On the Mathematical Modeling and Simulation of Crowd Motion

Marie-Therese Wolfram

Workshop on ’PDE models in the Social Sciences’ at Jussieu, Paris

Jan 24, 2014

Outline

1 Motivations and applicationsMicroscopic modelsMacroscopic approaches

2 Mean field model for fast exit scenariosMean field game modelsMathematical modelingAnalysis of the optimal control modelNumerical simulations

3 Understanding the Hughes model

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Pedestrian motion

• Empirical studied of human crowds startedabout 50 years ago.

• Nowadays there is a large literature on differentmicro- and macroscopic approaches available.

• Challenges: microscopic interactions notclearly defined, multiscale effects, finite sizeeffects,.....

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Mathematical modeling - microscopic level

1 Force based models: position of a particle is determined by forces acting on it e.g.Newton dynamics, stochastic differential equations (SDE), ....Newton equations of motion

dXi = Vi dt

dVi = Fi (X1, . . . ,XN ,V1, . . . ,VN )dt + σi dBti .

Xi = Xi (t) is the location of the i-th particle, Vi = Vi (t) its velocity, Fi the forcesacting on it, and dBi some additive noise.

2 Stochastic optimal control Each agent wants to minimize a cost functional

Ji (v1, v2, . . . , vN ) = E(

∫ T

0Li (Xi ,Vi ) + F (X1, . . . ,XN )dt)

under the constraint that

dXi = Vi dt + σi dBti .

where L and F denote the running cost.

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Mathematical modeling - microscopic level

1 Lattice based models: Consider a domain divided into a cells and each cell centerrepresents a possible position of a particle. Particle may jump from one cell toanother with a certain transition probability, e.g. cellular automata, ..

Time continuous, discrete in space random walk for the probability pi to find aparticle at a discrete lattice point xi :

∂pi

∂t= T −i+1pi+1 + T +

i−1pi−1 − (T +i + T −i )pi

$x_i+1$$x_i$

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Example: Social force model 1

• Each pedestrian changes his/her velocity Vi according to:

dVi

dt=

v0i e

0i − Vi

τi+

1

mi

∑i 6=j

fij︸︷︷︸interaction forces

,

where v0i is a desired velocity in direction e0

i , τi is the characteristic time and mi

the mass of the i-th particle.

• Interaction forces:

fij = Ai exp(Rij − dij

Bi) · nij︸ ︷︷ ︸

repulsion

+ k(Rij − dij ) · nij︸ ︷︷ ︸body force

− cijnij︸ ︷︷ ︸attraction

+ . . .

where Rij is the sum of the radii, dij the distance between two pedestrians andthe normalized vector pointing from pedestrian j to i .

1D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Phys. Rev. E. 51, 1995

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Nonlinear diffusion transportation models

In the macroscopic limit N →∞ one usually obtains a nonlinear transport-diffusionequations of the form

ρt = div(D(ρ)∇(E ′(ρ)− V + W ∗ ρ)︸ ︷︷ ︸:=v

).

• V = V (x) is an external potential energy (e.g. confinement,....),

• D = D(ρ) denotes the nonlinear diffusion/mobility

• E = E(ρ) an entropy/internal energy.

• W = W (x) is an interaction energy.

Classic fluid dynamic based models: Based on the conservation of mass

ρt = div(ρv)

with a particularly chosen velocity v (which depends on the specific modelingassumptions).Traffic flow: Lighthill-Whitman and Richards model,....Pedestrian motion: Colombo, Rossini, Venuti, .......Review on literature: Bellomo & Dogbe 2011.

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Hughes model for pedestrian flow 2

1 Speed of pedestrians depends on the density of the surrounding pedestrian flow

v = f (ρ)u, |u| = 1.

2 Pedestrians have a common sense of the task (called potential φ)

u = −∇φ|∇φ|

.

3 Pedestrians try to minimize their travel time, but want to avoid high densities

|∇φ| =1

g(ρ)f (ρ).

Saturation effects are included via the function f (ρ) = (ρmax − ρ), where ρmax denotesthe maximum density.

