A Semi-Lagrangian discretization of non linear fokker Planck equations

Transcription

1 A Semi-Lagrangian discretization of non linear fokker Planck equations E. Carlini Università Sapienza di Roma joint works with F.J. Silva RICAM, Linz November 2-25, 206 / 3

2 Outline A Semi-Lagrangian scheme for a nonlinear Fokker-Planck equation 2 Convergence Analysis 3 Numerical simulation 2 / 3

3 A nonlinear Fokker-Planck equation The nonlinear FP equation { t m 2 i,j x i x j (a ij [m](x, t)m) + div(b[m](x, t) m) = 0 in R d R + m(, 0) = m 0 ( ) in R d, () describes the evolution of the law of the diffusion process X(t) where dx(s ) = b[m](x(s ), s ))ds + σ[m](x(s ), s )dw (s ), X(s) = x, b[m] : R d R + R R d, given a vector field, depending non locally on m; σ[m] : R d R + R R d r, (A[m](x, t)) i,j = a ij [m](x, t) := (σ[m](x, t)σ[m](x, t) ) ij, is the diffusion matrix (possible degenerate),depending non locally on m; the density of the initial law is given by m 0 ; m 0 0 and R d m 0 (x)dx =. (2) 3 / 3

4 Some applications Non local interactions due to collective phenomena (biophysics, social behavior) Mean Filed Games: b[m](x, t) = DH(Dv[m](x, t)) where v[m] is the solution of a backward HJB { t v σ2 2 v + H(Dv) = f(x, m(t)) v(x, T ) = g(x, m(t ) Since b[m](x, t) depends on m(s) in all time 0 s T we gets an Implicit scheme: Fixed Point Iterations needed to solve it Hughes model b[m](x, t) = f 2 (m(x, t))dv[m](x, t) where v[m] is the solution of a stationary HJB Dv = f(m(x, t)) Since b[m](x, t) depends on m(s) only in time past 0 s t we gets an Explicit scheme 4 / 3

5 Weak solution For many problems of interest, () has only a formal meaning, henceforth we mean m is a weak solution of () if for all φ Cc (R d ) we have that d φ(x)dm(t)(x) dt R d = R d 2 a ij [m](x, t) xi x j φ(x) + b i [m](x, t)φ xi (x) dm(t)(x) i,j i [ = 2 tr(a[m](x, t)d2 φ(x)) + b[m](x, t) Dφ(x)] dm(t)(x) R d (3) 5 / 3

6 Numerical approximation of FP Some references Linear Chang and Cooper (970): finite difference scheme s.t. preserves positivity, equilibrium states and mass of the distribution (explicit version is stable under a parabolic type CFL condition) Kushner (976): finite difference via probabilistic method Naess and Johnsen (993) : Path Integration Method (time integration of the evolution probability density function, where the transition probability density is approximated by a Gaussian) Jordan, Kinderlehrer,Otto (998): variational scheme for FokkerPlanck equations for which the drift term is given by the gradient of a potential (JKO) (It preserves positivity, and mass of the distribution, costly in high dimension). Achdou,Camilli,Capuzzo Dolcetta (202): implicit finite difference scheme s.t. preserves positivity, and mass of the distribution. Non Linear Drozdov, Morillo (995): finite difference scheme s.t. preserves equilibrium states and mass of the distribution, (high order). Benamou, Carlier, Laborde (205): semi implicit variant of JKO 6 / 3

7 Numerical approximation based on SL for FP We propose a Semi-Lagrangian (SL) scheme for non linear FP equation s.t. it is first order accurate (numerically) it is explicit and it allows for large time steps it preserves the positivity of the density and conserves its integral equals to Ref. Silva, Camilli (203), Carlini and Silva (204,205) The scheme has been applied to numerically compute the solution of Mean Field Games model (with F. Silva) a regularized Hughes model for pedestrian flow (with A.Festa, F.Silva, M.T. Wolfram) Hughes model for pedestrian flow with different congestion penalty function (with A.Festa, F.Silva) 7 / 3

8 A Semi-discrete in time SL for a nonlinear d Fokker-Planck equation Given t, we define t k = k t, k = 0,..., N. We multiply the first equation in () by a smooth test function φ with compact support and integrate by parts to get: φ(x)dm(t k+ ) = φ(x)dm(t k ) (4) R R tk+ + [b[m](x, t)dφ(x) + 2 σ2 [m](x, t)d 2 φ(x)]dm(t)dt. t k R 8 / 3

9 Semi-discrete in time We first approximate (4) as φ(x)dm(t k+ ) = R [φ(x) + tb[m k ](x, t k )Dφ(x) + t 2 σ2 [m k ](x, t k )D 2 φ(x)]dm(t k+ ). R Note that the right hand side corresponds to a Taylor expansion. Hence we write φ(x)dm(t k+ ) = R [φ(x + tb[m k ](x, t k ) + tσ 2 [m k ](x, t k )dm(t k )+ 2 R [φ(x + tb[m k ](x, t k ) tσ 2 [m k ](x, t k )]dm(t k ). 2 R 9 / 3

