Numerical methods for variational principles in traffic
1. Numerical methods for variational principles in traffic
Guillaume Costeseque
joint work with J-P. Lebacque
Ecole des Ponts ParisTech, CERMICS & IFSTTAR, GRETTIA
S´eminaire Mod´elisation des r´eseaux de transport
October 16, 2013 - Marne-la-Vall´ee
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 1 / 49
2. Introduction
Variational principles in physical systems
Evolutionary systems in time and space
Position
Starting point (x0, t0)
Time
Target (xT , T)
Which is the (good) physical evolution?
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 2 / 49
3. Introduction
Variational principles in physical systems
Evolutionary systems in time and space
Variational methods ⇔ calculus of variations
Calculus of variations
Minimum principle (Pontryagin)
Hamilton−Jacobi−Bellman equation
Dynamic programming (Bellman)
Fermat principle in geometrical optics (1657)
Maupertuis principle in mechanics (1740)
... Euler-Lagrange-Jacobi principle of least action (1755-1840)
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 3 / 49
4. Introduction
Breakthrough in traffic monitoring
Traffic monitoring from GPS enabled devices
Floating car data (probe vehicles)
Cheaper (no dedicated infrastructure)
Increasing number (dense situations)
Accurate
Scientific challenge ⇒ data assimilation
[Mobile Millenium, 2008]
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 4 / 49
7. Macroscopic traffic flow models First order model
LWR model
Traffic state:
density of vehicles ρ(t, x) at time t and location x
flow speed v (mean spatial velocity of vehicles)
x x + ∆x
ρ(x, t)∆x
Q(x, t)∆t Q(x + ∆x, t)∆t
Scalar one dimensional conservation law
∂tρ + ∂x(ρv) = 0 Conservation of vehicles,
v = I(ρ, x) Fundamental diagram.
(1)
[Lighthill and Whitham, 1955], [Richards, 1956]
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 7 / 49
8. Macroscopic traffic flow models First order model
Fundamental diagram (FD)
Flow-density fundamental diagram F : ρ → ρI(ρ)
Empirical function with
ρmax the maximal or jam density,
ρc the critical density
Flux is increasing for ρ ≤ ρc: free-flow phase
Flux is decreasing for ρ ≥ ρc: congestion phase
0
Flow, F
ρmax
Density, ρ
0
Flow, F
ρmax
Density, ρ
0
Flow, F
ρmax
Density, ρ
[Garavello and Piccoli, 2006]
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 8 / 49
9. Macroscopic traffic flow models First order model
Kinematic waves theory
Conservation law
∂tρ + ∂xρ (∂ρF) = −∂xF
Characteristics= curves such that
˙x(t) = ∂ρF
along such curve, density evolves such that
˙ρ(t) = −∂xF
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 9 / 49
10. Macroscopic traffic flow models Second order models
Motivation for higher order models
Experimental evidences
fundamental diagram: multi-valued in congested case
[S. Fan, U. Illinois], NGSIM dataset
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 10 / 49
11. Macroscopic traffic flow models Second order models
Motivation for higher order models
Experimental evidences
fundamental diagram: multi-valued in congested case
phenomena not accounted for: bounded acceleration, capacity drop...
Need for models able to integrate measurements of different traffic
quantities (acceleration, fuel consumption, noise)
First order models: “mass” conservation but what about conservation
of momentum or energy?
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 11 / 49
12. Macroscopic traffic flow models Second order models
GSOM family
Generic Second Order Models (GSOM) family
∂tρ + ∂x(ρv) = 0 Conservation of vehicles,
∂t(ρI) + ∂x(ρvI) = ρϕ(I) Dynamics of the driver attribute I,
v = I(ρ, I) Fundamental diagram,
(2)
Specific driver attribute I
the driver aggressiveness,
the driver origin / destination,
the vehicle class,
...
Flow-density fundamental diagram
F : (ρ, I) → ρI(ρ, I).
