Some recent developments in the traffic flow variational formulation

Some recent developments in
the traffic flow variational formulation
Guillaume Costeseque
collaborations with J-P. Lebacque and J. Laval
Inria Sophia-Antipolis Méditerranée
Séminaire Modélisation des Réseaux de Transports
– IFSTTAR Marne-la-Vallée –
May 12, 2016
G. Costeseque HJ equations & traffic flow Marne-la-Vallée, May 12 2016 1 / 50

Motivation
HJ & Lax-Hopf formula
Hamilton-Jacobi equations: why and what for?
Smoothness of the solution (no shocks)
Physically meaningful quantity

Motivation
Analytical expression of the solution
Eﬃcient computational methods
Easy integration of GPS data
[Mazar´e et al, 2012]

Motivation
Analytical expression of the solution
Eﬃcient computational methods
Easy integration of GPS data
[Mazar´e et al, 2012]
Everything broken for network applications?

Motivation
Network model
Simple case study: generalized three-detector problem (Newell (1993))
N(t, x)

Motivation
Outline
1 Notations from traﬃc ﬂow modeling
2 Basic recalls on Lax-Hopf formula
3 Hamilton-Jacobi and source terms
4 Hamilton-Jacobi on networks

Notations from traﬃc ﬂow modeling
Outline

Convention for vehicle labeling
N
x
t
Flow

Three representations of traﬃc ﬂow
Moskowitz’ surface
Flow
x
t
N
x
See also [Makigami et al, 1971], [Laval and Leclercq, 2013]

Overview: conservation laws (CL) / Hamilton-Jacobi (HJ)
Eulerian Lagrangian
t − x t − n
CL
Variable Density ρ Spacing r
Equation ∂tρ + ∂x Q(ρ) = 0 ∂tr + ∂x V (r) = 0
HJ
Variable Label N Position X
N(t, x) =
+∞
x
ρ(t, ξ)dξ X(t, n) =
+∞
n
r(t, η)dη
Equation ∂tN + H (∂x N) = 0 ∂tX + V (∂x X) = 0
Hamiltonian H(p) = −Q(−p) V(p) = −V (−p)

Basic recalls on Lax-Hopf formula
Outline

Basic recalls on Lax-Hopf formula Lax-Hopf formula
Setting
Consider Cauchy problem
ut + H(Du) = 0, in Rn × (0, +∞),
u(., 0) = u0(.), on Rn.
(1)
Two formulas according to the smoothness of
the Hamiltonian H
the initial data u0

Lax-Hopf formulæ
Assumptions: case 1
(A1) H : Rn → R is convex
(A2) u0 : Rn → R is uniformly Lipschitz
Theorem (First Lax-Hopf formula)
If (A1)-(A2) hold true, then
u(x, t) := inf
z∈Rn
sup
y∈Rn
[u0(z) + y.(x − z) − tH(y)] (2)
is the unique uniformly continuous viscosity solution of (1).

Legendre-Fenchel transform
First Lax-Hopf formula (2) can be recast as
u(x, t) := inf
z∈Rn
u0(z) − tH∗ x − z
t
thanks to Legendre-Fenchel transform
L(z) = H∗
(z) := sup
y∈Rn
(y.z − H(y)) .
Proposition (Bi-conjugate)
If H is strictly convex, 1-coercive i.e. lim
|p|→∞
H(p)
|p|
= +∞,
then H∗ is also convex and
(H∗
)∗
= H.

Basic recalls on Lax-Hopf formula LWR in Eulerian
LWR in Eulerian (t, x)
Cumulative vehicles count (CVC) or Moskowitz surface N(t, x)
q = ∂tN and ρ = −∂x N
If density ρ satisfies the scalar (LWR) conservation law
∂tρ + ∂x Q(ρ) = 0
Then N satisfies the first order Hamilton-Jacobi equation
∂tN − Q(−∂x N) = 0 (3)

Legendre-Fenchel transform with Q concave (relative capacity)
M(q) = sup
ρ
[Q(ρ) − ρq]
M(q)
u
w
Density ρ
q
q
Flow F
w u
q
Transform M
−wρmax

(continued)
Lax-Hopf formula
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)

