Abstract
This chapter deals with models of living complex systems, chiefly human crowds, by methods of conservation laws and measure theory. We introduce a modeling framework which enables one to address both discrete and continuous dynamical systems in a unified manner using common phenomenological ideas and mathematical tools as well as to couple these two descriptions in a multiscale perspective. Furthermore, we present a basic theory of well-posedness and numerical approximation of initial-value problems and we discuss its implications on mathematical modeling.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Bibliography
L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, second edition, 2008.
L. Bruno, A. Tosin, P. Tricerri, and F. Venuti. Non-local first-order modelling of crowd dynamics: A multidimensional framework with applications. Appl. Math. Model., 35(1):426–445, 2011.
C. Canuto, F. Fagnani, and P. Tilli. An Eulerian approach to the analysis of Krause’s consensus models. SIAM J. Control Optim., 50(1):243–265, 2012.
R. M. Colombo and M. D. Rosini. Existence of nonclassical solutions in a pedestrian flow model. Nonlinear Anal. Real World Appl., 10(5):2716–2728, 2009.
V. Coscia and C. Canavesio. First-order macroscopic modelling of human crowd dynamics. Math. Models Methods Appl. Sci., 18(1 suppl.):1217–1247, 2008.
E. Cristiani, B. Piccoli, and A. Tosin. Multiscale Modeling of Pedestrian Dynamics. In preparation.
E. Cristiani, B. Piccoli, and A. Tosin. Modeling self-organization in pedestrians and animal groups from macroscopic and microscopic viewpoints. In G. Naldi, L. Pareschi, and G. Toscani, editors, Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, pages 337–364. Birkhäuser, Boston, 2010.
E. Cristiani, B. Piccoli, and A. Tosin. Multiscale modeling of granular flows with application to crowd dynamics. Multiscale Model. Simul., 9 (1):155–182, 2011.
E. Cristiani, B. Piccoli, and A. Tosin. How can macroscopic models reveal self-organization in traffic flow? In Proceedings of the 51st IEEE Conference on Decision and Control, Maui, Hawaii, USA, December 2012.
B. Maury, A. Roudneff-Chupin, and F. Santambrogio. A macroscopic crowd motion model of gradient flow type. Math. Models Methods Appl. Sci., 20(10):1787–1821, 2010.
B. Piccoli and F. Rossi. Transport equation with nonlocal velocity in Wasserstein spaces: convergence of numerical schemes. Acta Appl. Math., 124(1):73–105, 2013.
B. Piccoli and A. Tosin. Pedestrian flows in bounded domains with obstacles. Contin. Mech. Thermodyn., 21(2):85–107, 2009.
B. Piccoli and A. Tosin. Time-evolving measures and macroscopic modeling of pedestrian flow. Arch. Ration. Mech. Anal., 199(3):707–738, 2011.
E. Schröedinger. What is Life? Mind and Matter. Cambridge University Press, 1967.
A. Tosin and P. Frasca. Existence and approximation of probability measure solutions to models of collective behaviors. Netw. Heterog. Media, 6(3): 561–596, 2011.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 CISM, Udine
About this chapter
Cite this chapter
Tosin, A. (2014). Multiscale Crowd Dynamics Modeling and Theory. In: Muntean, A., Toschi, F. (eds) Collective Dynamics from Bacteria to Crowds. CISM International Centre for Mechanical Sciences, vol 553. Springer, Vienna. https://doi.org/10.1007/978-3-7091-1785-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-7091-1785-9_6
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-1784-2
Online ISBN: 978-3-7091-1785-9
eBook Packages: EngineeringEngineering (R0)