After successful completion of the course, students are able to research and work out a topic with given literature. They can present the results in a talk and discuss them with the fellow participants.

Many physical phenomena are too complex in order to be simulated in all their aspects using standard algorithms of numerical analysis. An example are optimisation problems, in which the evaluation of the goal functional requires the solution of a parameter dependent system of differential equations. In the course of the optimisation, this system has to be solved for many different values of these parameters. The goal of model order reduction is to identify the essential degrees of freedom that drive the system of differential equations. For the optimisation problem this means that in every optimisation step only a much smaller system needs to be solved. This greatly improves the performance of the method

In the course of this seminar, different aspects and algorithms will be investigated. The focus is on parameter-dependent PDES, i.e., PDEs for which the coefficients or the geometry, and thus the solution, depend on some external parameter.

The questions that arise are for example: For which values of the parameters should the PDE be solved? How can these so-called snapshots be used to efficiently get solutions for other parameter values? How large are the errors incurred by this procedure?


Possible topics include:

- reduced basis methods

- proper orthogonal decomposition

- greedy algorithms,

-a-posteriory error control

- best approximation theory and Kolmogorov n-widths

Supervised development of a chosen topic using the scientific literature (textbooks, scientific articles). Presentation of the students with feedback.

Successful presentation of the Topic

Numerical Analysis