tgf’11 moscow, september 28 - october 1, 2011 geodesics and shortest paths approach...

TGF’11 Moscow, September 28 - October 1, 2011Geodesics and Shortest Paths Approach in Pedestrian

Motions

M. Rascle, with B. Nkonga, F. Decoupigny (*) and G. Maignant (*)

Laboratoire JA Dieudonne, or (*): Laboratoire Espace,CNRS and Universite de Nice Sophia-Antipolis

Francehttp://math.unice.fr/~rascle/

October 15, 2011

M. Rascle (Universite de Nice) Geodesics and Shortest Paths in Pedestrian Motions October 15, 2011 1 / 43

1 Introduction2 Basic facts on Geodesics.

Variational Problem. DistanceOptimality condition: Euler-Lagrange equationExamplesLegendre transform. Extremality conditionsLink with Euler-Lagrange. Hamilton Jacobi equation

Distance to a curveAn application: mesh generation or refinementAn example from Geography

3 Pedestrian Flow: Principle of a fully Eulerian descriptionAssumptionsA natural algorithmComments.A few numerical tests: 1A few numerical tests: 2

4 Mixed Eulerian-Lagrangian descriptionPrincipleA few numerical tests. Comments.

5 Conclusion. ReferencesM. Rascle (Universite de Nice) Geodesics and Shortest Paths in Pedestrian Motions October 15, 2011 2 / 43

Introduction

MotivationsI I love eikonal or HJ equation, geodesics etc ... !I Previous joint work of mine on mesh generation, see belowI Contacts with people in geography, see belowI Hamilton-Jacobi theory provides a relevant global information: the

graph of the solution gives intuition, like a mountain landscapeI This theory is also relevant in traffic ...

WarningI No big scoop here! No attempt to use sophisticated modelingI Only (old) basic ideas and a few numerical simulations, using the

eikonal equation. See Hughes, 2002.I Very similar ideas: mathematical optimal transportationI Of course, many other aspects of pedestrian flows modeling do not

involve geodesics ...

”Distance” between two points y and zFor any curve X (t) = (X1(t),X2(t)), 0 ≤ t ≤ T from pointy = (y1, y2) (origin) to point z = (z1, z2) (destination), define:

L[X ] :=

0L(X (t), X (t)) dt (2.1)

The cost L(x , v) = L(X (t), X (t)) ≥ 0 is the Lagrangian. We set

dL(y , z) = infL[X ] , X (0) = y , X (T ) = z. (2.2)

Indeed, the infimum, taken on all such curves from y = (y1, y2) toz = (z1, z2), depends on y , z and (non locally) on the choice of L.

Distance? We always have :

dL(y , z) ≥ 0 and dL(x , z) ≤ dL(x , y) + dL(y , z), (2.3)

with equality iff ... but dL(x , y) = dL(y , x) iff L(x , v) ≡ L(x ,−v)(symmetry). How about a steep ascent?

Optimality condition: Euler Lagrange equationIf x = X (.) is optimal between y and z , then for all X + δX (.)between y and z , L[X + δX ] ≥ L[X ]. So, at first order,∫ T

0[∂xL(X , X ) · δX + ∂vL(X , X ) · ˙δX ] dt ≤ 0. (2.4)

Integrating by parts, and using ±δX , we obtain for all δX vanishingat y and z:∫ T

0[∂xL(X , X )− d

dt( ∂vL(X , X ))] · δX dt + [∂vL ·δX ]T0 = 0, (2.5)

where v = (v1, v2) = ˙X (.), x = X (.) and the gradients ∂xL, ∂vL are2D vectors.

Therefore, we obtain the Euler-Lagrange equation

dt( ∂vL(X , X )) = ∂xL(X , X ). (2.6)

ExamplesExample 1: Straight line. L(x , v) ≡ L(v) = 1

2 |v |2. By (2.6),

dt( ∂vL(X , X )) =

dt(v) = ∂xL(X , X ) = 0. (2.7)

Therefore v = X = z−yT : the optimal velocity is constant:

propagation in straight line.