2Hughes, R. A continuum theory for the flow of pedestrians, Transportation Research Part B, 36, 507-535, 2002

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Hughes model for pedestrian flow

Hughes’ model for pedestrian flow (with g(ρ) = 1):

∂tρ− div(ρf 2(ρ)∇φ) = 0

|∇φ| =1

f (ρ)

Analytic issues:

• fully coupled system; nonlinear hyperbolic conservation law.

• density dependent stationary Hamilton Jacobi equation ⇒ φ ∈ C 0,1 only.

Let us consider the regularized system:

∂tρε − div((ρεf 2(ρε)∇φε) = ε∆ρε

−δ1∆φε + |∇φε| =1

f (ρε) + δ2.

1D : solution ρε converges to an entropy solution for ε→ 0, but δ1 > 0, δ2 > 0 !3

3M. Di Francesco, P.A. Markowich, J.-F. Pietschmann and MTW, On Hughes model for pedestrian flow: theone-dimensional case, JDE, 2011

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Mean field games 4

Microscopic modelN-player stochastic differential game

infVi∈A

E[∫ T

0f (t,Xi ,Vi , ρ)dt + g(ρ,Xi , t = T )

]dXi = Vi dt + σdBi , Xi (t = 0) = x .

Transient macroscopic modelCalculate Nash equilibrium, limiting equations as N →∞ gives time dependent meanfield game: Find (φ, ρ) such that

∂φ

∂t+ ν∆φ− H(x ,∇φ) = 0

∂ρ

∂t− ν∆ρ− div(

∂H

∂p(x ,∇φ)ρ) = 0,

with the initial and end conditions

φ(x ,T ) = g [ρ(x ,T )], ρ(x , 0) = ρ0(x),

where H is the Legendre transform of the running cost f .

4P.-L. Lions, J.-M. Lasry, Mean field games, Japan. J. Math., 2, 229-260, 2007

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Link to optimal control problems

If the running cost f has the form

f (x , t, v , ρ) = L(x , t, v)ρ,

then the MFG can be written as an optimal control problem. For example let usconsider the kinetic energy f (x , t, v) = 1

2ρ|v |2, then

infv

[1

2

∫ T

0

∫Ωρ|v |2 dxdt + g(ρ(T ))

]under the constraint that

∂ρ

∂t= ν∆ρ− div(ρv), ρ(x , 0) = ρ0(x).

The formal optimality condition is v = ∇φ and therefore the adjoint equation reads as

∂φ

∂t+ ν∆φ−

1

2|φ|2 = 0

with the terminal condition φ(x ,T ) = g ′(ρ(T )).

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An optimal control approach for fast exit scenarios 5

• Let us consider an evacuation or fast exit scenario, i.e. a room with one or severalexits from which a groups wants to leave as fast as possible.

• Each individual tries to find the optimal trajectory to the exit, taking into accountthe distance to the exit, the density of people and other costs.

Figure: Fast-exit experiment conducted at the TU Delft

5M. Burger, M. Di Francesco, P.A. Markowich, MTW, Mean field games with nonlinear mobilities in pedestriandynamics, accepted at DCDS, 2014.

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Fast exit of particles

• Let X (t) denote the trajectory of a particle, which wants to leave the domain Ωas fast as possible. The exit time is defined as:

Texit (X ) = supt > 0 | X (t) ∈ Ω.

• Fastest path is chosen such that

1

2

∫ Texit

0|V (t)|2 dt +

α

2Texit (X )→ min

(X ,V ).

subject to X (t) = V (t), X (0) = X0.

• For the Dirac measure µ = δX (t) and a final time T sufficiently large we can write:

Texit =

∫ T

0

∫Ω

dδX (t)dt.

• This yields the equivalence between the continuum formulation and the particleformulation, i.e.∫ T

0

∫Ω|v(x , t)|2dµdt =

∫ T

0

∫Ω|v(x , t)|2dδX (t)dt =

∫ Texit

0|v(X (t), t)|2dt,

by mapping the Eulerian to the Lagrangian coordinates.

