Despite tremendous advances in computing power over the past decades, many numerical computations, such as the simulation of large climate models, can still be extremely costly. In such cases, many-query applications like the solution of inverse problems or the optimization of a mathematical model for a certain set of design parameters, are often still infeasible. In other cases, the need for real-time solutions under small computational resources might render numerical simulation impossible.
In this course, we will be concerned with so-called model order reduction (MOR) techniques, which build on top of existing numerical models and exploit their inherent low-rank dynamics to construct efficient surrogate models that can be simulated orders of magnitude faster than the original model. At the same time, our goal will be to rigorously control the approximation error incurred by replacing the original model by its surrogate.
MOR techniques have been developed somewhat independently in multiple application fields. While certain core ideas are shared, the mathematical tools and language used in different branches of MOR can be quite different. With this course, our is goal to give a solid overview over some of the most important MOR approaches and work out their similarities and differences. The first part of the course will be dedicated to classical projection-based techniques for linear time-invariant systems of ODEs. We will consider both state-space based balancing methods as well as interpolatory methods for approximating the system's transfer function. The second part of the course will cover reduced basis methods for parametric systems of PDEs. We will cover both the classical theory for elliptic, parameter-separable problems, as well as various extensions of the approach towards nonlinear or time-dependent systems. In the third and final part we will cover data-driven MOR methods that build a reduce-order model only from simulation/measurement data without requiring access to a full-order model. In particular, we will discuss dynamic mode decomposition and the Loewner framework.
- Lehrende/r: Hendrik Kleikamp
- Lehrende/r: Stephan Rave