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 12 / 49
13. Macroscopic traffic flow models Second order models
Kinematic waves or 1-waves:
similar to the seminal LWR model
density variations at speed ν = ∂ρI(ρ, I)
driver attribute I is continuous
Contact discontinuities or 2-waves:
variations of driver attribute I at speed ν = I(ρ, I)
the flow speed v is constant.
[Lebacque et al., 2007]
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 13 / 49
14. Macroscopic traffic flow models Second order models
Examples of GSOM models
LWR model= a GSOM model with no specific driver attribute
The LWR model with bounded acceleration [Lebacque, 2002-2003],
[Leclercq, 2007] = a GSOM model with driver attribute the speed of
vehicles.
The ARZ model (for Aw, Rascle and Zhang) with driver attribute
I = v − Ve(ρ) and
I(ρ, I) = I + Ve(ρ)
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 14 / 49
15. Macroscopic traffic flow models Second order models
Examples of GSOM models
(continued)
Multi-commodity models (multi-class, multi-lanes)
[Jin and Zhang, 2004],
[Bagnerini and Rascle, 2003],
[Herty, Kirchner, Moutari and Rascle, 2008],
[Klar, Greenberg and Rascle, 2003].
The Colombo 1-phase model with no driver attribute in fluid situation
and driver attribute I a non-trivial scalar in congested situation.
The stochastic GSOM model of [Khoshyaran and Lebacque, 2009]
with driver attribute I a random variable such that I = I(N, t, ω).
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 15 / 49
16. Variational principles in traffic
Outline
1 Macroscopic traffic flow models
2 Variational principles in traffic
LWR model
GSOM family
3 Numerical method
4 Conclusion
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 16 / 49
17. Variational principles in traffic LWR model
Key surface
Three-dimensional representation of traffic flow
Eulerian (x, t), Lagrangian (n, t), T-coordinates (x, n)
See [Makigami et al, 1971], [Laval and Leclercq, 2013]
Moskowitz surface
[Leclercq, Th´eorie du trafic, ENTPE]
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 17 / 49
18. Variational principles in traffic LWR model
LWR in Eulerian (x, t)
Cumulative vehicles count (CVC) or Moskowitz surface N(x, t)
f = ∂tN and ρ = −∂xN
If density ρ satisfies the scalar (LWR) conservation law
∂tρ + ∂xF(ρ) = 0
Then N satisfies the first order Hamilton-Jacobi equation
∂tN − F(−∂xN) = 0 (3)
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 18 / 49
19. Variational principles in traffic LWR model
LWR in Eulerian (x, t)
Legendre-Fenchel transform with F concave (relative capacity)
M(q) = sup
ρ
[F(ρ) − ρq]
M(q)
u
w
Density ρ
q
q
Flow F
w u
q
Transform M
−wρmax
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 19 / 49
20. Variational principles in traffic LWR model
LWR in Eulerian (x, t)
(continued)
Lax-Hopf formula (representation formula) [Daganzo, 2006]
N(T, xT ) = min
u(.),(t0,x0)
T
t0
M(u(τ))dτ + N(t0, x0),
˙X = u
u ∈ U
X(t0) = x0, X(T ) = xT
(t0, x0) ∈ J
(4) Time
Space
J
(T, xT )˙X(τ)
(t0, x0)
Viability theory [Claudel and Bayen, 2010]
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 20 / 49
21. Variational principles in traffic LWR model
LWR in Eulerian (x, t)
(Historical note)
Dynamic programming [Daganzo, 2006] for triangular FD
(u and w free and congested speeds)
Flow, F
w
u
0 ρmax
Density, ρ
u
x
w
t
Time
Space
(t, x)
Minimum principle [Newell, 1993]
N(t, x) = min N t −
x − xu
u
, xu ,
N t −
x − xw
w
, xw + ρmax(xw − x) ,
(5)
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 21 / 49
22. Variational principles in traffic LWR model
LWR in Lagrangian (n, t)
Consider X(t, n) the location of vehicle n at time t ≥ 0
v = ∂tX and r = −∂nX
If the spacing r := 1/ρ satisfies the LWR model (Lagrangian coord.)
