Faster is Slower Effect B. Maury DMA, Ecole Normale Suprieure & - - PowerPoint PPT Presentation

Faster is Slower Effect

• B. Maury

DMA, Ecole Normale Supérieure & Laboratoire de Mathématiques d’Orsay With F. Al Reda, S. Faure, A. Roudneff-Chupin, F. Santambrogio, J. Venel Marseille, June 5 th 2019

Parisi et al. 15’ Garcimartin et al. 14’ Zuriguel et al. 16’

Observed phenomena : Capacity Drop and Faster is Slower effects

Nicolas et al. 16’

Experimental evidence (and counter-evidence…)

Complementary CDF for time lapses

Experimental evidence (and counter-evidence…)

Related phenomenon : role of an obstacle

Faster is Slower Effect in other contexts (or: the best is the enemy of the good)

For general systems : counter-effective increase of the forcing term (possibly above a certain threshold) Examples Expiratory Flow Limitation Hurtling Droplet

Gouttes, bulles, perles et ondes

David Quéré, Françoise Brochard-Wyart et Pierre-Gilles de Gennes

Faster is Slower Effect in other contexts (or: the best is the enemy of the good)

In the context of the respiratory system : Expiratory Flow Limitation (EFL) Poiseuille’s Law :

Q = P/R(P)

Flux

Q0 = R(P) − PR0(P) R(P)2

R0 ≤ R/P

R0 ≥ R/P

Maximal flux

Faster is Slower Effect in other contexts (or: the best is the enemy of the good)

mi dui dt = mi τ (Ui − ui) +

• j̸=i

fij +

• k

f w

ik,

Helbing’s Social Force Model :

Reproduces the FiS effect with an additional friction term

Cellular Automata (Schadschneider, Seyfried, Klüpfel … )

Von Neumann

Again, friction makes it work: in case of a conflict, there is a non-zero probability that no competitor moves

Faster is Slower Effect in crowd motion

Faster is Slower Effect in crowd motion

N.B. all computations performed with Python Package cromosim pypi.org/project/cromosim/ (S. Faure & B.M.) Mathematical aspects described in Crowds in Equations, B.M. & S. Faure, World Scientific

Alternative standpoint, an underlying bizarre Laplace operator

Faster is Slower Effect

ty U = U(x)

Spontaneous velocity Feasible densities

Macroscopic model (with A. Roudneff Chupin & F. Santambrogio)

K = {ρ ∈ P(Ω) , 0 ≤ ρ ≤ 1 a.e.}

• ∂ρ

∂t + ∇ · (ρu) = 0 u = PCρU, Cone of feasible velocities : nonnegative divergence wherever

re ρ = 1

• u + ∇p

= U −∇ · u ≤ p ≥

• ω

u · ∇p = 0,

The projection can be formulated as a unilateral Darcy problem

Macroscopic setting : mathematical issues

The problem takes the form of a conservation law ∂ρ ∂t + ∇ · F(ρ) = 0, where F(ρ) = ρPCρ(U) is non-local, non-smooth in both ways : the velocity field u is simply L2, and its dependence upon ρ is highly non-smooth. − → Standard theory is not applicable

• ˜

ρk+1 = (id + τU)# ρk transport (prediction), ρk+1 = PK

• ˜

ρk+1 projection (correction),

Idea : extend Moreau’s catching up algorithm to measures

• Th. (with A. Roudneff-Chupin & F. Santanbrogio) :

the evolution problems admits a solution (in the Wasserstein sense)

Evacuation of a room

• u + ∇p

= U −∇ · u ≤ p ≥

• ω

u · ∇p = 0,

−∆p = −∇ · U > 0,

p = 0 p = 0

At the exit :

u · n = U · n − ∂p ∂n ≥ U · n

People exit faster as they would if they were alone : no capacity drop, no clogging.

u = PCρU.

−∆pβ = −∇ · βU >

p = 0

p = 0

Faster is Slower effect ?

• −∆q = 0

in Ω, q = 1

• n Γout

q = 0

• n Γup

∂q ∂n = 0

• n Γw

is U · ∇q

The gradient of J is speed correction factor where q solves the adjoint problem :

J(β) = Z

Γout

uβ · n = Z

Γout

βUβ · n − Z

Γout

∂pβ ∂n

β = β(x)

for « reasonable » choices of U : Faster is Faster effect …

U · rq 0

x

N individuals, centered at

at q1, q2, . . ., qN ∈ R2.

K = {q ∈ R2N, Dij(q) = |qj − qi| − 2r ≥ 0 ∀i ̸= j}. Set of feasible configurations

U = (U1, . . . , UN)

Spontaneous velocities

u = dq dt = PCqU,

Microscopic level : A granular (purely selfish) model (with J. Venel)

Cq = {v , Dij(q) = 0 ⇒ eij · (vj − vi) ≥ 0}

r r eij −eij qj qi Dij

Gradient flow framework

Dissatisfaction functional

Ψ(q) =

• i

D(qi) + IK(q).

