.
Voraussetzungen
Keine
Inhalt
The course offers an introduction to the subject of statistical inversion, where, in its most basic form, the goal is to study how to estimate model parameters from data. We will introduce mathematical concepts and computational tools for systematically treating these inverse problems in a Bayesian framework, including an assessment of how uncertainties affect the solution. In the first part of the course, we will study the Bayesian framework for finite-dimensional inverse problems. While the first part will introduce some machine-learning ideas, the second part will address how machine learning is impacting, and has the potential to impact further on, the subject of inverse problems. In the final part of the course, we will generalize the Bayesian inverse problem theory to a Banach space setting and discuss sampling strategies for accessing the Bayesian posterior.
Topics covered include:
- Bayesian Inverse Problems and Well-Posedness
- The Linear-Gaussian Setting
- Optimization Perspective on Bayesian Inverse Problems
- Gaussian Approximation
- Markov Chain Monte Carlo
- Blending Inverse Problems and Machine-Learning
- Bayesian Inversion in Banach spaces
Empfehlungen
Die Inhalte der Module 'M-MATH-101321 - Einführung in die Stochastik' und 'M-MATH-103214 – Numerische Mathematik 1+2' sowie 'M-MATH-106053 — Stochastic Simulation' werden empfohlen.
Arbeitsaufwand
Gesamter Arbeitsaufwand: 120 Stunden
Präsenzzeit: 45 Stunden
- Lehrveranstaltung einschließlich studienbegleitender Modulprüfung
Selbststudium: 75 Stunden
- Vertiefung der Studieninhalte durch häusliche Nachbearbeitung des Vorlesungsinhaltes
- Bearbeitung von Übungsaufgaben
- Vertiefung der Studieninhalte anhand geeigneter Literatur und Internetrecherche
- Vorbereitung auf die studienbegleitende Modulprüfung