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Fast exit of particles

• Hence the minimization for the particle problem can be written as a continuumproblem

IT (µ, v) =1

2

∫ T

0

∫Ω|v(x , t)|2dµdt +

α

2

∫ T

0

∫Ω

dµdt,

subject to ∂tµ+∇ · (µv) = 0, µ |t=0= δX0.

• For a stochastic particle dµ = ρ dx and a final time T sufficiently large, theminimization reads as

IT (ρ, v) =1

2

∫ T

0

∫Ωρ(x , t) |v(x , t)|2 dxdt +

1

2

∫ T

0

∫Ωρ(x , t) dxdt,

subject to ∂tρ+∇ · (ρv) = σ2

2∆ρ, ρ(x , 0) = ρ0(x).

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Optimality conditions

• Formal optimality conditions; calculated via the Lagrangian with dual variable φ:

LT (ρ, v , φ) = IT (ρ, v) +

∫ T

0

∫Ω

(∂tρ+∇ · (vρ)−σ2

2∆ρ)φ dx dt.

• For the optimal solution we have

0 = ∂v LT (ρ, v , φ) = ρv − ρ∇φ

0 = ∂ρLT (ρ, v , φ) =1

2|v |2 +

α

2− ∂tφ− v · ∇φ−

σ2

2∆φ,

with the additional terminal condition φ(x ,T ) = 0.

• Inserting v = ∇φ we obtain the following system (with MFG structure):

∂tρ+∇ · (ρ∇φ)−σ2

2∆ρ = 0

∂tφ+1

2|∇φ|2 +

σ2

2∆φ =

α

2.

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Mean field games and crowding

We consider the following generalization of the optimal control problem:

IT (ρ, v) =1

2

∫ T

0

∫Ω

F (ρ)|v(x , t)|2dx dt +1

2

∫ T

0

∫Ω

E(ρ) dx dt,

subject to

∂tρ+∇ · (G(ρ)v) =σ2

2∆ρ, with initial condition ρ(x , t = 0) = ρ0(x).

Motivation:

• G = G(ρ) corresponds to a nonlinear mobility, e.g. G(ρ) = ρ(ρmax − ρ).

• F = F (ρ) corresponds to transport costs created by large densities. For example:

F (ρ)→∞ as ρ→ ρmax.

• E = E(ρ) may model active avoidance of jams, in particular by penalizing largedensity regions.

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Relation to the classical model by Hughes

• Let H(ρ) = G 2

F= ρf (ρ)2, E(ρ) = αρ and σ = 0. Then we obtain the following

optimality conditions:

∂tρ+∇ · (ρ f (ρ)2∇φ) = 0

∂tφ+f (ρ)

2(f (ρ) + 2ρf ′(ρ))|∇φ|2 =

α

2

• For large T we expect equilibration of φ backward in time, hence we consider

∂tρ+∇ · (ρ f (ρ)2∇φ) = 0

(f (ρ) + 2ρf ′(ρ))|∇φ|2 =α

f (ρ),

which has a similar structure as the Hughes model

∂tρ+∇ · (ρ f (ρ)2∇φ) = 0

|∇φ| =1

f (ρ).

• Setting f (ρ) = ρmax − ρ and α = 1 yields f (ρ) + 2ρf ′(ρ) = ρmax − 3ρ, thus forsmall densities the behavior is similar, but the singular point is ρ = ρmax

3.

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Boundary conditions

On the Neumann boundary ΓN we clearly have no outflux, hence naturally

(−σ2

2∇ρ+ j) · n = 0.

At an exit ΓE : the outflux depends on how fast people can leave the room.

(−σ2

2∇ρ+ j) · n = βρ︸︷︷︸

outflow

.

Boundary conditions for the adjoint variable φ can be obtained via Lagrangian andresult in

σ2

2∇φ · n + βφ = 0 on ΓE and −

σ2

2∇φ · n = 0 on ΓN .

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Analysis of the optimal control model

Let ρmax > 0 denote the maximum density and Υ = [0, ρmax]. Let F = G = H whichsatisfy the following assumptions:

(A1) F = F (ρ) ∈ C 1(R), F bounded, E = E(ρ) ∈ C 1(R) and F (ρ) ≥ 0, E(ρ) ≥ 0 forρ ∈ Υ.