∂tr + ∂nV(r) = 0
with the speed-spacing FD V : r → I (1/r) ,
Then X satisfies the first order Hamilton-Jacobi equation
∂tX − V(−∂nX) = 0. (6)
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 22 / 49
23. Variational principles in traffic LWR model
LWR in Lagrangian (n, t)
(continued)
Legendre-Fenchel transform with V concave
M(u) = sup
r
[V(r) − ru] .
Lax-Hopf formula
X(T, nT ) = min
u(.),(t0,n0)
T
t0
M(u(τ))dτ + X(t0, n0),
˙N = u
u ∈ U
N(t0) = n0, N(T) = nT
(t0, n0) ∈ J
(7)
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 23 / 49
24. Variational principles in traffic LWR model
LWR in Lagrangian (n, t)
(continued)
Dynamic programming for triangular FD
1/ρcrit
Speed, V
u
−wρmax
Spacing, r
1/ρmax
−wρmax
n
t
(t, n)
Time
Label
Minimum principle ⇒ car following model [Newell, 2002]
X(t, n) = min X(t0, n) + u(t − t0),
X(t0, n + wρmax(t − t0)) + w(t − t0) .
(8)
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 24 / 49
25. Variational principles in traffic LWR model
Numerical methods for LWR
Dynamic programming [Daganzo, 2006]
Minimization of a cost function over a computational grid
Computational cost proportional to the complexity of the grid
u
w
Position
Time
∆x
∆t
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 25 / 49
26. Variational principles in traffic LWR model
Numerical methods for LWR
Dynamic programming [Daganzo, 2006]
Minimization of a cost function over a computational grid
Computational cost proportional to the complexity of the grid
Exact for piecewise affine (PWA) value conditions and piecewise
affine FD (but not for arbitrary concave FDs)
Possibility to integrate internal boundary conditions but uneasy
Possibility to integrate space/time dependent FDs (space/time
dependent cost)
[Mazar´e et al, 2012]
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 26 / 49
27. Variational principles in traffic LWR model
Numerical methods for LWR
(continued)
Lax-Hopf algorithm [Claudel and Bayen, 2010]
Minimization of closed form partial solutions (grid-free)
Computational cost proportional to the number of initial and
boundary condition blocks
Exact for PWA value conditions and arbitrary concave FDs
Possible integration of internal boundary conditions
Integration of space-time varying fundamental diagrams: to be done
[Mazar´e et al, 2012]
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 27 / 49
28. Variational principles in traffic LWR model
Examples of data assimilation
Eulerian coordi-
nates (x, t) [Mazar´e et al, 2012]
Lagrangian coordinates (n, t)
[Han et al, 2012]
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 28 / 49
29. Variational principles in traffic GSOM family
GSOM in Eulerian (x, t)
From [Li and Zhang, 2013], system of coupled scalar conservation
laws
∂tρ + ∂xf(ρ, s) = 0 Conservation of vehicles,
∂ts + ∂xg(ρ, s) = 0 Dynamics around the equilibrium.
(9)
GSOM family for s = ρI and ϕ = 0
Variational representations for cumulative quantities
Nρ :=
+∞
x
ρ(y, t)dy and Ns :=
+∞
x
s(y, t)dy,
if characteristics system (coupled ODEs) is satisfied...
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 29 / 49
30. Variational principles in traffic GSOM family
GSOM in Lagrangian (n, t)
From [Lebacque and Khoshyaran, 2013], GSOM in Lagrangian
∂tr + ∂N v = 0 Conservation of vehicles,
∂tI = ϕ(N, I, t) Dynamics of I,
v = W(N, r, t) := V(r, I(N, t)) Fundamental diagram.