10 Fully-discrete in space Given x > 0 consider a space grid, for i =,..., M E i = [x i 2 x, x i + 2 x], m i,k := E i dm(t k ). By the standard rectangular quadrature formula, we get φ(x j )m j,k+ = j Z 2 φ(φ + j [m k)])m j,k + φ(φ j 2 [m k)])m j,k, j Z j Z (5) (6) where, for µ R M, j Z, k = 0,..., N, we have defined Φ ± j [µ] := x j + t b[µ](x j, t k ) ± tσ[µ](x j, t k ). (7) 0 / 3

11 Interpretation of the scheme by means of characteristics Note that Φ defined in (7) (with µ j = m(x j, t k )) can be interpreted as a single Euler step approximation of dx(s) = b[m] (X(s), s)) ds + σ[m] (X(s), s)) dw (s), s (t k, t k+ ), X(t k ) = x j, (8) with a random walk discretization of the Brownian motion W ( ). Indeed, considering a random value Z in R such that f P(Z = ) = P(Z = ) = 2 (9) the function Φ ± j [m k] corresponds to one realization of x j + tb[m k ](x j, t k ) ± 2 tσ[m k ](x j, t k )Z. / 3

12 2 -,5 - -0,5 0 0,5,5 2 Fully-discrete in space Figure: P-basis function, β i Given i Z and setting in (6) φ = β i we have for all i Z m i,k+ = G(m k, i, k) = ( ) 2 j Z β i (Φ + j [m k]) + β i (Φ j [m k]) m i,0 = E i m 0 (x)dx m j,k (0) Non-negative : m i,k 0 for k = 0,..., N, i Z Mass conservative : x i m i,k = for k = 0,..., N Generalizable to any dimension Generalizable to handle Dirichlet and Neumann Boundary conditions Generalizable to handle degeneracy of the diffusion matrix 2 / 3

13 Boundary Condition Let us consider () in Ω R 2, we denote by ˆn(x) the outward normal to Ω at x Ω. Homogeneous Neumann condition on Γ N Ω J[m](x, t) ˆn = 0 on Γ N, () where J i [m](x, t) = b i [m](x, t)m k x k (a i,k [m](x, t)m) Dirichlet boundary condition on Γ D := Ω \ Γ N m = 0 on Γ D. (2) The idea is to reflect the discrete characteristics when they intersect Γ N and to cut them at Γ D. 3 / 3

14 Boundary Condition: Dirichlet, d = 2 Let us define, for l =, 2 ĥ l i,± = inf{γ > 0 ; x + γb[m k ](x i, t k ) ± 2dγσ l [m k ](x i, t k ) Γ D } h, and redefine Φ l i,± as Φ l i,± := x i + ĥl,± i,k b[m k](x i, t k ) + Ref. Falcone, Ferretti(203) 2dĥl i,± σ l[m k ](x i, t k ). 4 / 3

15 Boundary Condition: Neumann, d = 2 In order to consider the reflection when exiting at Γ N, we use a symmetrized Euler scheme: we project the discrete characteristics on Ω using the operator P : R d Ω, defined as z, if z Ω, P (z) := 2w z, if z / Ω, (3) where w := argmin z w. w Ω Ref. M. Bossy and E. Gobet and D.Talay, (2004) 5 / 3

16 Unstructured grids Let T be a triangulation of Ω, with x the maximum diameter of the triangles. We call {β i ; i =,..., M} the set of affine shape functions related to the triangular mesh, such that β i (x j ) = δ i,j and i β i(x) = for each x Ω. Median dual control volume E i := E i,t, T T :x i T E i,t := {x T : β j (x) β i (x) j i}. We approximate the solution m by a sequence {m k } k, such that m i,k dm(t k ) E i 6 / 3

17 Consistency Let us denote m x, t a proper extension in R d [0, T ] of the discrete approximation m i,k and P a propability space, metrized by the Kantorowich-Rubinstein distance: { } d (µ, µ 2 ) := sup f(x)d(µ (x) µ 2 (x)) ; Df. R d Proposition (Weak consistency) Assuming that b and σ are continuous and Lipschitz w.r. to x, for every φ C0 (R) if ( x n, t n ) is s.t. as n ( x n, t n ) 0 and xn t n 0, and m xn, t n m in C(0, T ; P (R d )), then for k n such that t k n t lim φ(x) [dm xn, t n n (t kn +) dm xn, t n (0)] R d t [ = 2 tr(a[m](x, s)d2 φ) + b[m](x, s) Dφ ] dm(s). R 0 7 / 3

18 Stability Proposition Suppose that ( x) 2 = O( t), b and σ continuous functions s.t. there exists C > 0 b(x, t, µ) C x σ(x, t, µ) C x, for any x, t, µ. Then, there exists a constant C > 0 (independent of ( x, t)) such that: Proposition R d x 2 dm x, t (t) C t [0, T ]. (4) Suppose that ( x) 2 = O( t). Then, there exists a constant C > 0 (independent of ( x, t)) such that: d (m x, t (t ), m x, t (t 2 )) C t t 2 2 t, t 2 [0, T ]. (5) 8 / 3