Example 2: Broken line. Now L(x , v) = 12c(x) |v |

2, with

c(x) = c1 > 0 (or c2 > 0) for x2 < 0 (> 0): refraction law ofDescartes.

Example 3: Straight line, again, but now L(x , v) ≡ L(v) = |v |.Again, by (2.6),

dt( ∂vL(X , X )) =

|v |) = 0, (2.8)

but now, only the direction of v(t) = X (t) is fixed, not its length (Lis not strictly convex in v).

Legendre transform. Hamiltonian

Legendre transform in v , for any x . Define the Hamiltonian:

H(x , p) := supvp.v − L(x , v). (2.9)

Example 1: L(x , v) ≡ L(v) = 12 |v |

H(x , p) := supvp.v − 1

2|v |2 − 1

2|p|2+

2|p|2 =

2|p|2 := H(p).

(2.10)Here H ≡ L and the supremum in v is reached for v = p

Example 2: L(x , v) = 12c(x) |v |

H(x , p) := supvp.v − 1

2c(x)|v |2 − c(x)

2|p|2+

2|p|2 =

2|p|2.

(2.11)

Extremality conditions. Link with Euler-Lagrange

Extremality conditions. For all x (position), v (”speed”) and p satisfy:

H(x , p) + L(x , v) ≥ p.v , (2.12)

and for all x , equality holds iff p and v are linked by the followingextremality conditions:

H(x , p) + L(x , v) = p.v ⇔ p = ∂vL(x , v)⇔ v = ∂pH(x , p), (2.13)

e.g. in example 2 above, at supremum in v , v = c(x)p and

L(x , v) = H(x , p) =1

2p · v =

2c(x)|v |2. (2.14)

When relations (2.13) hold, then

∂xL(x , v) = −∂xH(x , p). (2.15)

Link with Euler-Lagrange. Hamiltonian system

Since v = X , we rewrite the Euler-Lagrange equation under the formdXdt = v

ddt (∂vL(X , v)) = ∂xL(X , v).

(2.16)

Now, using (2.15)and (2.13), this is nothing but the celebratedHamiltonian characteristic system:

dXdt = ∂pH(X , p) = v

dpdt = −∂xH(X , p) = ∂xL(X , v),

(2.17)

satisfied by X (.) and p = p(.). Obviously, H(X (.), p(.)), and e.g. inexample 2, v ≡ C

√c(x), is constant along any trajectory.

Hamilton Jacobi equation

Now a tricky calculation allows to relate this system to the following(evolution) Hamilton Jacobi equation:

∂tΦ + H(x , ∂xΦ(x , t)) = 0. (2.18)

Namely, p(t) := ∂xΦ(X (t), t) is a (smooth) solution to system(2.17), as long as Φ is smooth. So, we are led to solve either (2.18) ...

or ... its stationary version:

H(x , ∂xϕ(x)) = f (x), (2.19)

... with suitable initial and/or boundary conditions.

A typical example is to study the zero-level curves of Φ, orequivalently the t-level curves of ϕ, in the case whereΦ(x , t) = ϕ(x)− t, of course with ad hoc data.

Distance to a curve

Prototype: as in the above Example 2, L(x , v) ≡ L(v) = 12c(x) |v |

but now c(x) > 0 is arbitrary. We recall that then H(x , p) = c(x)2 |p|

2,and that here, by (2.13),

v = c(x)p ⇔ H(x , p) + L(x , v) = p · v ⇔ p =v

c(x). (2.20)

Now, let (C ) be a smooth given curve, e.g. (C ) = x2 = (x1)2.We want to compute for any point x :

dL(x ,C ) = infy ,XL[X ] , X (0) := y ∈ (C ) ,X (T ) = x, (2.21)

where L is as in (2.1) and y arbitrary on (C ).

Only change: now, in the integration by parts to get (2.6), for t = 0,the vector ∂vL(y , X (0)) must be normal to curve (C ) at point y : yis ”the foot of the perpendicular from x to (C )”.