Hamilton-Jacobi and source terms
Outline

Hamilton-Jacobi and source terms Problem Formulation
Long homogeneous corridor with numerous entrances and exits
Net lateral freeway “inflow” rate φ
Flow
inflow
outflow
= φ
∂tρ + ∂x H(ρ) = φ,
k = g on Γ
(6)
∂tN − H (−∂x N) = Φ,
N = G on Γ,
(7)
where
Φ(t, x) = −
x
0
φ(t, y)dy
G(t, x) =
Γ
g(t, x)dΓ, (t, x) ∈ Γ

Hamilton-Jacobi and source terms Problem Formulation
Some remarks
The ﬂow reads q = Nt − Φ and the cumulative count curves are
N(t, x) =
t
0
q(s, x)ds
usual N-curve
+
t
0
x
0
φ(s, y)dsdy
net number of vehicles “entering”
If φ = φ(k) then
Φ(t, x) = ˜Φ(t, x, −Nx ) = −
x
0
φ(−Nx (t, y))dy, (8)
This means that (7) becomes the more general HJ equation
Nt − ˜H(t, x, −Nx ) = 0
where ˜H(t, x, k) = H(k) + ˜Φ(t, x, k).

Hamilton-Jacobi and source terms Exogenous inﬂow
Variational problem
Lax-Hopf formula:
N(P) = min
B∈ΓP , ξ∈VBP
f (B, ξ) (9)
f (B, ξ) := G(B) +
t
tB
R(s, ξ(s), ξ′
(s)) ds
P ≡ (t, x) “target” point
B ≡ (tB , y) on the boundary ΓP
ξ ∈ VBP set of valid paths B → P
R(·) Legendre transform of ˜H
R(t, x, v) = sup
k
˜H(t, x, k) − vk .

Assume a triangular ﬂow-density diagram
The function f (B, ξ) to be minimized reads
f (B, ξ) = G(B) + (t − tB)Q − (x − y)K +
t
tB
Φ(s, ξ(s)) ds
=:J
(10)
where Q = capacity, K = critical density.
J = net number of vehicles leaving the area
A(ξ) below the curve x = ξ(t)
J = −
t
tB
ξ(s)
y
φ(s, x) dxds, (11)

Initial value problems with constant density
Assume
N(0, x) = G(x) = −k0x (g(x) = k0) ,
φ(t, x) = a,
(12)
min f (B ≡ (y, 0), ξ) reached for a path
(i) maximizing A(ξ) when a > 0
(ii) or minimizing A(ξ) when a < 0
f (y) = c0 + c1y + c2y2 with c2 > 0
Explicit solution:
N(t, x) =
f (y∗), t > K−k0
a > 0
min{f (xU), f (xD)}, otherwise

Extended Riemann problem (ERP)
Consider
(g(x), φ(x)) =
(kU, aU), x ≤ x0
(kD, aD), x > x0,
(13)
Assuming G(x0) = 0
G(x) =
(x0 − x)kU , x ≤ x0
(x0 − x)kD , x > x0
(14)
J-integral = weighted average of the portion of
A(ξ) upstream and downstream of x = x0 with
weighs aU and aD

Extended Riemann problem (ERP)
(continued)
Same minimization of J(ξ)
f (y) = G(y) + tQ − (x0 − y)K + J(y) with
J(y) =
min{j1(y), j2(y), j3(y)}, y > x0
min{j4(y), j5(y), j6(y)}, y ≤ x0
Possible minima for the components of f (y)
y = yi ∈ ΓP
, i = 1, . . . 6
Semi-explicit solution (9 candidates):
N(t, x0) = min
y∈Y
f (y)
Y = {xU, x0, xD, y∗
1 , . . . y∗
6 }

Hamilton-Jacobi and source terms Numerical solution methods
Godunov’s method
Basis of the well known Cell Transmission (CT) model
Time and space increments ∆t and ∆x = u∆t
Numerical approximation of the density
kj
i = k(j∆t, i∆x) (15)
Discrete approximation of the conservation law (6):
kj+1
i − kj
i
∆t
+
qj
i+1 − qj
i
∆x
= φ(kj
i ) (16)
with (CT rule)
qj
i = min{Q, ukj
i , (κ − kj
i+1)w} (17)

Example
Consider an empty freeway at t = 0 with
g(x) = 0, (18a)
φ(k) = ax − buk, a, b > 0. (18b)
Exact solution (method of characteristics):
k(t, x) =
a
b2u
bx − 1 + (1 − b(x − tu))e−btu
(19)
provided k(t, x) ≤ K (Laval, Leclercq (2010))

