Numerical methods for variational principles in traffic

Numerical methods for variational principles in traffic
Guillaume Costeseque
joint work with J-P. Lebacque
Ecole des Ponts ParisTech, CERMICS & IFSTTAR, GRETTIA
Séminaire Modélisation des réseaux de transport
October 16, 2013 - Marne-la-Vallée
G. Costeseque (Université ParisEst) Variational principles: numerics Marne-la-Vallée, October 2013 1 / 49

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?

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)

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]

Introduction
Outline
1 Macroscopic traffic flow models
2 Variational principles in traffic
3 Numerical method
4 Conclusion

Macroscopic traﬃc ﬂow models
Outline
3 Numerical method
4 Conclusion

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]

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]

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

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

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?

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).

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]

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)

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).

Variational principles in traﬃc
Outline
LWR model
GSOM family
3 Numerical method
4 Conclusion

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, Théorie du trafic, ENTPE]

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)

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

(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]

(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)

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)

(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)

(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)

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

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)
[Mazaré et al, 2012]

(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]

Examples of data assimilation
Eulerian coordi-
nates (x, t) [Mazar´e et al, 2012]
Lagrangian coordinates (n, t)
[Han et al, 2012]

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...

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.

(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

(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)

(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

Numerical method
Outline
3 Numerical method
Methodology
Elementary blocks
Numerical example
4 Conclusion

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

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.

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

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.

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.

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

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

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

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

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)

Numerical result

Conclusion
Outline
3 Numerical method
4 Conclusion

Conclusion
High interest of variational theory in traﬁc
Semi-explicit computational algorithms
Data assimilation
Diﬃculty when Hamiltonian depends on time / space / vehicle

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?

Conclusion
The End
Thanks for your attention
guillaume.costeseque@cermics.enpc.fr
guillaume.costeseque@ifsttar.fr

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.

Numerical methods for variational principles in traffic

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