Distance to a curve:

Figure

With the above function H, the (signed) distance dL(x ,C ) is thesolution to the stationary eikonal equation:

H(x , ∂xϕ) = 12 in R2

ϕ ≡ 0 on curve(C ).

(2.22)

Then the normal derivative of the solution satisfies: H(x , ∂ϕ∂ν (x)) = 12 ,

since the tangential derivative is zero.

Non uniqueness: caustics, shocks...Uniqueness of the viscositysolution (P.L. Lions ...) ... Level set method (Osher-Sethian ...)...

Numerical approximations: e.g. fast marching or sweeping ...

This equation is a powerful curve mover!

An application: mesh generation or refinement

z=c+ !c

A B C Dx

G(x)=G1(x), h(x)=h1(x), Vmin=.001

G(x)=G2(x), h(x)=h1(x), Vmin=.001

G(x)=G1(x), h(x)=h2(x), Vmin=10^(!8)

G(x)=G1(x), h(x)=h1(x), Vmin=10^(!6)

dgr=0.04, qmin=0.265, 978 Triangles first order, nx=201 ny=201, dt=0.0025 cfl=1

Examples of optimal path problems from geography

Example 1: Presentation modele cheminement. Problem: optimizethe location of parking lots in a national park, in order to restrict thenumber of pedestrian paths.

Idea: (Decoupigny) identify a cost on each path, depending on theslope and of the interest of each site, find the geodesics, comparewith existing paths, make recommendations

Example 2: (not shown here) Similar problem: preliminary study ( afew years ago) of the impact of the construction of a street car at Nice

Joint works in progress, with Decoupigny and Maignant: improvednumerical algorithms for the above example; finer description ofpedestrian traffic on a Plaza of Nice

Graphe de cheminement

potentiel

Plus court

Graphe de cheminement

potentiel

Plus facile

Parking du Tanet

Parking de la station du TanetParking du Lac Vert Parking du Dreieck

Lac Vert Parking de la réserve naturelle Du Tanet-Gazon du Faing

Lac des Truites

Lac Noir

Lac Blanc

Diffusion de l’automate cellulaire

Espace potentiel de cheminement

Graphe MNT de la réserve du Tanet-Gazon du Faing

Diffusion pédestres au départ des parkings sur la réserve naturelle du Tanet

Pour des temps de promenade de 80 minutes(Parking - site = 20 mm et site à site = 60 mm)

Plus Court

Multiplication des sentiers

Plus Facile

Intensification sur les

sentiers

Parking

"Plus Facile""Plus Court"

Structure du p-graphe MNT

Parking

Point "d'arrêt" (ou sites naturels, curiosités naturelles)

P-graphe sentier

Espace théorique

Parking plus attractif sans

fermeture

0 5 10 km

Grd Ballon

Marstein

Col de la Schlucht

Ballon

d'Alsace

Petit Ballon

Lac Blanc

Lac Noir

Tanet station et lac Vert

GaschneyHohneck

Tanet station et lac Vert

Grd Ballon

Marstein

Col de la Schlucht

Ballon

d'Alsace

Petit Ballon

Lac Blanc

Hautes Huttes

RN Gazon

du Faing

GaschneyHohneck

0 5 10 km

Evaluation prospectif du projet de fermeture de la route des crêtes

Parking plus attractif après la fermeture

Avec aménagement de parkingsPas d’aménagement de parkings

Pedestrian movmentsimulations

Path and impacts of thepedestrian movments

Cellular automates and graph theory

Localization of pedestrian impacts in natural environment

National Park of Mercantour : Vallées des Merveilles

Area : PACA

MNT Alpes Maritimes © IGN

Simulation des cheminements pédestres au départ des parkings

pour une promenade de trois heures

Intégration des résultats dans un SIG vectoriel : MapInfo

Cheminements de promenade d’accès à la Vallée des Merveilles

Gestion des cheminements pédestres dans un SIG

Fully Eulerian Pedestrian Flow: assumptions

Consider (Figure) a plaza or a big hall with four doors Dk , k = 1..4,all being both entrance and exit doors.