Example
(continued)
Comparison of numerical solutions (∆t = 40 s) and the exact solution
Main diﬀerence = the ﬂow estimates

Example
(continued)
Density RMSE (numerical VS exact solution) for varying ∆t:
Both converge as ∆t → 0
Accuracy of ERP > CT rule

Example
(continued)
Optimal candidate that minimizes f (y) at each time step
Diﬀerence when k → K
Most accurate optimal
candidate y∗
1

Variational networks
Dt
u
-w
vi
di
ti
timeti
i
Si = area
Dx
space

Dt
u
-w
vi
di
ti
timeti
i
Si = area
Dx
space
Only three wave speeds with costs
L(vi ) =
⎧
⎪⎨
⎪⎩
wκ, vi = −w
Q, vi = 0
0, vi = u
(20)
The cost on each link:
ci = L(vi )τi + Ji .
Ji = contribution of the J-integral in
the cost of each link i

Dt
u
-w
vi
di
ti
timeti
i
Si = area
Dx
space
Advantage: free of numerical errors
(when inﬂows are exogenous)
Drawback:
cumbersome to implement unless u
w is
an integer
merge models expressed in terms of
ﬂows or densities rather than N values:
additional computational layer needed

Hamilton-Jacobi on networks
Outline

A special network = junction
HJN
HJ1
HJ2
HJ3
HJ4
HJ5
Network

A special network = junction
HJN
HJ1
HJ2
HJ3
HJ4
HJ5
Network
HJ4
HJ2
HJ1
HJ3
Junction J

Space dependent Hamiltonian
Consider HJ equation posed on a junction J
ut + H(x, ux ) = 0, on J × (0, +∞),
u(t = 0, x) = g(x), on J
(21)
Extension of Lax-Hopf formula(s)?
No simple linear solutions for (21)
No deﬁnition of convexity for discontinuous functions

Hamilton-Jacobi on networks Literature with application to traﬃc
Junction models
Classical approaches for CL:
Macroscopic modeling on (homogeneous) sections
Coupling conditions at (pointwise) junction
For instance, consider
⎧
⎪⎨
⎪⎩
ρt + (Q(ρ))x = 0, scalar conservation law,
ρ(., t = 0) = ρ0(.), initial conditions,
ψ(ρ(x = 0−, t), ρ(x = 0+, t)) = 0, coupling condition.
(22)
See Garavello, Piccoli [4], Lebacque, Khoshyaran [6] and Bressan et al. [1]

Hamilton-Jacobi on networks Literature with application to traffic
Examples of junction models
Model with internal state (= buffer(s))
Bressan & Nguyen (NHM 2015) [2]
ρ → Q(ρ) strictly concave
advection of γij (t, x) turning ratios from (i) to (j)
(GSOM model with passive attribute)
internal dynamics of the buffers (ODEs): queue lengths
Extended Link Transmission Model
Jin (TR-B 2015) [5]
Link Transmission Model (LTM) Yperman (2005, 2007)
Triangular diagram
Q(ρ) = min {uρ, w(ρmax − ρ)} for any ρ ∈ [0, ρmax]
Commodity = turning ratios γij(t)
Definition of boundary supply and demand functions

Hamilton-Jacobi on networks Settings
First remarks
If N solves
Nt + H (Nx ) = 0
then ¯N = N + c for any c ∈ R is also a solution

First remarks
If N solves
Nt + H (Nx ) = 0
then ¯N = N + c for any c ∈ R is also a solution
Ni
(t, x)
N0
(t)
Nj
(t, x)
(j)
(i)
No a priori relationship between initial
conditions
Nk (t, x) consistent along the same
branch Jk and
∂tN0
(t) =
i
∂tNi
t, x = 0−
=
j
∂tNj
t, x = 0+

Key idea
Assume that H is piecewise linear (triangular FD)
Nt + H (Nx ) = 0
with
H(p) = max{H+
(p)
supply
, H−
(p)
demand
}

Key idea
Assume that H is piecewise linear (triangular FD)
Nt + H (Nx ) = 0
with
H(p) = max{H+
(p)
supply
, H−
(p)
demand
}
Partial solutions N+ and N− that solve resp.
⎧
⎪⎨
⎪⎩
N+
t + H+ (N+
x ) = 0,
N−
t + H− (N−
x ) = 0
such that N = min N−
, N+
Upstream demand advected by waves moving forward
Downstream supply transported by waves moving backward

