pierrecardaliaguet université paris dauphine€¦ · thefollow-the-leadermodel d dt u iptq vpu i...
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From heterogeneous microscopic traffic flow models tomacroscopic models
Pierre CardaliaguetUniversité Paris Dauphine
Joint work with N. Forcadel (INSA Rouen)
May 18, 2020
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The problem of modelling traffic flow
§ By which laws do vehicles interact with each other?§ Temporal evolution of traffic density?
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What we address here
§ Traffic on a single line§ No overtaking
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Two classical models of traffic flow
We study traffic flow models on a single straight road (without overtake).
Two kinds of models:
1) Microscopic models: e.g., the follow-the-leader model is a system ofODEs
ddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z.
2) Macroscopic models: e.g., the Lighthill-Whitham-Richards (LWR)model is the scalar conservation law
Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q,
(M. J. Lighthill and G. B. Whitham (1955), P. I. Richards (1956))
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The follow-the-leader model
i=0 i=1 i=2i=-1i=-2
U-2(t)U
-2(t) U
-1(t) U
0(t) U
1(t) U
2(t)
ddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z,
where§ Uiptq denotes the position of car i P Z at time t ě 0,§ Cars are ordered: Uiptq ď Ui`1ptq for all t, i ,§ The velocity V “ V ppq ě 0 of car i depends in an increasing way onthe distance p of car i to car i ` 1.
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Vz(h)
hz0
0
V zmax
h
Figure: Typical shape of the optimal velocity function V .
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Properties of the follow-the-leader model
ddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z,
§ Compute trajectories of each vehicle
§ Can be extended to multiple lanes
§ At the core of most micro-simulators
§ Good for simulation
(cf. Seibold, B. (2015). A mathematical introduction to traffic flow theory.IPAM Tutorials.)
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The LWR Model
The Lighthill-Whitham-Richards (LWR) model is the scalar conservation law
Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q,
where§ ρ is the density of vehicles on the road,§ v : R` Ñ R`. The map f pρq “ ρvpρq is the so-called “fundamentaldiagram"
Properties of the LWR model:§ Describe aggregate quantities via PDE§ Natural framework for traveling waves and shocks
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The fundamental diagram f pρq “ ρvpρq
(after Seo, T., Kawasaki, Y., Kusakabe, T., & Asakura, Y. (2019))
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Goal of the talk
§ Discuss how to derive the LWR model
Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q,
from the follow-the-leader modelddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z.
§ Well-known when all the vehicles are identical, i.e., V does not dependon i . Then f pρq “ ρvpρq “ ρV p1ρq (Aw, Klar, Materne, and Rascle (2002))
§ The fact that the vehicles are identical is a very restrictive (andunatural) assumption.
§ Main contribution: we address the case where the vehicles are different:ddt Uiptq “ VipUi`1ptq ´ Uiptqq, t ě 0,@i P Z,
where the distribution of the Vi is “well distributed".
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From an homogeneous traffic flow...ddt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z, f pρq “ ρV p1ρq
i=0 i=1 i=2i=-1i=-2
U-2(t)U
-2(t) U
-1(t) U
0(t) U
1(t) U
2(t)
... to an heterogenous one:ddt Uiptq “ VipUi`1ptq ´ Uiptqq, t ě 0,@i P Z,
i=0 i=1 i=2i=-1i=-2
U-2(t)U
-2(t) U
-1(t) U
0(t) U
1(t) U
2(t)
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Outline
Heuristic arguments in the homogeneous case
Main results
Ideas of proof
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Outline
Heuristic arguments in the homogeneous case
Main results
Ideas of proof
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Outline
Heuristic arguments in the homogeneous case
Main results
Ideas of proof
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Outline
Heuristic arguments in the homogeneous case
Main results
Ideas of proof
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Heuristic derivation of the macro model from the micro oneThe microscopic model: d
dt Uiptq “ V pUi`1ptq ´ Uiptqq, t ě 0,@i P Z,The macroscopic model (LWR): Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q
§ Consider the distribution of vehicles Rptq “ÿ
iPZδUi ptq.
§ After an hyperbolic scaling px , tq Ñ pε´1x , ε´1tq, we obtainρεptq “ ε
ÿ
iPZδεUi pε´1tq.
§ Then, for any test function φ P C8c pRq,
ddt
ż
Rφpxqρεpdx , tq “ d
dt εÿ
iPZφpεUipε
´1tqq
“ εÿ
iPZφx pεUipε
´1tqq ddt Uipε
´1tq
“ εÿ
iPZφx pεUipε
´1tqq V`
Upi`1qpε´1tq ´ Uipε
´1tq˘
§ Next we show that Upi`1qpε´1tq ´ Uipε
´1tq » 1`
ρεpUipε´1tq
˘
.
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§ Proof that Upi`1qpε´1tq ´ Uipε
´1tq » 1`
ρεpUipε´1tq
˘
:
Indeed, if x “ εUipε´1tqq and dx “ εpUpi`1qpε
´1tq ´ Uipε´1tqq, then,
ρεprx , x ` dxq, tq “ ε cardtj P Z, εUjpε´1tq P rx , x ` dxqq “ ε,
so that
ρεpx , tq » ρεprx , x ` dxq, tq pdxq´1 “ ε pdxq´1
“ pUpi`1qpε´1tq ´ Uipε
´1tqq´1.
