Modeling Crowd Dynamics

by Felix Dietrich (01.07.2015)

**In the last two decades the interest in crowd dynamics has increased, following the advent of cheap computing power and larger and more frequent crowd events in urban areas. Experiments with humans are both difficult and expensive and can therefore be applied in only a small number of cases, such as airplane or ship development. Building models of pedestrians and simulating the system enables computer experiments that operate faster and on a much larger scale (see Figure 1 for a simulation and analysis of the entrance queue to a festival).**

Pedestrians are influenced by a large number of variables, from surrounding pedestrians, road and weather conditions up to personal experiences. This is why numerous models for both crowds and single pedestrians exist. They can roughly be classified into microscopic and macroscopic models, ranging from explanations of individual stepping behavior to modified models of fluid dynamics for large and dense crowds. When performing numerical analysis with computers, the definition of a system on a microscopic instead of a macroscopic scale poses severe limitations on how often the system can be queried for a macroscopic analysis. Many existing numerical techniques can be employed on systems with an inherent scale gap, either spatially or in time. They work as follows: first, scale up the model to construct a computationally less demanding one. Then analyze the model on the new scale that allows a much larger number of queries, and finally draw conclusions to the original model. The existing multi-scale methods are worth investigating, since the properties of a system in crowd dynamics often are on a different scale than the models that try to capture them.

We try to answer the following questions: Which methods for scale transitions in dynamical systems theory and numerics are useful in crowd dynamics? How can they be used to analyze models?

Up to this point, the contributions of the author summarized the state of the art [1] and closed certain gaps in the types of models [3,4]. Further work will concentrate on numerical analysis and upscaling of a more general class of systems [2]. A first goal of the thesis is to establish a more rigorous and generalized understanding of crowd models embedded in dynamical systems theory. On this basis, the methods for scale transition and multiscale analysis outlined above will be tested for their ability to work with these models. The last step is to actually apply the methods in order to analyze models in crowd dynamics.

**References**

[1] Felix Dietrich and Gerta Köster. Gradient navigation model for pedestrian dynamics. Physical Review E, 89(6):062801, 2014.

[2] Felix Dietrich, Gerta Köster, and Hans-Joachim Bungartz. Numerical model construction with closed observables. arXiv, 1506.04793:v1, 2015.

[3] Felix Dietrich, Gerta Köster, Michael Seitz, and Isabella von Sivers. Bridging the gap: From cellular automata to differential equation models for pedestrian dynamics. Journal of Computational Science, 5(5):841--846, 2014.

[4] Michael J. Seitz, Felix Dietrich, and Gerta Köster. The effect of stepping on pedestrian trajectories. Physica A: Statistical Mechanics and its Applications, 421:594--604, 2015.

**seit 10/2013**- Promotionsstudium Mathematik, Arbeitsgebiet: Numerische Analyse Dynamischer Systeme, Personenstromdynamik, Technische Universität München
**2013-2014**- M.Sc. Mathematik, Technische Universität München
**2011-2013**- B.Sc. Mathematik, Technische Universität München
**2008-2011**- B.Sc. Scientific Computing, Hochschule München

- * Stemmer Imaging GmbH - Preis für eine hervorragende Abschlussarbeit in der Bildverarbeitung (2012)
- * Max Weber-Programm (2009-2014)
- * Studienstiftung des deutschen Volkes (2009-2014)

- * F. Dietrich, G. Koester, H.-J. Bungartz, Numerical Model Construction with Closed Observables. arXiv pre-print, 06/2015
- * M. J. Seitz, F. Dietrich, G. Koester, A study of pedestrian stepping behaviour for crowd simulation. The Conference in Pedestrian and Evacuation Dynamics 2014, 09/2014.
- * F. Dietrich, G. Koester, M. J. Seitz, I. von Sivers, Bridging the gap: From cellular automata to differential equation models for pedestrian dynamics. Journal of Computational Science, 5, 841-8738, 09/2014.
- * F. Dietrich, G. Koester, Gradient navigation model for pedestrian dynamics. Physical Review E, 89, 062801, 01/2014.
- *P. A. Pellett, F. Dietrich, J. Bewersdorf, J. E. Rothman, G. Lavieu, Inter-Golgi transport mediated by COPI-containing vesicles carrying small cargoes. eLife, 2, e01296-e01296, 10/2013.