A Semi-Lagrangian scheme for a regularized version of the Hughes model for pedestrian flow

Transcription

1 A Semi-Lagrangian scheme for a regularized version of the Hughes model for pedestrian flow Adriano FESTA Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences (ÖAW) Linz, Austria SMAI Toulouse, 23th March 2016 work in collaboration with E. Carlini, F. Silva and M-T. Wolfram

2 Motivations Pedestrian dynamics - Complex behaviors Micro-meso-macro models - Interaction rules Basic issue for crowd management Example: Sziget Festival in Budapest

3 Hughes Model

4 Hughes model: assumptions In 2002 R. Hughes proposed a macropscopic model for pedestrian dynamics, based on: 1 The total number of individuals is conserved in time and the speed of individuals is link to the density of the surrounding pedestrian flow. 2 Individuals have a common goal, e.g. reaching a certain location in space. 3 Pedestrians want to minimize their estimated travel time but try to avoid regions of high density.

5 Hughes model: original formulation Those considerations lead to the following PDE system: t m(x, t) div(m(x, t) f 2 (m(x, t)) u(x, t)) = 0, 1 u(x, t) = f (m(x, t)), where x Ω is the position in space, t (0, T ] the time, m = m(x, t) the pedestrian density, u the weighted shortest distance to a target. An example for f is f (m) = 1 m.

6 Hughes model: a regularization We consider a regularized version in a d-dimentional closed domain (d N + ) t m(x, t) ε m(x, t) div(m(x, t) f 2 (m(x, t)) u(x, t)) = 0, ε u(x, t) u(x, 1 t) 2 = 2f 2 (m(x, t)) + δ. for small parameters ε and δ in R +.

7 Hughes model: boundary conditions Possible choices for the pedestrian density m at the exit are: a given fixed outflow (Neumann BC) an outflux which depends on the density (Robin BC) or a prescribed pedestrian density (Dirichlet BC). We choose: let T denote the common target/goal of the crowd m(x, 0) = m 0 (t), on Ω {0}, m(x, t) = 0, on T (0, T ), u(x, t) = 0, on T (0, T ), u(x, t) = g(x) on Ω \ T (0, T ), (ε m + f 2 (m) u m)(x, t) ˆn(x) = 0, on Ω \ T (0, T ), where ˆn denotes the outer normal vector to the boundary.

8 Trajectory interpretation

9 Trajectory interpretation: HJ equation Given a process α (F s -measurable for all s and admissible), and x Ω, we define u(x, t) = inf α y x,α (s) = x+ s 0 α(r)dr + 2εW (s) for all s > 0, and τ x,α := inf{s > 0 ; y x,α (s) Ω}, where W is a d-dimensional Brownian motion adapted to F. The function u(x, t) defined as { ( τx,α E 0 [ 1 2 α(s) 2 + (2f 2 (m(y x,α (s), t)) + δ) 1 is the solution of the HJ equation associated. ] ds +g(y x,α (τ x,α ))) },

10 Trajectory interpretation: FP equation Let us consider the Stochastic Differential Equation (SDE) dx (t) = b(x (t), µ(x (t), t), t) dt + 2ε dw (t), for all t 0, X (0) = X 0, where b : R d R R + R d is a vector-valued function, X 0 is a random vector in R d, independent of the Brownian motion W ( ), with density m 0, and µ(, t) is the density of X (t). It can be shown that the eq. above admits a unique solution and that µ is the unique classical solution of the nonlinear FP equation µ ε µ + div(b(x, µ, t)µ) = 0 in R d [0, [, µ(, 0) = m 0 ( ) in R d.

11 Approximate the system

12 Discrete space Let us suppose Ω = (0, L) d. Given t > 0 and x > 0, let us set (x i, t k ) := (i x, k t), where i {0,..., M} d and k = 0,..., N, and for a given A Ω set G x (A) := {i {0,..., M} d : x i A}. We call B(G x (A)) and B(G x, t (A)) the spaces of grid functions defined respectively on {x i : i G x (A)} and {(x i, t k ), i G x (A), k = 0,..., N}.

13 Discretizing the HJB equation We define the following linear interpolation operator on Ω I [u]( ) := u(x i )β i ( ) for u B(G x (Ω)). i G x (Ω) We approximate y x,α (h) by x hα + εhdz, where Z is a random vector in R d so then: defining h l,± x,α := inf{γ > 0 ; x γα ± εγd e l Ω} h, yx,α l,± := x hx,αα l,± ± εhx,αd l,± e l,

14 A SL-scheme for the HJB equation For v B(G x (Ω)) define W (v, i) := min α A { 1 2d [ ]} d I [v](y l,± i,α ) + ĥl,± i,α 2 α 2 + ĥ l,± i,α F (x i, t). l=1 where F (x, t) := 1/(2f 2 (m(x, t)) + δ). Find u B(G x (Ω)) such that { ui = W (u, i) for all i G x (Ω), u i = g(x i ) for all i G x ( Ω).

15 Some hints for a fast resolution Note that, alternatively, the previous problem can be written in the form: Find u B(G x (Ω)) such that where, if i G x (Ω), 0 = sup {(B α u) i c(α) i } α A (B α v) i = v i 1 2d c(α) i = 1 2d l=1 j G x (Ω), l=1,...,d i G x (Ω), d [ ] 1 2 hl,± x i,α α 2 + hx l,± i,αp(x i, µ), This suggests a policy iteration method. [ β j (y l,+ i,α ) + β j(y l, i,α ) ] v j,

16 Policy iteration method: fundamentals Lemma The previous problem admits a unique solution u x,h [µ]. Moreover, the sequence, for an alpha 0 arbitrary in A, v k = (B αk 1 ) 1 c(α k 1 ), α k argmax α A { B α v k c(α) }, k 1, is well-defined. Moreover for all i G x (Ω), the sequence vi k is non-increasing, converging to u x,h i [µ], and for every limit point α x,h [µ] of α k we have 0 = (B α x,h [µ] u x,h [µ]) i c(α x,h [µ]) i i G x (Ω).