§ As ρεptq “ εř
iPZ δεUi pε´1tq, we have
ddt
ż
Rφpxqρεpdx , tq » ε
ÿ
iPZφx pεUipε
´1tqqVˆ
1ρεpεUipε´1tq, tq
˙
“
ż
Rφx pxqV
ˆ
1ρεpx , tq
˙
ρεpdx , tq.
§ So ρε solves pLWRq Btρε ` pρεvpρεqqx “ 0 in the sense of
distribution with vpsq “ V p1sq.
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Some references
Rigorous derivation of the macroscopic model from the microscopic one:§ For one type of vehicles:
§ Argall, Cheleshkin, Greenberg, Hinde, and Lin (2002)§ Aw, Klar, Materne, and Rascle (2002)§ Di Francesco and Rosini (2015)§ Goatin and Rossi (2017)§ Holden and Risebro (2018)
§ For several types of cars:§ Chiabaut, Leclercq, and Buisson (2010)
(random model, heuristic derivation)§ Forcadel and Salazar (2015)
(periodic setting)
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Another heuristic derivation (through Hamilton-Jacobi)
The microscopic model: ddt Uiptq “ VipUi`1ptq ´ Uiptqq, t ě 0,@i P Z,
The macroscopic model (LWR): Btρ` pρvpρqqx “ 0 in Rˆ p0,`8q
§ Let up¨, tq be the piecewise affine map such that upi , tq “ Uiptq for alli P Z.
§ We consider the hyperbolic scaling: uεpx , tq “ εupε´1x , ε´1tq.§ If ρεptq “ ε
ÿ
iPZδεUi pε´1tq, one can show that ρεptq “ Bx puεq´1p¨, tq.
§ If x “ εi ,
Btuεpx , tq “ddt
`
εUipε´1tq
˘
“ddt Uipε
´1tqq “ VipUi`1pε´1tq ´ Uipε
´1tqq
“ Vrxεs`
ε´1puεpx ` ε, tq ´ uεpx , tqq˘
» VrxεspBxuεpx , tqq.
§ So uε “solves” the HJ equation: Btuεpx , tq “ VrxεspBxuεpx , tqqfrom which one expect to derive (LWR).
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Some references (cont’d)
The proof based on Hamilton-Jacobi is related to the analysis of theFrenkel-Kontorova models:
§ Aubry (1983), Aubry and Le Daeron (1983),§ Forcadel, Imbert, and Monneau (2009)
Our work is within the framework of (stochastic) homogenization of HJequations:
§ Lions, Papanicolaou, and Varadhan (1987): periodic setting§ Souganidis (1999), Rezakhanlou and Tarver (2000): convergence§ Armstrong, C., Souganidis: convergence rate§ Subsequent works by Armstrong, Ciomaga, Davini, Feldman, Kosygina,Lin, Lions, C., Nolen, Novikov, Schwab, Seeger, Smart, Souganidis,Tran, Varadhan, Yilmaz, Zeitouni...
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Outline
Heuristic arguments in the homogeneous case
Main results
Ideas of proof
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A random microscopic model
We consider a random version of the follow-the-leader model:
ddt Uiptq “ VZi pUi`1ptq ´ Uiptqq, t ě 0,@i P Z,
where§ Uiptq denotes the position of car i at time t,§ Cars are ordered: Uiptq ď Ui`1ptq for all t, i ,§ The velocity V “ VZi ppq of car i depends on the distance p of car i tocar i ` 1 and on the “type” Zi of car i
§ The types are pZiq are I.I.D. random variables.
(cf. N. Chiabaut, L. Leclercq, and C. Buisson (2010))
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Assumptions
On the optimal velocity map V : Z ˆ R` Ñ R`, we assume the following:pH1q The map pz , pq Ñ Vzppq is uniformly continuous on Z ˆ R` and
p Ñ Vzppq is Lipschitz continuous, uniformly with respect to z P Z;pH2q For any z P Z, there exists hz
0 ą 0 (depending in a measurable way onz) such that Vzppq “ 0 for all p P r0, hz
0s;pH3q For any z P Z, p Ñ Vzppq is increasing in rhz
0,`8q;pH4q There exists Vmax ą 0 and, for any z P Z, there exists V z
max ď Vmax,such that limpÑ`8 Vzppq “ V z
max.
pH5q If we set Vmax :“ infzPZ V zmax, then lim
θÑVmax´E”
V´1Z0pθq
ı
“ `8.
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Vz(h)
hz0
0
V zmax
h
Figure: Schematic representation of the optimal velocity functions.
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Main result (1)
Scaling: For ε ą 0, we consider an initial condition pUε,0i q such that there
exists a Lipschitz continuous function u0 : RÑ R with
limεÑ0, εiÑx
εUε,0i “ u0pxq,
locally uniformly with respect to x . Let pUεi q be the solution of
ddt Uε
i ptq “ VZi pUεi`1ptq ´ Uε
i ptqq, t ě 0,@i P Z.
with initial condition pUε,0i q.
