Title: Reduced Basis Methods - Prospects and Challenges

The Reduced Basis Method (RBM) has become widely accepted and frequently used numerical method for model reduction of parameterized partial differential equations (PPDE) in cases where the PPDE has to be solved very often for many parameters (multi-query, examples are optimization or statistics), extremely fast (real-time, an example is optimal control) and/or with strong limitations concerning memory (e.g. in embedded systems).

From a mathematical point of view, we want that both the computation of a reduced system (typically done “offline”) and the reduced approximation itself (to be computed “online”) are certified in the sense that the error with respect to the exact solution of the PPDE can be controlled. This leads to the question, which kind of PPDEs can be reduced by the RBM at all.

In this talk, we will review some types of PPDEs (even from industrial problems) that can be treated very well using the RBM and indicate current prospects and challenges for example for transport or hyperbolic problems.

Karsten Urban is a professor at Ulm University in Germany. He studied Mathematics and Computer Science at Bonn University and RWTH Aachen, and obtained his Ph.D. in Mathematics at RWTH Aachen 1995 (supervisor: Wolfgang Dahmen).