17 Discretization of the Fokker-Plank equation We define E i = [x 1 i 1 2 x, x 1 i x]... [x d i 1 2 x, x d i x], m i,k := 1 ( x) d E i m(x, t k )dx. j Z d, k = 0,..., N 1 and l = 1,..., d. We define also Φ l,± j,k [µ] := x j + t b(x j, µ, t k ) ± 2dε te l.

18 SL-scheme for FP equation Given i Z d setting φ = β i we have the following explicit scheme for m i,k : m i,k+1 = G(m k, i, k) k = 0,..., N 1, i Z d, m i,0 E m 0 (x)dx = i ( x) d i Z d, in which the nonlinear operator G is defined by G(w, i, k) := 1 2d for every w B(Z d ). d j Z d l=1 ( ) ( )) (β i Φ l,+ j,k [w j] + β i Φ l, j,k [w j] w j,

19 Proposition (Weak consistency - in the full space) Assume that m : R d [0, T ] R + is regular enough. Then, assuming that b is Lipschitz, for every φ C0 ( R d ) and k = 0,..., N φ(x)g x, t (m k, x, t k )dx = φ(x)m(x, t k )dx R d R d tk+1 + b(x, m(t, x), t) φ(x)m(x, t)dxdt t k R d tk+1 + ε φ(x)m(x, t)dxdt + O( x + ( t) 2 ). R d t k

20 Boundary conditions (Φ r ) l,± i [b k, µ] := (x j + ν ± i b(x j, µ, t k ) ± ) 2dε te l, { ν ± i := min max{h 0 s.t. ( P(Φ l,± i )[b k, µ] :=P Ω x i + t b(x j, µ, t k ) ± ) 2dε te l, x i + t b(x j, µ, t k ) ± 2dεh T }, t where P Ω : R d Ω is a projection into Ω following z, if z Ω P Ω (z) := 2w z, w := argmin z w, if z / Ω. w Ω }.

21 A Semi-Lagrangian scheme for the system. Given x, t, h we define the following discrete scheme for k = 0,..., N 1: v i,k = W x,h [m k ](v k, i) i G x (Ω), v i,k = g(x i ), i G x ( Ω), where m k+1 i = G x, t ( x v k, m k, i) i G x (Ω), m i,0 = E i m 0 (x)dx, i G x (Ω). x v i := 1 [ ] v(x i + x e 1 ) v(x i x e 1 ),..., v(x i + x e d ) v(x i x e d ). 2 x

22 Pedestrian Flux Simulations

23 Test 1: two exits We model a crowd in a room Ω := (0, 1) 2 with the initial distribution { k x [1/3, 2/3] 2 m 0 := 0 otherwise. for k R. ε = 0.001, f (x, m) := 1 {x Ω (0, [0.13, 0.27])} (1 m(x, t)), T := 2 {x Ω (1, [0.49, 0.51])}. Here tests with δ := 10 6, x = t = h = 0.08 and k = 0.7, k = 0.9.

24 Influence of the regularization parameter ε We fix k = 0.7 ε t e left exit right exit % % % % % % % % % % % % % % Examples: test with ε = and ε = 0.01.

25 Test 2: the utility of the barriers and turnstiles Domain Ω = [0, 1] 2 \ Γ where Γ := { x [0, 1] 2 s.t. min(0.1 x 0.5, 0.02 y cs ) 0, s N } Here a test with k = 0.7, a = 0.5, c = 0.1 and diffusion ε =

26 Test 2: compare barriers and diffusion ɛ = δ 1 c=0.1 c=0.2 c=0.3 no bar k c=0.1 c=0.2 c=0.3 no bar Table : Comparison of the evacuation time varying the parameters of the test.

27 Test 3: the renovation in Les Halles in Paris Salle d exchange RER of Les Halles, Paris. In the 2014 (left) and in 2016 (project).

28 Test3: the exchange hall Evacuation scenario starting from a crowd of pedestrians placed in the center of the hall: simplified example - no new pedestrians from the platforms below - no use of the lifts - f (x, m) := 1 m(x,t) l(x) where l = 2 (turnstiles), 1 (elsewhere). compare before/after renovation.

29 Test3: Evacuation times from the exchange hall m 0 := { 0.7 x (d1 1 3, d ) (d 2 1 3, d ) \ Γ 0 otherwise. where d 1, d 2 are the dimensions of the domain and Γ are the constraints. k 2014 configuration 2016 configuration improvement % % % % % %

30 E. Carlini and F. J. Silva. A fully discrete semi-lagrangian scheme for a first order MFG problem. SIAM Journal on Numerical Analysis, 52(1):45 67, E. Carlini A. Festa, F. J. Silva and M-T. Wolfram. A Semi-Lagrangian scheme for a regularized version of the Hughes model for pedestrian flow Submitted, M. Di Francesco, P. A. Markowich, J.-F. Pietschmann, and M.-T. Wolfram. On the Hughes model for pedestrian flow: The one-dimensional case. Journal of Differential Equations, 250(3): , R. L. Hughes. A continuum theory for the flow of pedestrians. Transportation Research Part B: Methodological, 36(6): , Aimé Lachapelle and Marie-Therese Wolfram. On a mean field game approach modeling congestion in pedestrian crowds. Transportation research part B: methodological, 45(10): , Thank you!