Multiscale Methods for Crowd Dynamics -...

Multiscale Methods for Crowd DynamicsIndividuality vs. Collectivity

Andrea Tosin

Istituto per le Applicazioni del Calcolo “M. Picone”Consiglio Nazionale delle Ricerche

Rome, Italy

[email protected]://www.iac.cnr.it/˜tosin

Department of Mathematical SciencesPolitecnico di Torino

October 11, 2013

Andrea Tosin, IAC-CNR (Rome, Italy) Multiscale Methods for Crowd Dynamics, Individuality vs. Collectivity 1/9

Individuality vs. Collectivity

Microscopic (Particle-Based) Models

“Social force” model (Helbing et al., 1995)xi = vi

vi =v0,i − vi

N∑j=1

∇Uij(xj − xi) + . . .

“Contact handling” model (Maury and Venel, 2007)

xi = PC(X)(Vdes(xi))

Vdes : Rd → Rd pedestrian desired velocity

PC(X) projection operator on the space ofadmissible velocities

vi =v0,i − vi

N∑j=1

∇Uij(xj − xi) + . . .

vi =v0,i − vi

N∑j=1

∇Uij(xj − xi) + . . .

Macroscopic (Density-Based) Models

First order model (Hughes 2002, Colombo et al. 2005,Bruno-Venuti 2007, Coscia-Canavesio 2008, . . . )

∂t+∇ · (ρV (ρ)ν) = 0

V (ρ) speed-density relationship

ν preferred direction of movement

Second order model (Bellomo-Dogbe 2008, Coscia-Canavesio 2008, Twarogowska et al. 2013)∂ρ

∂t+∇ · (ρv) = 0

∂t(ρv) +∇ · (ρv ⊗ v) =

ρV (ρ)ν − ρvτ

−∇P (ρ)

∂t+∇ · (ρV (ρ)ν) = 0

∂t+∇ · (ρv) = 0

∂t(ρv) +∇ · (ρv ⊗ v) =

−∇P (ρ)

∂t+∇ · (ρV (ρ)ν) = 0

∂t+∇ · (ρv) = 0

∂t(ρv) +∇ · (ρv ⊗ v) =

−∇P (ρ)

Multiscale Descriptive Approach by Flow Maps and Measures

Get inspiration from a simple first order particle model:

xi = Vdes(xi) +∑

xj∈SR,α(xi)

K(xj − xi) R α

Vdes(xi)dt

K(xj−xi)dt

Label pedestrians by means of their initial position:

Xt : Rd → Rd (flow map)

x = Xt(ξ) : position at time t > 0 of the walker who initially was in ξ ∈ Rd

Fix a walker ξ and rewrite the particle model using the flow map:

Xt(ξ) = Vdes(Xt(ξ)) +

∫X−1t (SR,α(Xt(ξ)))

K(Xt(η)−Xt(ξ)) dµ0(η)

xi = Vdes(xi) +∑

xj∈SR,α(xi)

K(xj − xi) R α

Vdes(xi)dt

K(xj−xi)dt

xi = Vdes(xi) +∑

xj∈SR,α(xi)

K(xj − xi) R α

Vdes(xi)dt

K(xj−xi)dt

xi = Vdes(xi) +∑

xj∈SR,α(xi)

K(xj − xi) R α

Vdes(xi)dt

K(xj−xi)dt

Transport µ0 by means of Xt, i.e., µt := Xt#µ0 , to discover:∂µ

∂t+∇ · (µv[µ]) = 0

v[µt](x) = Vdes(x) +

∫SR,α(x)

K(y − x) dµt(y)x ∈ Rd, t > 0

Description compatible with both a discrete and a continuous view of the crowd:

discrete: µ0 =

N∑i=1

δξi , continuous: µ0 = ρ0Ld

xi = Vdes(xi) +∑

xj∈SR,α(xi)

K(xj − xi) R α

Vdes(xi)dt

K(xj−xi)dt

Transport µ0 by means of Xt, i.e., µt := Xt#µ0 , to discover:∂µ

∂t+∇ · (µv[µ]) = 0

v[µt](x) = Vdes(x) +

∫SR,α(x)

K(y − x) dµt(y)x ∈ Rd, t > 0

Description compatible with both a discrete and a continuous view of the crowd:

discrete: µ0 =

N∑i=1

δξi , continuous: µ0 = ρ0Ld

Discrete and Continuous: So Close. . .

Comparison between discrete and continuous models in terms of statisticaldistributions of pedestrians:

µdiscr0 =

N∑i=1

δξi , µcont0 =

Ld(Ω)Ld

KN W1(µdiscr

0 , µcont0 ) ≤

diam(Ω)

Ld(Ω)Ld(Ω \KN )

N→∞−−−−→ 0

Continuous dependence estimate (for smooth Vdes and K):

W1(µdiscrt , µcont

t ) ≤ CW1(µdiscr0 , µcont

0 ) ∀ t ∈ (0, T ]

hence µdiscrt ≡ µcont

t for all t in the limit N →∞.

µdiscr0 =

N∑i=1

δξi , µcont0 =

Ld(Ω)Ld

KN W1(µdiscr

0 , µcont0 ) ≤

diam(Ω)

Ld(Ω)Ld(Ω \KN )

N→∞−−−−→ 0

0 ) ∀ t ∈ (0, T ]

µdiscr0 =

N∑i=1

δξi , µcont0 =

Ld(Ω)Ld

KN W1(µdiscr

0 , µcont0 ) ≤

diam(Ω)

Ld(Ω)Ld(Ω \KN )

N→∞−−−−→ 0

0 ) ∀ t ∈ (0, T ]

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 1023 gas molecules.For finite N physical mass distributions matter.

1 45 57 223 281 446 561

N [ped]

L = 10 mL = 50 m

L = 100 m

SIMAI Activity Group on Complex Systems – http://sisco.simai.eu

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.

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