Individual dissatisfaction (distance to the exit) Non overlapping constraint

IK(q) =

• if q ∈ K

+∞ if q / ∈ K

dq dt ∈ −∂Ψ(q)

∂Ψ(q) = {v, Ψ(q) + v · h ≤ Ψ(q + h) ∀h}

Micro-macro similarities

spontaneous velocities

U = (U1, . . . , UN). Gij = ∇Dij(x) = (0, . . . , 0, −eij, 0, . . . , 0, eij, 0, . . . , 0) ∈ R2N

Constraint when i and j are in contact

Gij · u ≥ 0

r r eij −eij xj xi Dij

• u −
• i∼j

pij Gij = U, −Gij · u ≤ ∀i ∼ j, p ≥ 0, Gij · u > 0 = ⇒ pij = 0.

• u + ∇p

= U −∇ · u ≤ p ≥

• ω

u · ∇p = 0,

• u + B⋆p

= U, Bu ≤ 0, p ≥ 0, p · Bu = 0.

Micro Macro

Remark : « Standard » laplacian on the primal network

uij = −cij(pj − pi)

pj

pi

Ohm / Poiseuille / Fick law Kirchhoff law

X

j∼i

uij = 0 X

j∼i

cij(pi − pj) = 0

Discrete harmonicity Maximum principle

u = krp

Macroscopic

r · u = 0

r · krp = 0

Discrete Poisson problem

−∆p = −∇ · U > 0, Discrete

× B = ⎛ ⎜ ⎜ ⎝ 1 −1 0 · · 1 −1 · · · · · · · · · · · · 1 −1 ⎞ ⎟ ⎟ ⎠

             

2 −1 · · −1 2 −1 · · −1 · · · · · · · · · · 2 −1 · · −1 2

             

BB? =

....

In 1d : In higher dimensions : a bit different

BB?p = BU

Continuous

2

• 1
• 1

2

• 1
• 1

2 2

• 1
• 1

2

• 1
• 1

Here : non-standard laplacian on the dual network

Pressures defined at edges (i.e. contact points)

Here : non-standard laplacian on the dual network

Pressures defined at edges (i.e. contact points)

i j k

The matrix is not diagonally dominant in general + some extra diagonal terms are positive

Consequence : no maximum principle

BB?p = BU > 0

does not imply

p > 0

• +

+ + + 4 -disk system, glued together

Faster is slower effect in the microscopic situation ?

speed correction factors where q solves the adjoint problem : β = (βi)i

BB?p = B(β U)

J(β) = −B?p · ni

ni

The gradient of J is U B?q

BB?q = Bni

+ +

• -

No reason for J to be positive: some individuals may accelerate the egress of the blue guy by reducing their speed

−∆pβ = −∇ · βU >

p = 0 p = 0

Faster is slower effect ?

• −∆q = 0

in Ω, q = 1

• n Γout

q = 0

• n Γup

∂q ∂n = 0

• n Γw

is U · ∇q

The gradient of J is speed correction factor where q solves the adjoint problem :

J(β) = Z

Γout

uβ · n = Z

Γout

βUβ · n − Z

Γout

∂pβ ∂n

β = β(x)

for « reasonable » choices of U : Faster is faster effect …

U · rq 0

x

Red : FiF persons Blue : FiS persons

Red : FiF persons Blue : FiS persons

Computation : S. Faure

Positive outcome : it is possible to speed up evacuation by cooling down the crowd (Slower is Faster effect)

βi ∈ [0, 1]

Ψ(q) =

N

• i=1

βiD(qi) + IK(q)

5 10 15 20 25 30 35 40 45 5 10 15 20 25 30 35 40 45 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 Mean frustration Dissatisfaction

Mathematical formulation is given Find such that where Example :

Game theoretical extension of the granular model (with F. Al Reda)

Hierarchical situation

If the graph is acyclic, which is reasonable to assume in case of an evacuation, the game is replaced by a hierarchical optimization inhibition based model

Hierarchical situation

If the graph is acyclic, which is reasonable to assume in case of an evacuation, the game is replaced by a hierarchical optimization inhibition based model

Hierarchical situation

If the graph is acyclic, which is reasonable to assume in case of an evacuation, the game is replaced by a hierarchical optimization inhibition based model

Hierarchical situation

If the graph is acyclic, which is reasonable to assume in case of an evacuation, the game is replaced by a hierarchical optimization inhibition based model

13 12 11 10 9 8 7 6 5 4 3 2 1

If the graph is acyclic, which is reasonable to assume in case of an evacuation, the game is replaced by a hierarchical optimization inhibition based model

Hierarchical situation

Validation

Power law distribution Computations

Inhibition based model (with F. Al Reda)

Long time computation (periodic setting)

Selfish model Individuals do not push (reduction of the velocity) Same with an

• bstacle

1 2 3

( Calculs : Fatima Al Reda)

Calculs: F. Al Reda

Inhibition based model (with F. Al Reda)