We want to study the limit
upx , tq :“ limεÑ0, εpi ,sqÑpx ,tq
εUεi psq
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Main result (2)
Theorem (C.-Forcadel)Under assumptions pH1q ´ pH5q, the limit
upx , tq :“ limεÑ0, εpi ,sqÑpx ,tq
εUεi psq
exists a.s., locally uniformly in px , tq, and u is the unique (deterministic)viscosity solution of
"
Btu “ F pBxuq in Rˆs0,`8rupx , 0q “ u0pxq in R
where the effective velocity F : r0,`8q Ñ r0,Vmaxq is the continuous andincreasing map defined by
§ F ppq “ 0 if p ď h0 where h0 :“ ErhZ00 s,
§ and F ppq is the unique solution to ErV´1Z0pF ppqqs “ p if p ą h0.
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Link with the Lighthill-Whitham-Richards (LWR) modelWe consider the (rescaled) empirical density of cars:
ρεptq “ εÿ
iPZδεUε
i ptεq, t ě 0.
Corollary [Convergence to the LWR model]As εÑ 0, ρεptq converges, a.s., in distribution and locally uniformly in time,to the density of cars
ρptq :“ up¨, tq7dx ,
where u is the solution of the limit HJ equation. If, in addition, there existsC ą 0 such that
C´1 ď Bxu0pxq ď C ,
then ρ has an absolutely continuous density which is locally bounded and isthe entropy solution of the LWR model
pLWRq Btρ` Bx pρvpρqq “ 0 in Rˆ R`,
with vpρq “ F p1ρq.
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Sketch of proof of the corollary
§ Let ϕ P C0c pRq. Then, for any t 1 ě 0,
ż
Rϕpxqρεpdx , t 1q “ ε
ÿ
iPZϕpεUε
i pt 1εqq “ż
RϕpεUε
rxεspt1εqqdx .
§ As εprxεs, t 1εq Ñ px , tq as εÑ 0 and t 1 Ñ t, the main Theoremimplies:
limεÑ0, t1Ñt
ż
Rϕpxqρεpdx , tq “
ż
Rϕpupx , tqqdx
“
ż
Rϕpxqdpup¨, tq7dxq “
ż
Rϕpxqρpdx , tq.
This proves that ρεptq converges locally uniformly in time and in thesense of measures to ρptq :“ up¨, tq7dx .
§ This implies that ρ is an entropy solution to (LWR) (Caselles (1992)).
l
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Outline
Heuristic arguments in the homogeneous case
Main results
Ideas of proof
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Preliminary results on the micro model
Lemma [Uniform bounds]Let Ui be a solution of
pMicroq ddt Uiptq “ VZi pUi`1ptq ´ Uiptqq, t ě 0,@i P Z.
Then, for all t ě 0,0 ď Uiptq ´ Uip0q ď Vmaxt.
We also have the following comparison principle:Proposition [Comparison]Let Ui and Ui be two solutions of (Micro) such that there exists i0 P Z with
Uip0q ď Uip0q @i ě i0.
ThenUiptq ď Uiptq @t ě 0 and i ě i0.
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Construction of the effective velocity
Recall that h0 :“ ErhZ00 s. Given p ą h0, we consider the solution Up to the
problem with linear initial condition:
ddt Up
i ptq “ VZi pUpi`1ptq ´ Up
i ptqq, t ě 0, Upi p0q “ p i @i ě 0.
Proposition [Convergence for linear initial conditions]There exists Ω0 P F with PpΩ0q “ 1 such that for every p ě 0, i P N andω P Ω0
limtÑ`8
Upi ptqt “ F ppq @i ě 0,
where the continuous and non-decreasing map F : R` Ñ R` is defined by§ F ppq “ 0 if p ď h0 where h0 :“ ErhZ0
0 s,§ ErV´1
Z0pF ppqqs “ p if p ą h0.
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Main argument of the proof of the proposition: CorrectorsGiven θ P p0,Vmaxq, we consider the random sequence pcθi qiě0 defined by
cθ0 “ 0, cθi`1 “ cθi ` V´1Zipθq i ě 0.
In other words,VZi pcθi`1 ´ cθi q “ θ @i ě 0.
Thus, if we set Uθi ptq “ cθi ` tθ, we have
ddt Uθ
i ptq “ θ “ VZi pcθi`1 ´ cθi q “ VZi pUθi`1ptq ´ Uθ
i ptqq.
So the pUθi q are the correctors of the problem.
By the law of large numbers, there exists Ω0 with PpΩ0q “ 1 such that forevery ω P Ω0, we have
cθii “
1i
i´1ÿ
j“0V´1
Zjpθq Ñ E
”
V´1Z0pθq
ı
as i Ñ `8.
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Open problems
‚ Convergence rate
‚ Models with local perturbations(cf. Forcadel-Salazar-Zaydan (2017),deterministic setting)
‚ Models with several roads(cf. Forcadel-Salazar (2019), twooutgoing roads)