(10)
Position X(N, t) :=
t
−∞
v(N, τ)dτ satisfies the HJ equation
∂tX − W(N, −∂N X, t) = 0, (11)
And I(N, t) solves the ODE
∂tI(N, t) = ϕ(N, I, t),
I(N, 0) = i0(N), for any N.
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 30 / 49
31. Variational principles in traffic GSOM family
GSOM in Lagrangian (n, t)
(continued)
Legendre-Fenchel transform of W according to r
M(N, c, t) = sup
r∈R
{W(N, r, t) − cr}
M(N, p, t)
pq
W(N, q, t)
W(N, r, t)
q r
p
p
u
c
Transform M
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 31 / 49
32. Variational principles in traffic GSOM family
GSOM in Lagrangian (n, t)
(continued)
Lax-Hopf formula
X(NT , T) = min
u(.),(N0,t0)
T
t0
M(N, u, t)dt + ξ(N0, t0),
˙N = u
u ∈ U
N(t0) = N0, N(T) = NT
(N0, t0) ∈ J
(12)
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 32 / 49
33. Variational principles in traffic GSOM family
GSOM in Lagrangian (n, t)
(continued)
Optimal trajectories = characteristics
˙N = ∂rW(N, r, t),
˙r = −∂N W(N, r, t),
(13)
System of ODEs to solve
Difficulty: not straight lines in the general case
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 33 / 49
34. Numerical method
Outline
1 Macroscopic traffic flow models
2 Variational principles in traffic
3 Numerical method
Methodology
Elementary blocks
Numerical example
4 Conclusion
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 34 / 49
35. Numerical method Methodology
General ideas
First key element: Lax-Hopf formula
Computations only for the characteristics
X(NT , T) = min
(N0,r0,t0)
T
t0
M(N, ∂rW(N, r, t), t)dt + ξ(N0, t0),
˙N(t) = ∂rW(N, r, t)
˙r(t) = −∂N W(N, r, t)
N(t0) = N0, r(t0) = r0, N(T) = NT
(N0, r0, t0) ∈ K
(14)
K is the set of initial/boundary values
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 35 / 49
36. Numerical method Methodology
General ideas
(continued)
Second key element: inf-morphism prop. [Aubin et al, 2011]
Consider a union of sets (initial and boundary conditions)
K =
l
Kl,
then the global minimum is
X(NT , T) = min
l
Xl(NT , T), (15)
with Xl partial solution to sub-problem Kl.
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 36 / 49
37. Numerical method Elementary blocks
Assumptions
Piecewise affine value conditions
the initial condition: positions of vehicles at time t = t0,
the “upstream” boundary condition: trajectory of the first vehicle
N = N0 traveling on the section,
and internal boundary conditions: cumulative vehicles counts at fixed
location X = x0.
Finite horizon problems (N, t) ∈ [N0, Nmax] × [t0, tmax]
No relaxation ϕ = 0
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 37 / 49
38. Numerical method Elementary blocks
Initial conditions
Discretize the set of N into [np, np+1] of length ∆n
Assume that ∆n small enough such that
Simplified dynamic
ϕ(N, I, t) = ϕp(I, t), for any N ∈ [np, np+1].
Initial data are piecewise constant
I(N, t0) = I0,p,
r(N, t0) = r0,p.
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 38 / 49
39. Numerical method Elementary blocks
Sub-algorithm for initial block [np, np+1] × {t0}
(i) Initialize X to +∞
(ii) Number of characteristics to compute
(iii) Compute N(t) of each characteristic while t ≤ tmax and N ≤ Nmax
(iv) Calculate the (exact) solution Xp all along each characteristic
(v) Compute the exact value at any point within the characteristics fan
(simple translation)
(vi) In a rarefaction interpolate the value of X at each point within the
influence domain.
