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Tensorapproximationen für hochdimensionale Probleme
Low-rank Tensor Discretization for High-Dimensional Problems

Modul MA5904

Dieses Modul wird durch Fakultät für Mathematik bereitgestellt.

Diese Modulbeschreibung enthält neben den eigentlichen Beschreibungen der Inhalte, Lernergebnisse, Lehr- und Lernmethoden und Prüfungsformen auch Verweise auf die aktuellen Lehrveranstaltungen und Termine für die Modulprüfung in den jeweiligen Abschnitten.

Basisdaten

MA5904 ist ein Semestermodul in Englisch auf Master-Niveau das unregelmäßig angeboten wird.

Die Gültigkeit des Moduls ist von WS 2016/7 bis SS 2017.

GesamtaufwandPräsenzveranstaltungenUmfang (ECTS)
90 h 30 h 3 CP

Inhalte, Lernergebnisse und Voraussetzungen

Inhalt

- Tensor formats: Canonical format, Tucker format, hierarchical Tucker format, tensor trains
- Tensor approximations in hierarchical Tucker and tensor train format
- Tensor operations in hierarchical Tucker and tensor train format
- Generalized cross approximation
- Alternating least squares method and the solution of low-rank linear systems
- Applications to approximation of high-dimensional partial differential equations and parametric partial differential equations
- Programming exercises in Matlab

Lernergebnisse

Upon successful completion of this module, students are able to
- understand the concept of low-rank tensor approximation, algorithms for the approximation of high-dimensional tensors, and the implementation of basic tensor operations.
- compare various low-rank tensor formats and algorithms,
- design Matlab programs for the approximation of multi-variate functions, the solution of high-dimensional partial differential equations or the quantification of uncertainties based on a low-rank tensor representation of the solution.

Voraussetzungen

Basic knowledge on numerical mathematics and linear algebra.

Lehrveranstaltungen, Lern- und Lehrmethoden und Literaturhinweise

Lern- und Lehrmethoden

The module is offered as lectures with a few accompanying practice sessions. In the lectures, the content will be presented with blackbord notes and discussions with the students. Small exercises will be provided for each lecture with which students can deepen their understanding and check their progress. The lecture will also give an overview of the relevant literature and guide the students in their further study of the subject. Approximately three sessions will be dedicated to practical programming exercises where students can apply their acquainted knowledge to create small Matlab programs based on low-rank tensor functions.

Medienformen

blackboard, exercise sheets

Literatur

Wolfgang Hackbusch, Tensor Spaces and Numerical
Tensor Calculus, Springer, 2012.

Modulprüfung

Beschreibung der Prüfungs- und Studienleistungen

In the oral examination (approximately 30 minutes) students show their knowledge on the theoretical foundations of low-rank tensor approximation, algorithmic understanding as well as their ability to apply their knowledge to applications (including a discussion of the results of the programming exercises).

Wiederholbarkeit

Eine Wiederholungsmöglichkeit wird am Semesterende angeboten.

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