Multiscale Methods for Crowd Dynamics

Multiscale Methods for Crowd Dynamics Individuality vs. Collectivity Andrea Tosin Istituto per le Applicazioni del Calcolo M. Picone Consiglio Nazionale delle Ricerche Rome, Italy a.tosin@iac.cnr.it http://www.iac.cnr.it/ tosin Department of Mathematical Sciences Politecnico di Torino October 11, 2013 Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 1/9

Individuality vs. Collectivity Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 2/9

Microscopic (Particle-Based) Models Social force model (Helbing et al., 1995) ẋ i = v i v0,i vi N v i = U ij(x j x i) +... τ j=1 Contact handling model (Maury and Venel, 2007) ẋ i = P C(X) (V des(x i)) V des : R d R d pedestrian desired velocity P C(X) projection operator on the space of admissible velocities Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 3/9

Microscopic (Particle-Based) Models Social force model (Helbing et al., 1995) ẋ i = v i v0,i vi N v i = U ij(x j x i) +... τ j=1 Contact handling model (Maury and Venel, 2007) ẋ i = P C(X) (V des(x i)) V des : R d R d pedestrian desired velocity P C(X) projection operator on the space of admissible velocities Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 3/9

Microscopic (Particle-Based) Models Social force model (Helbing et al., 1995) ẋ i = v i v0,i vi N v i = U ij(x j x i) +... τ j=1 Contact handling model (Maury and Venel, 2007) ẋ i = P C(X) (V des(x i)) V des : R d R d pedestrian desired velocity P C(X) projection operator on the space of admissible velocities Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 3/9

Macroscopic (Density-Based) Models First order model (Hughes 2002, Colombo et al. 2005, Bruno-Venuti 2007, Coscia-Canavesio 2008,... ) ρ + (ρv (ρ)ν) = 0 t V (ρ) speed-density relationship ν preferred direction of movement Second order model (Bellomo-Dogbé 2008, Coscia- Canavesio 2008, Twarogowska et al. 2013) ρ t + (ρv) = 0 ρv (ρ)ν ρv (ρv) + (ρv v) = P (ρ) t τ Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 4/9

Macroscopic (Density-Based) Models First order model (Hughes 2002, Colombo et al. 2005, Bruno-Venuti 2007, Coscia-Canavesio 2008,... ) ρ + (ρv (ρ)ν) = 0 t V (ρ) speed-density relationship ν preferred direction of movement Second order model (Bellomo-Dogbé 2008, Coscia- Canavesio 2008, Twarogowska et al. 2013) ρ t + (ρv) = 0 ρv (ρ)ν ρv (ρv) + (ρv v) = P (ρ) t τ Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 4/9

Macroscopic (Density-Based) Models First order model (Hughes 2002, Colombo et al. 2005, Bruno-Venuti 2007, Coscia-Canavesio 2008,... ) ρ + (ρv (ρ)ν) = 0 t V (ρ) speed-density relationship ν preferred direction of movement Second order model (Bellomo-Dogbé 2008, Coscia- Canavesio 2008, Twarogowska et al. 2013) ρ t + (ρv) = 0 ρv (ρ)ν ρv (ρv) + (ρv v) = P (ρ) t τ Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 4/9

Multiscale Descriptive Approach by Flow Maps and Measures Get inspiration from a simple first order particle model: V des (x i )dt ẋ i = V des(x i) + K(x j x i) x j S R,α (x i ) R x j x i α dx i K(x j x i )dt Label pedestrians by means of their initial position: X t : R d R d (flow map) x = X t(ξ) : position at time t > 0 of the walker who initially was in ξ R d Fix a walker ξ and rewrite the particle model using the flow map: Ẋ t(ξ) = V des(x t(ξ)) + K(X t(η) X t(ξ)) dµ 0(η) X 1 t (S R,α (X t (ξ))) Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 5/9

Multiscale Descriptive Approach by Flow Maps and Measures Get inspiration from a simple first order particle model: V des (x i )dt ẋ i = V des(x i) + K(x j x i) x j S R,α (x i ) R x j x i α dx i K(x j x i )dt Label pedestrians by means of their initial position: X t : R d R d (flow map) x = X t(ξ) : position at time t > 0 of the walker who initially was in ξ R d Fix a walker ξ and rewrite the particle model using the flow map: Ẋ t(ξ) = V des(x t(ξ)) + K(X t(η) X t(ξ)) dµ 0(η) X 1 t (S R,α (X t (ξ))) Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 5/9

