In this course, students will learn the fundamentals of mathematical modeling.
A model is an idealized representation of the real world and mathematical modeling is the process by which the complexity of real-world phenomena is simplified such that they can be described, more abstractly, in mathematical terms. The hope is that this abstraction of the real world can then help us comprehend the essence of the phenomena we wish to understand and/or predict.
As such, the concept of mathematical modeling is common to virtually all empirical sciences—not only natural and engineering sciences but also human sciences such as economics, psychology, sociology, etc.
The lectures introduce several types of mathematical modeling strategies: optimization, modeling through dynamical systems (be they continuous, discrete or of cellular automaton-type) and, if time allows, probabilistic models.
The introductory part starts with the concept of optimization and the mathematical methods needed to tackle such problems. The middle and main part explores the behaviour of both discrete and continuous dynamical systems. The focus will lie on stability analysis for such systems as well as on the relationship between discrete and continuous models (continuum limits and discretisation methods). In the last part (if time permits), techniques for modeling probabilistic phenomena and stochastic processes as well as the concept of a Monte Carlo simulation will be introduced.