Hamilton-Jacobi on networks Mathematical expression
Junction model
Optimization junction model (Lebacque’s talk)
Lebacque, Khoshyaran (2005) [6]
max
⎡
⎣
i
φi (qi ) +
j
ψj (rj )
⎤
⎦
s.t.
0 ≤ qi ∀i
qi ≤ δi ∀i
0 ≤ rj ∀j
rj ≤ σj ∀j
0 = rj − i γij qi ∀j
(23)
where φi , ψj are concave, non-decreasing

Example of optimization junction models
Herty and Klar (2003)
Holden and Risebro (1995)
Coclite, Garavello, Piccoli (2005)
Daganzo’s merge model (1995) [3]
⎧
⎪⎨
⎪⎩
φi (qi ) = Nmax qi −
q2
i
2pi qi,max
ψ = 0
where pi is the priority of ﬂow coming from road i and Nmax = φ′
i (0)

Solution of the optimization model
Lebacque, Khoshyaran (2005)
Karush-Kuhn-Tucker optimality conditions:
For any incoming road i
φ′
i (qi )+
k
skγik −λi = 0, λi ≥ 0, qi ≤ δi and λi (qi −δi ) = 0,
and for any outgoing road j
ψ′
j (rj ) − sj − λj = 0, λj ≥ 0, ri ≤ σj and λj (rj − σj ) = 0,
where (sj , λj ) = Karush-Kuhn-Tucker coeﬃcients (or Lagrange multipliers)

Solution of the optimization model
Lebacque, Khoshyaran (2005)
⎧
⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
qi = Γ[0,δi ] (φ′
i )−1
−
k
γiksk , for any i,
rj = Γ[0,σj ] (ψ′
j )−1
(sj ) , for any j,
(24)
ΓK is the projection operator on the set K
i qi
i γijδi
σj rj
sj

Model equations
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
Ni
t + Hi (Ni
x ) = 0, for any x ̸= 0,
∂tNi (t, x−) = qi (t),
∂tNj (t, x+) = rj (t),
at x = 0,
Ni (t = 0, x) = Ni
0(x),
∂tNi (t, x = ξi ) = ∆i (t),
∂tNj (t, x = χj ) = Σj(t)

Algorithm
Inf-morphism property: compute partial solutions for
initial conditions
upstream boundary conditions
downstream boundary conditions
internal boundary conditions
1 Propagate demands forward
through a junction, assume that the downstream supplies are maximal
2 Propagate supplies backward
through a junction, assume that the upstream demands are maximal

Hamilton-Jacobi on networks Application to a simple junction
Spatial discontinuity
Time
Space
Density
Flow
Downstream condition
(u)
(d)
Upstream condition
Initial condition

Hamilton-Jacobi on networks Application to a simple junction
Numerical results

Conclusion and perspectives
Final remarks
In a nutshell:
No explicit solution right now
Only speciﬁc cases
Importance of the supply/demand functions
General optimization problem at the junction
Perspectives:
Lane changing behaviors
Estimation on networks
Stationary states

References
Some references I
A. Bressan, S. Canic, M. Garavello, M. Herty, and B. Piccoli, Flows
on networks: recent results and perspectives, EMS Surveys in Mathematical
Sciences, (2014).
A. Bressan and K. T. Nguyen, Conservation law models for traffic flow on a
network of roads, to appear, (2014).
C. F. Daganzo, The cell transmission model, part ii: network traffic,
Transportation Research Part B: Methodological, 29 (1995), pp. 79–93.
M. Garavello and B. Piccoli, Traffic flow on networks, American institute of
mathematical sciences Springfield, MO, USA, 2006.
W.-L. Jin, Continuous formulations and analytical properties of the link
transmission model, Transportation Research Part B: Methodological, 74 (2015),
pp. 88–103.

References
Some references II
J.-P. Lebacque and M. M. Khoshyaran, First-order macroscopic traffic flow
models: Intersection modeling, network modeling, in Transportation and Traffic
Theory. Flow, Dynamics and Human Interaction. 16th International Symposium on
Transportation and Traffic Theory, 2005.

References
Thanks for your attention
guillaume.costeseque@inria.fr

Some recent developments in the traffic flow variational formulation

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