Existence of minimizes is guaranteed if

(A2) E = E(ρ) is convex.

To ensure that the minimizes satisfy ρ ∈ Υ, we need the following assumption on F :

(A3) F (0) > 0 if ρ ∈ Υ and F = 0 otherwise.

Uniqueness holds for:

(A4) F = F (ρ) is concave.

We consider the optimization problem on the set V × Q, i.e. IT (ρ, v) : V × Q → R,where V and Q are defined as follows

V = L2(0,T ; H1(Ω)) ∩ H1(0,T ; H−1(Ω)) and Q = L2(Ω× (0,T )).

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Abstract problem formulation in different variables

• Then the optimization problem reads as

min(ρ,v)∈V×Q

IT (ρ, v) such that ∂tρ =σ2

2∆ρ− div(F (ρ)v),

respectively in momentum formulation

min(ρ,j)∈V×Q

IT (ρ, j) such that ∂tρ =σ2

2∆ρ− div(j).

• In the case of non-concave H we use the variable w =√

F (ρ)v . Then theminimization problem becomes:

min(ρ,w)∈V×Q

1

2

∫ T

0

∫Ω

(|w |2 + E(ρ)) dxdt such that ∂tρ =σ2

2∆ρ− div(

√F (ρ)w).

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Existence results

Theorem (Existence in the general case)

Let ρ0 ∈ L2(Ω). Let (A1) and (A2) be satisfied, σ > 0 and w =√

F (ρ)v. Then thevariational problem has at least a weak solution (ρ,w) ∈ V × Q with initial conditionρ0. If in addition (A3) is satisfied, then ρ ∈ Υ.

Idea of the proof: Let (ρk ,wk ) ∈ V ×Q be a minimizing sequence. Both sequences ρk

and wk are bounded, hence we can extract a weakly convergent sub-sequence withlimit (ρ, w). Using the Lemma of Aubin and Lions we can show that (ρ, w) isadmissible and the lower semicontinuity of the objective functional implies the it isindeed a minimizer.

Theorem (Existence for concave mobility)

Let (A1), (A2), (A3), and (A4) be satisfied, σ > 0 and j = F (ρ)v. Then thevariational problem has at least one minimize (ρ, j) ∈ L∞(Ω× (0,T ))× Q such thatρ(x) ∈ Υ for almost every x ∈ Ω. If E is strictly convex the minimize is unique.

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Steepest descent

We solve the optimal control problem using a steepest descent method given by

1 Solve the forward equation for ρ = ρ(x , t)

∂tρ =σ2

2∆ρ−∇ · (F (ρ)v)

(−σ2

2∆ρ+∇ · F (ρ)v) · n = βρ on ΓE and (−

σ2

2∆ρ+∇ · F (ρ)v) · n = 0 on ΓN ,

using Newton’s method for the implicit Euler discretization and a mixed hybridDG method for the discretization in space.

2 Calculate the backward evolution of the adjoint variable φ using the density ρ

− ∂tφ−σ2

2∆φ− G ′(ρ)v · ∇φ = −

1

2F ′(ρ)|v |2 −

1

2E ′(ρ)

−σ2

2∇φ · n − βφ = 0 on ΓE and −

σ2

2∇φ · n = 0 on ΓN ,

using an implicit in time discretization and a mixed hybrid DG method.

3 Update the velocity v = v − τ(F (ρ)v − G(ρ)∇φ).

4 Go to (1) until convergence of the functional.

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Fast exit for three groups

(a) Solution of the classical Hughes model (b) Solution of the mean field optimal control approach

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Understanding the Hughes model

• Let us consider N particles with position Xk = Xk (t) and the empirical density

ρN (t) =1

N

N∑k=1

δ(x − Xk (t)).

• To define the cost functional in a proper way we introduce the smoothedapproximation ρN

g by

ρNg (t) = (ρN ∗ g)(x , t) =

1

N

N∑k=1

g(x − Xk (t)),

where g is a sufficiently smooth positive kernel.