Multiscale Descriptive Approach by Flow Maps and Measures Get inspiration from a simple first order particle model: V des (x i )dt ẋ i = V des(x i) + K(x j x i) x j S R,α (x i ) R x j x i α dx i K(x j x i )dt Label pedestrians by means of their initial position: X t : R d R d (flow map) x = X t(ξ) : position at time t > 0 of the walker who initially was in ξ R d Fix a walker ξ and rewrite the particle model using the flow map: Ẋ t(ξ) = V des(x t(ξ)) + K(X t(η) X t(ξ)) dµ 0(η) X 1 t (S R,α (X t (ξ))) Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 5/9

Multiscale Descriptive Approach by Flow Maps and Measures Get inspiration from a simple first order particle model: V des (x i )dt ẋ i = V des(x i) + K(x j x i) x j S R,α (x i ) R x j x i α dx i K(x j x i )dt Transport µ 0 by means of X t, i.e., µ t := X t#µ 0, to discover: µ t + (µv[µ]) = 0 x R d, t > 0 v[µ t](x) = V des(x) + K(y x) dµ t(y) S R,α (x) Description compatible with both a discrete and a continuous view of the crowd: N discrete: µ 0 = δ ξi, continuous: µ 0 = ρ 0L d i=1 Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 5/9

Multiscale Descriptive Approach by Flow Maps and Measures Get inspiration from a simple first order particle model: V des (x i )dt ẋ i = V des(x i) + K(x j x i) x j S R,α (x i ) R x j x i α dx i K(x j x i )dt Transport µ 0 by means of X t, i.e., µ t := X t#µ 0, to discover: µ t + (µv[µ]) = 0 x R d, t > 0 v[µ t](x) = V des(x) + K(y x) dµ t(y) S R,α (x) Description compatible with both a discrete and a continuous view of the crowd: N discrete: µ 0 = δ ξi, continuous: µ 0 = ρ 0L d i=1 Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 5/9

Discrete and Continuous: So Close... Comparison between discrete and continuous models in terms of statistical distributions of pedestrians: µ discr 0 = 1 N N δ ξi, i=1 µ cont 0 = 1 L d (Ω) Ld Ω h K N W 1(µ discr 0, µ cont d 0 ) 2 h + diam(ω) L d (Ω) Ld (Ω \ K N ) N 0 Continuous dependence estimate (for smooth V des and K): hence µ discr t W 1(µ discr t, µ cont ) C W 1(µ discr 0, µ cont 0 ) t (0, T ] t µ cont t for all t in the limit N. Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 6/9

Discrete and Continuous: So Close... Comparison between discrete and continuous models in terms of statistical distributions of pedestrians: µ discr 0 = 1 N N δ ξi, i=1 µ cont 0 = 1 L d (Ω) Ld Ω h K N W 1(µ discr 0, µ cont d 0 ) 2 h + diam(ω) L d (Ω) Ld (Ω \ K N ) N 0 Continuous dependence estimate (for smooth V des and K): hence µ discr t W 1(µ discr t, µ cont ) C W 1(µ discr 0, µ cont 0 ) t (0, T ] t µ cont t for all t in the limit N. Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 6/9

Discrete and Continuous: So Close... Comparison between discrete and continuous models in terms of statistical distributions of pedestrians: µ discr 0 = 1 N N δ ξi, i=1 µ cont 0 = 1 L d (Ω) Ld Ω h K N W 1(µ discr 0, µ cont d 0 ) 2 h + diam(ω) L d (Ω) Ld (Ω \ K N ) N 0 Continuous dependence estimate (for smooth V des and K): hence µ discr t W 1(µ discr t, µ cont ) C W 1(µ discr 0, µ cont 0 ) t (0, T ] t µ cont t for all t in the limit N. Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 6/9

Discrete and Continuous:... So Far Does the limit N really make sense for crowds? Often it does not: pedestrians in a crowd are not as many as 10 23 gas molecules. For finite N physical mass distributions matter. B L x B b 1.34 1.2 L = 10 m L = 50 m L = 100 m 1 u(n) [m/s] 0.8 0.6 0.4 0.2 0 1 45 57 223 281 446 561 N [ped] Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 7/9

SIMAI Activity Group on Complex Systems http://sisco.simai.eu Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 8/9

References N. Bellomo, B. Piccoli, and A. Tosin. Modeling crowd dynamics from a complex system viewpoint. Math. Models Methods Appl. Sci., 22(supp02):1230004 (29 pages), 2012. 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. A. Corbetta and A. Tosin. Bridging discrete and continuous differential models of crowd dynamics: A fundamental-diagram-aided comparative study. Preprint: arxiv:1306.2472, 2013. A. Corbetta, A. Tosin, and L. Bruno. From individual behaviors to an evaluation of the collective evolution of crowds along footbridges. Preprint: arxiv:1212.3711, 2012. E. Cristiani, B. Piccoli, and A. Tosin. Multiscale Modeling of Pedestrian Dynamics. In preparation. 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. 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. 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. Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 9/9