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# Low-rank Tensor Discretization for High-Dimensional Problems

## Module MA5904

This Module is offered by TUM Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

### Basic Information

MA5904 is a semester module in English language at Master’s level which is offered irregular.

This module description is valid from WS 2016/7 to SS 2017.

90 h 30 h 3 CP

### Content, Learning Outcome and Preconditions

#### Content

- 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

#### Learning Outcome

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.

#### Preconditions

Basic knowledge on numerical mathematics and linear algebra.

### Courses, Learning and Teaching Methods and Literature

#### Learning and Teaching Methods

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.

#### Media

blackboard, exercise sheets

#### Literature

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

### Module Exam

#### Description of exams and course work

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).

#### Exam Repetition

The exam may be repeated at the end of the semester.

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