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 39 / 49
40. Numerical method Elementary blocks
PWA initial conditions
Domain of influence of the initial condition
Couples for initial conditions (N, r0(N))
r0,p
0
I0,p
N0
N
np np+1 Nmax
t
tmax
2
1
t0
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 40 / 49
41. Numerical method Elementary blocks
Upstream boundary conditions
Domain of influence of the upstream boundary condition
Couples for initial conditions (N0, r0(t))
N0
t0 N
Nmax
t
tmax
2 1
0
r0,q
tq+1
tq
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 41 / 49
42. Numerical method Elementary blocks
Internal boundary conditions
Domain of influence of the internal boundary condition
Triplet for initial conditions (N(t), r0(t), v0(t))
r0,p
1
0
2
np np+1N0
N
Nmax
t
tmax
t0
r0,p
2
0
1
np np+1N0
N
Nmax
t
tmax
t0
Under-critical case Over-critical case
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 42 / 49
43. Numerical method Numerical example
Fundamental Diagram and Driver Attribute
0 20 40 60 80 100 120 140 160 180 200
−5
0
5
10
15
20
25
30
35
Headway r (m)
SpeedW(m/s)
Fundamental diagram W(N,r,t)
I(N,t)=1
I(N,t)=2
I(N,t)=3
0 5 10 15 20 25 30
1
1.5
2
2.5
3
Label N
Initial conditions I(N,t
0
)
DriverattributeI
0
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 43 / 49
44. Numerical method Numerical example
Initial and Boundaries Conditions
0 5 10 15 20 25 30
15
20
25
30
35
40
45
50
55
60
Label N
Initial conditions r(N,t
0
)Headwayr
0
(m)
0 5 10 15 20 25 30
−800
−700
−600
−500
−400
−300
−200
−100
0
Label N
Initial positions X(N,t
0
)
PositionX(m)
0 50 100 150 200 250 300
10
20
30
40
50
60
70
80
90
Time t (s)
Headwayr
0
(m)
Headway r(N0
,t)
0 50 100 150 200 250 300
0
1000
2000
3000
4000
5000
Time t (s)
Position X(N
0
,t)
PositionX
0
(m)
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 44 / 49
45. Numerical method Numerical example
Numerical result
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 45 / 49
47. Conclusion
High interest of variational theory in trafic
Semi-explicit computational algorithms
Data assimilation
Difficulty when Hamiltonian depends on time / space / vehicle
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 47 / 49
48. Conclusion
High interest of variational theory in trafic
Semi-explicit computational algorithms
Data assimilation
Difficulty when Hamiltonian depends on time / space / vehicle
Open questions:
Extend the algorithm for non-zero dynamics ϕ(I) = 0
Confront the algorithm with real data (NGSIM/MOCoPo datasets)
“Simple” formula for time/space dependent Hamiltonians?
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 47 / 49
49. Conclusion
The End
Thanks for your attention
guillaume.costeseque@cermics.enpc.fr
guillaume.costeseque@ifsttar.fr
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 48 / 49
50. Complements
Some references
G. Costeseque, J.P. Lebacque, A variational formulation for higher
order macroscopic traffic flow models: numerical investigation,
Working paper, (2013).
C.F. Daganzo, On the variational theory of traffic flow:
well-posedness, duality and applications, Networks and Heterogeneous
Media, AIMS, 1 (2006), pp. 601-619.
J.A. Laval, L. Leclercq, The Hamilton-Jacobi partial differential
equation and the three representations of traffic flow, Transportation
Research Part B, 52 (2013), pp. 17-30.
J.P. Lebacque, M.M. Khoshyaran, A variationnal formulation for
higher order macroscopic traffic flow model of the GSOM family,
Procedia-Social and Behavioral Sciences, 80 (2013), pp. 370-394.
G. Costeseque (Universit´e ParisEst) Variational principles: numerics Marne-la-Vall´ee, October 2013 49 / 49