Four populations of density ρk(x , t), k = 1..4 at point x and time t,going to exit at Dk . Let: ρn(x) :=

∑k ρ

nk be the total density.

For each equation in ρk , see below, the other entrance doorsDj , j 6= k , only play a role in boundary data, as entering fluxes.

At each time tn, each population k responds to the distance:

dnk (x ,Dk) = inf

y ,XLnk [X ] , X (0) := y ∈ Dk ,X (T ) = x, (3.1)

where the associated cost depends on the total density:

Lnk(x , v) =

2c(x)|v |2, with c(x) := cn

k (x) = f (ρn(x)), (3.2)

A natural algorithm

... where f (.) is a given function of ρ, small for large densities

By (2.13), the associated stationay boundary value problem is now:H(x , ∂xdn

k ) :=f (ρn(x)

2|∂xdn

k |2 =1

2in Ω

dnk (x) ≡ 0 on door Dk .

In principle, the algorithm is as follows:

Step 1: assuming ρ := ρn is known at time tn, compute each distancedk := dn

k ( (we drop the index n), by solving the above problem

Step 2: by extremality condition (2.13), the corresponding velocityfield vk is (note the - sign):

vk = − 1

2 f (ρ(x)) ∂xdk (3.4)

Algorithm : sequel ...

Step 3: for each k, refresh ρk from tn to tn+1 by approximating thesolution at time tn+1 to the Initial Value Problem (IVP):

∂tρk +∇.(ρk vk) = 0 in Ω× (tn, tn+1),

ρk(x , tn) = ρnk(x) in Ω,

... with the classical entering boundary conditions: the flux ρ v ateach point x of the boundary ∂Ω is imposed whenever vn

k · ν < 0,ν(x) being the exterior normal vector at point x . Practically, the fluxρk · vk is only entering in Ω through the other doors Dj , j 6= k .

Step 4: knowing the partial densities ρn+1k at time tn+1, refresh the

total density ρn+1. Go to step 1.

Comments. First questions.

Again, very basic model, not even new, cf Hughes ... but sostraightforward!

The algorithm is thus completetely ”natural”, yet rather expensive

Propagation: for each door Dk , the corresponding distancedk(x) is computed backwards : further and further from Dk .

In contrast, every k− pedestrian travels towards Dk .

Yet, each distance dk provides meaningful relevant information,probably too expensive and too detailed ... Only refresh occasionally?Only use a coarse grid ? Or combine with discrete populations as innext section?

No maximal density is imposed. However, by (2.14), thedependence in ρ penalizes dense regions, since c = f (ρ) is smallfor large densities.

A few numerical tests: 1Here, a curve-shaped hall, with a dense region in the middle and agiven (fixed) c(x) = f (ρ(x))Figures below show the distance to two doors out of four,computed on an unstructured mesh, via an algorithm of Abgrall et all,solver of the evolution HJ equation

... and below are the distances to the last two doors.Quiz: distance to which door?

With these ideas, without computing the densities, where are theoptimal trajectories from SW to NE door? and conversely?

Answer: compute the sum of the distances to these two doors, showits level curves

With these ideas, without computing the densities, where are theoptimal trajectories from SW to NE door? and conversely?

Answer: compute the sum of the distances to these two doors, showits level curves

A few numerical tests: 2

A corridor or a platform with two exit doors (right), with given incomingflux (left). Plot of the density at a given time t, with higher incomingdensity in right picture, data similar to Xia, Wong, Zhang, Shu, C.-W. andLam, 2008.

Mixed Eulerian-Lagrangian description: principleWe considereach discrete population k, going to door Dk , and weassume that their individual velocities are given from a continuousvelocity field.Obvious prototype: at each time t, e.g. t = tn, we assume that

vk(x) := vk(x , t) = −c(x , t) ∂xdk(x ,Dk), (4.1)

where dk(x ,Dk) = ϕk is the solution to (3.3), and again

c(x) := c(x , t) = f (ρ(x , t)) (4.2)

is a function of a continuous approximation of the total discretedensity ρ =

∑k ρk at time t, as in steps 1 and 2 of our above fully

Eulerian algorithm. The resulting semi-discrete problem at time t is,for each individual j in population k:

Xj(t) = vj(t)

vj(t) = −c(Xj(t), t) ∂xdk(Xj(t),Dk).