19 Convergence Theorem Suppose that ( x) 2 = O( t), b and σcontinuous functions Lipschtiz w.r. to x and there exists C > 0 such that for any x, t, µ b(x, t, µ) C x σ(x, t, µ) C x, then m x, t m in C([0, T ], P ), where m is solution of () and m x, t is a sequence of solution of (0) (extended to a general dimension d). 9 / 3

20 Interacting species Let m and m 2 be the densities of two interacting species t m = div(m ( E (m ) + U (m, m 2 ))) t m 2 = div(m 2 ( E 2 (m 2 ) + U 2 (m, m 2 ))) b.c. internal energy E (m) = E 2 (m) = 3 2 m2 K ε (repulsive self-interactions) cross interaction potential U (m, m 2 ) = W m + W 2 m 2 U (m, m 2 ) = W 2 m + W 22 m 2 Ref. J.D.Benamou, G. Carlier, M.Laborde 20 / 3

21 Interacting species Let us set W = 3 x 2 2,W 2 = 3 x 2 2 W 2 = x 2 2,W 22 = x 2 2 we solve the system on Ω = [.5,.5] [.5,.5] and [0, T ] = [0, 3], we set h = 0.03 and t = h/2 Homogenous Neumann Condition 2 / 3

22 Interacting species t = 0 t = 0.6 t = t =.8 t = 2.4 t = 3 22 / 3

23 Interacting species without self-interactions (E = E 2 = 0 Figure: t = 3 23 / 3

24 Interacting species without selfinteractions (E = E 2 = 0) t = 0 t = 0.6 t = t =.8 t = 2.4 t = 3 24 / 3

25 Interacting species without selfinteractions (E = E 2 = 0 Figure: t = 3 25 / 3

26 The Hughes model In 2002 R. Hughes proposed a macroscopic model for pedestrian dynamics given by t m(x, t) div(m(x, t) f 2 (m(x, t))du(x, t)) = 0, Du(x, t) = f(m(x, t)), (6) where x Ω, t (0, T ], T R +, m corresponds to the density of a crowd trying to exit a domain as fast as possible u the weighted shortest distance to a target, for example an exit. Typical choice for f: f(m) = m where corresponds to the maximum scaled pedestrian density. Few analytic results are available, all of them restricted to spatial dimension one. In this case the dependence of the drift on m is local!, and the convergence analysis does not apply. 26 / 3

27 A modified version of the Hughes model (Joint work with A.Festa,F.J Silva, M.T. Wolframm) A modified version corresponds to t m(x, t) ε m(x, t) div(m(x, t) f 2 (m(x, t))du(x, t)) = 0, ε u(x, t) + 2 Du(x, t) 2 = 2f 2 (m(x, t)) + δ, (7) in Ω (0, T ). δ > 0 prevents the blow-up of the cost when approaching the maximum density one. ε > 0 represents a small diffusion. Ref: Di Francesco M, Markowich P A, Pietschmann F P, Wolfram M T (20) 27 / 3

28 Exit scenario with two doors Figure: Density contour lines at time t = 0.3, ε = The exits are marked with the letter E. The position of the two exits induces an asymmetric splitting of the crowd 28 / 3

29 Exit scenario with two doors Figure: Density contour lines at time t = 0.3 (left) and t =.2 (right). M 0 = 0.7 and ε = Initially, a large part of the crowd chooses to move to the right exit. After a while this exit gets congested, inducing a part of the population to change objective. 29 / 3

30 References Conclusions: we have presented a new SL scheme for a non linear FP main advantage: it is explicit, it allows large time steps, it preserves the positivity and it conserves the mass On going work convergence analysis SL scheme for non linear FP convergence analysis SL scheme for HJB on bounded domain with Neumann end Dirichlet condition Open questions high order SL scheme for FP 30 / 3

31 References E.Carlini, F.J. Silva, A Semi-Lagrangian scheme for a degenerate second order Mean Field Game system, DCDS-A, ( 205) E.Carlini, F.J. Silva, A Semi-Lagrangian scheme for the Fokker-Planck equation, 2nd IFAC Workshop on Control of Systems Governed E.Carlini, A. Festa, F.J. Silva, M.T. Wolfram, A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow, Dynamic Games and Applications, Springer, (206) 3 / 3

32 References E.Carlini, F.J. Silva, A Semi-Lagrangian scheme for a degenerate second order Mean Field Game system, DCDS-A, ( 205) E.Carlini, F.J. Silva, A Semi-Lagrangian scheme for the Fokker-Planck equation, 2nd IFAC Workshop on Control of Systems Governed E.Carlini, A. Festa, F.J. Silva, M.T. Wolfram, A Semi-Lagrangian scheme for a modified version of the Hughes model for pedestrian flow, Dynamic Games and Applications, Springer, (206) THANK YOU 3 / 3