Comptes Rendus
Defective Laplacians and paradoxical phenomena in crowd motion modeling
Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 615-646.

In both continuous and discrete settings, Laplace operators appear quite commonly in the modeling of natural phenomena, in several context: diffusion, heat propagation, porous media, fluid flows through pipes, electricity.... In these contexts, the discrete Laplace operator enjoys all the properties of its continuous counterpart, in particular: self-adjointness, variational formulation, stochastic interpretation, mean value property, maximum principle, ...In a first part, we give a detailed description of the correspondence between these mathematical properties and modeling considerations, in contexts where the continuous and the discrete settings perfectly match. In a second part, we describe a pathological situation, in the context of granular crowd motion models. Accounting for the non-overlapping constraint between hard discs leads to a particular operator acting on a field of Lagrange multipliers, defined on the dual graph of the contact network. This operator is defective in a certain sense: although it is the microscopic counterpart of the macroscopic Laplace operator, this discrete operator indeed lacks some properties, in particular the maximum principle. We investigate here how this very defectivity may explain some paradoxical phenomena that are observed in crowd motions and granular materials, phenomena that are not reproduced by macroscopic models.

Aux niveaux continu et discret, l’opérateur de Laplace intervient de façon très courante dans la modélisation de phénomènes naturels, dans de nombreux contextes : diffusion, propagation de la chaleur, milieux poreux, écoulement de fluides dans des conduits, électricité.... Dans ces contextes, le laplacien discret possède toutes les propriétés de son pendant continu : caractère auto-adjoint, structure variationnelle, interprétation stochastique, propriété de la valeur moyenne, principe du maximum, .... Dans une première partie, nous proposons une description détaillée des liens entre ces propriétés et les aspects de modélisations, dans des contextes où les notion continues et discrètes se correspondent parfaitement. Dans une seconde partie, nous décrivons une situation pathologique, dans le contexte de la modélisation de foules d’un point de vue granulaire. La prise en compte de la contrainte de non recouvrement entre grains rigides conduit à un opérateur particulier qui agit sur les champs de multiplicateurs de Lagrange, définis sur le graphe dual du réseau de contacts. Cet opérateur est déficient dans un certain sens  : bien qu’il apparaisse comme le pendant discret du laplacien continu, il ne vérifie pas certaines des propriétés usuelles du laplacien, en particulier le principe du maximum. Nous explorons comment cette déficience permet d’expliquer certains effets paradoxaux observés en mouvements de foules, que les modèles continus ne reproduisent pas.

Received:
Revised:
Accepted:
Online First:
Published online:
DOI: 10.5802/crmeca.205
Keywords: Discrete Laplace operator, maximum principle, crowd motion, faster-is-slower effect
Mot clés : Laplacien discret, principe du maximum, mouvements de foules, effet faster-is-slower

Bertrand Maury 1

1 LMO, Université Paris-Saclay, F-91405 Orsay cedex & DMA, Ecole Normale Supérieure, PSL University, Paris
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{CRMECA_2023__351_S1_615_0,
     author = {Bertrand Maury},
     title = {Defective {Laplacians} and paradoxical phenomena in crowd motion modeling},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {615--646},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {351},
     number = {S1},
     year = {2023},
     doi = {10.5802/crmeca.205},
     language = {en},
}
TY  - JOUR
AU  - Bertrand Maury
TI  - Defective Laplacians and paradoxical phenomena in crowd motion modeling
JO  - Comptes Rendus. Mécanique
PY  - 2023
SP  - 615
EP  - 646
VL  - 351
IS  - S1
PB  - Académie des sciences, Paris
DO  - 10.5802/crmeca.205
LA  - en
ID  - CRMECA_2023__351_S1_615_0
ER  - 
%0 Journal Article
%A Bertrand Maury
%T Defective Laplacians and paradoxical phenomena in crowd motion modeling
%J Comptes Rendus. Mécanique
%D 2023
%P 615-646
%V 351
%N S1
%I Académie des sciences, Paris
%R 10.5802/crmeca.205
%G en
%F CRMECA_2023__351_S1_615_0
Bertrand Maury. Defective Laplacians and paradoxical phenomena in crowd motion modeling. Comptes Rendus. Mécanique, Volume 351 (2023) no. S1, pp. 615-646. doi : 10.5802/crmeca.205. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.205/

[1] Bertrand Maury The Respiratory System in Equations, MS&A. Modeling, Simulation and Applications, 7, Springer, Milano, 2013 | DOI | MR | Zbl

[2] Carl A. Whitfield; Peter Latimer; Alex Horsley; Jim M. Wild; Guilhem J. Collier; Oliver E. Jensen Spectral graph theory efficiently characterizes ventilation heterogeneity in lung airway networks, J. R. Soc. Interface, Volume 17 (2020), 20200253 | DOI

[3] Jérôme Droniou Finite volume schemes for diffusion equations: Introduction to and review of modern methods, Math. Models Methods Appl. Sci., Volume 24 (2014) no. 8, pp. 1453-1455 | Zbl

[4] José A. Ferreira; Mario Grassi; Elías Gudiño; Paula de Oliveira A new look to non-Fickian diffusion, Appl. Math. Modelling, Volume 39 (2015) no. 1, pp. 194-204 | DOI | MR | Zbl

[5] Jacques-Louis Lions; Enrico Magenes Non-Homogeneous Boundary Value Problems and Applications, Grundlehren der Mathematischen Wissenschaften, 181, Springer, Berlin, Heidelberg, New York, 1972 | Zbl