We freeze the empirical density ρN and look for the optimal trajectory of each particle,i.e.

C(X ; ρ(t)) = min(X ,V )

1

2

∫ T +t

t

|V (s)|2

G(ρNg (ξ(s; t))

ds +1

2Texit (X ,V ),

subject to dξds

= V (s) and ξ(0) = X (t).

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Understanding the Hughes model

• For sufficiently large T > Texit :

Texit (X ,V )− t =

∫ T +t

t

∫Ωδξ(s) dx ds.

Since the macroscopic version of ρN (t) converges to the mean field ρ(t) (at least inthe weak-* sense of measures as N →∞), we replace ρN

g (t) by ρ(t) and obtain:

C(X ; ρ(t)) = min(µ,w)

J(µ,w) =1

2

∫ T +t

t

∫Ω

(w2(x , s)

G(ρ(ξ(s; t)))+ 1) dµds,

subject to ∂sµ+∇ · (µw) = 0 with µ(t = 0) = δX .

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Understanding the Hughes model

• The formal optimality conditions can be calculated via the Lagrange functional

0 = ∂w LX = w − G(ρ)∇ψ,

0 = ∂µLX = ∂sψ +1

2G(ρ(t))|∇ψ|2 −

1

2.

with ψ(T + t) = 0, ψ(t) = −φ• For T → 0 the behavior at s = t represents the long-time behavior of the HJE.

Hence we expect φ = −ψ(t) to solve

G(ρ(t))|∇ψ|2 = 1.

Then we recover the Hughes model by choosing G(ρ) = f (ρ)2 i.e.

∂tρ+∇ · (ρf (ρ)2∇φ) = 0,

|∇φ| =1

f (ρ).

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Numerical simulations of the particle model

Consider N particles located at positions Xi = Xi (t), i = 1, . . .N. In every time stepwe update their position by:

1 Determine the empirical density

ρNg (x , t) =

1

N

N∑i=1

g(x − Xi (t)),

where g denotes a Gaussian to approximate the Dirac deltas.

2 Solve the Eikonal equation6

|∇φ| =1

f (ρNg )

with boundary conditions φ = 0 at the exit and φ = 1 at obstacles.

3 Update each position by Xi (t + ∆t) = Xi (t) + ∆t(ρNg f (ρN

g )2∇φ).

6J. Qian, Y.-T. Zhang, H.-K. Zhao, Fast sweeping methods for Eikonal equations on triangular mesh, SIAM J.Numer. Anal. 45(1), 2007

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Future work

• Analysis of the problem for more general functions E , F and G ⇒ non convexoptimization problems

• Simulations in 2D, numerical solvers for non-convex optimization problems.

• Other generalization of the Hughes model for pedestrian flow, e.g. include localvision.

• Multiscale problems in crowd dynamics: crowd-leader interactions or the motionof small social groups in large crowds.

• Inverse problems: How many leaders are necessary to guide a crowd efficiently ?

Thank you very much for your attention !

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Bibliography

M. Burger, M. Di Francesco, P.A. Markowich, M.T. WolframMean field games with nonlinear mobilities in pedestrian dynamics, accepted atDCDS, 2014

M. DiFrancesco, P.A. Markowich, J.F. Pietschmann, M.T. Wolfram,On Hughes model for pedestrian flow: the one-dimensional case, J. DifferentialEquations 250, 2011

D. Helbing and P. MolnarSocial force model for pedestrian dynamics, Phys. Rev. E. 51, 1995

R. HughesA continuum theory for the flow of pedestrians, Trans. Res.: Part B 36, 2002

J. Qian, Y.-T. Zhang, H.-K. ZhaoFast sweeping methods for Eikonal equations on triangular mesh, SIAM J.Numer. Anal. 45(1), 2007

P.-L. Lions, J.-M. LasryMean field games, Japan. J. Math. 2, 2007

A. Lachapelle, M.T. WolframOn a mean field game approach modeling congestion and aversion in pedestriancrowds, Trans. Res.: Part B 45(10), 2011

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