Mathematical models play an indispensable, yet often hidden role in our daily life and our understanding of nature. A typical example is the weather forecast, in which complex systems of differential equations, that describe the basic laws of thermo- and hydrodynamics, are simulated. Other models can be found in epidemiology (describing the spreading of infectious diseases), finance and economics (e.g., models for minimization of costs or maximization of profits under constraints, or the determination of fair option prices), or engineering (e.g., artificial neural networks, search engines).
In this presentation, I will derive a few simple mathematical models and discuss their usefulness, but also limitations. I will demonstrate how such models can be simulated to obtain approximate solutions for forecasting certain dynamics in the future. Moreover, instead of only simulating a future behavior, one may want to control and optimize the model to obtain a desired solution behavior. I will show a few such applications of control in engineering applications.
In order to make all the above-mentioned applications computationally feasible, model reduction is often a necessary step to find an approximate model describing the dynamics with a significantly lower number of variables. In the last part of the presentation, I will present some ongoing research in the field of model reduction in which an underlying reproducing kernel Hilbert space structure is exploited.