[6] Francesca Rapetti; Alain Bossavit Whitney Forms of Higher Degree, SIAM J. Numer. Anal., Volume 47 (2009) no. 3, pp. 2369-2386 | DOI | MR | Zbl

[7] Joseph L. Doob Classical Potential Theory and Its Probabilistic Counterpart, Grundlehren der Mathematischen Wissenschaften, 262, Springer, 1984 | DOI | Zbl

[8] David Gilbarg; Neil S. Trudinger Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, 224, Springer, 2001 | DOI | Zbl

[9] Ioannis Karatzas; Steven Shreve Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113, Springer, 1991 | Zbl

[10] Björn Gustafsson Lectures on Balayage, Clifford algebras and potential theory. Proceedings of the summer school, Mekrijärvi, Finland, June 24–28, 2002 (Sirkka-Liisa Eriksson, ed.) (Report series. Department of Mathematics, University of Joensuu), Volume 7, University of Joensuu, Department of Mathematics, Joensuu, 2002, pp. 17-63 | Zbl

[11] Mervin E. Muller Some Continuous Monte Carlo Methods for the Dirichlet Problem, Ann. Math. Stat., Volume 27 (1956) no. 3, pp. 569-589 | DOI | MR | Zbl

[12] Russel Lyons; Yuval Peres Probability on Trees and Networks, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, 2016 | DOI | Zbl

[13] Roland Glowinski; Bertrand Maury Fluid-Particle Flow: a Symmetric Formulation, C. R. Acad. Sci. Paris Sér. I Math., Volume 324 (1997), pp. 1079-1084 | MR | Zbl

[14] Roland Glowinski; Tsorng-Whay Pan; Todd I. Hesla; Daniel D. Joseph A distributed Lagrange multiplier/fictitious domain method for particulate flows, Int. J. Multiphase Flow, Volume 25 (1999) no. 5, pp. 755-794 | DOI | MR | Zbl

[15] Bertrand Maury A time-stepping scheme for inelastic collisions. Numerical handling of the nonoverlapping constraint, Numer. Math., Volume 102 (2006), pp. 649-679 | DOI | MR | Zbl

[16] Bertrand Maury; Antony Preux Pressureless Euler equations with maximal density constraint : a time-splitting scheme, Topological Optimization and Optimal Transport: In the Applied Sciences (Maïtine Bergounioux et al., eds.), Walter de Gruyter, 2017, pp. 333-355 | DOI | Zbl

[17] Pierre Degond; Piotr Minakowski; Laurent Navoret; Ewelina Zatorska Finite volume approximations of the Euler system with variable congestion, Comput. Fluids, Volume 169 (2018), pp. 23-39 | DOI | MR | Zbl

[18] Roberta Bianchini; Charlotte Perrin Soft congestion approximation to the one-dimensional constrained Euler equations, Nonlinearity, Volume 34 (2021) no. 10, pp. 6901-6929 | DOI | MR | Zbl

[19] Bertrand Maury; Aude Roudneff-Chupin; Fillipo Santambrogio A macroscopic Crowd Motion Model of the gradient-flow type, Math. Models Methods Appl. Sci., Volume 20 (2010) no. 10, pp. 1787-1821 | DOI | MR | Zbl

[20] Raymond Trémolières; Jacques-Louis Lions; Roland Glowinski Numerical Analysis of Variational Inequalities, Studies in Mathematics and its Applications, 8, North-Holland, Amsterdam, New York, 2000 | Zbl

[21] Félicien Bourdin; Bertrand Maury Multibody and macroscopic impact laws: a Convex Analysis standpoint (Giacomo Ed Albi; Sara Merino-Aceituno; Alessia Nota; Mattia Zanella, eds.) (Trails in Kinetic Theory: Foundational Aspects and Numerical Methods), Springer, 2021, pp. 97-139 | DOI

[22] Bertrand Maury; Juliette Venel A discrete contact model for crowd motion, ESAIM, Math. Model. Numer. Anal., Volume 45 (2011) no. 1, pp. 145-168 | DOI | Numdam | MR | Zbl

[23] Elvezia Cepolina; Nick Tyler Understanding Capacity Drop for designing pedestrian environments, Proceedings of Walk 21: Everyday Walking Culture (2005)

[24] Boris Andreianov; Carlotta Donadello; Massimiliano D. Rosini Crowd dynamics and conservation laws with nonlocal constraints and capacity drop, Math. Models Methods Appl. Sci., Volume 24 (2014) no. 13, pp. 2685-2722 | DOI | MR | Zbl

[25] Boris Andreianov; Carlotta Donadello; Ulrich Razafison; Massimiliano D. Rosini Qualitative behaviour and numerical approximation of solutions to conservation laws with non-local point constraints on the flux and modeling of crowd dynamics at the bottlenecks, ESAIM, Math. Model. Numer. Anal., Volume 50 (2016) no. 5, pp. 1269-1287 | DOI | Numdam | MR | Zbl

[26] Carlos Gershenson; Dirk Helbing When slower is faster, Complexity, Volume 21 (2015), pp. 9-15 | DOI

[27] Bertrand Maury; Sylvain Faure Crowds in Equations. An introduction to the microscopic modeling of crowds, Advanced Textbooks in Mathematics, World Scientific, Hackensack, 2019 | Zbl

[28] Jean-Jacques Moreau Décomposition orthogonale d’un espace Hilbertien selon deux cônes mutuellement polaires, C. R. Acad. Sci. Paris, Volume 255 (1962), pp. 238-240 | Zbl

Cited by Sources:

Comments - Policy