... Principle (sequel)

We obviously approximate this ODE system, e.g. by the explicit Eulerscheme. At each time step, the function c(.) is defined by (4.2), andeither we refresh the density, (at each time step? more rarely, e.g. ateach time of visualization?), or we keep the (obsolete) initial density...

When refreshing the density, obvious problem: pass from the discretepositions to a continuous density. Simple choice: count the number ofparticles in each numerical cell.

Another crucial ingredient concerns the issue of concentration:evacuation, panic ... Should we enforce a maximal density in themodel (choice 1), or is it enough (choice 2) to use the repulsiveproperties of high density regions in the eikonal equation?

More precisely, we have made a couple of comparisons between:

Choice 1: cf Maury, Venel (and Santambrogio et al on moretheoretical side), eikonal equation with a constant c and a constrainton the maximal (total) density with corresponding Lagrangemultipliers and a numerical solution based on Uzawa algorithm. Verynice mathematically and numerically, but expensive. Its”incompressible” feature in congested regions is perhaps (?) a goodcartoon for lane formation.

Choice 2: as above, use eikonal equation with a variable coefficientc(x) = f (ρ(x)), to prevent the density from becoming too large. Canbe a regularization of choice, with a suitable function f . We haveonly tested it with a smooth f , not steep enough for large densities

Choice 3 (not shown here): combine choices 1 and 2: some tests,not shown here.

A few numerical tests: I.Two doors, SW and NE. Initial data. Brief comparison of choices 1 and 2

A few numerical tests: I.Choice 1 (left) vs choice 2 (right). Snapshots at computationalmidtimeFor comparison, the speeds are normalized in both algorithms.

A few numerical tests: I.Choice 1 (left) vs choice 2 (right). Snapshots at computational finaltimeSame normalization. Here, more lane formation with choice 1.

A few numerical tests: II.

1. moviebn: two doors, choice 2: moderate density, eikonalequation with variable c(x) = f (ρ(x)), f gentle. We added a term(T − t)V (x), where V is an attractive short range potential, and T adeadline constraint.

2. movieBbn: two doors, choice 2: Idem, but no potential term,and c(x) = f (ρ(x)) + c1 or c2, like a glue zone in the middle, andsame (too) gentle f as above

3. if time allows, movie Fourmis2: Idem, choice 1, random initialvelocities, lane formation

4. if time allows still more: Snapshots of the evacuation of a movietheater, choice 1. Master project (4 weeks): Liana Amaya Morenoand Polina Marinovahttp://math.unice.fr/~rascle/

Conclusion. References

Conclusion: 1) work in (slow) progress,2) .. to be compared with competing approaches.

A few references: Besides the above-mentioned pioneering work:L. Hughes, Transp Research B, 2002,a fairly incomplete list of names or references relevant here includes:

on Hamilton Jacobi, uniqueness and viscosity solutions: the books ofL. C. Evans: Partial Differential Equations, 1997, of P.L. Lions:Hamilton-Jacobi ..., 1983

on the level set method: the original paper of S. Osher and J. A.Sethian, Proc. Nat. Acad. Sci, 1996

on numerical schemes:fast marching: books of Sethian, of Osher-Fedwick, 2002 ...fast sweeping: e.g. in the context of pedestrian flows: Xia, Wong,Zhang, Shu, C.-W. and Lam, 2008, or Huang, Wong, Shu, Lam, 2009...

Thanks for your attention.M. Rascle (Universite de Nice) Geodesics and Shortest Paths in Pedestrian Motions October 15, 2011 43 / 43

tgf’11 moscow, september 28 - october 1, 2011 geodesics and shortest paths approach...

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