DIMACS
New Brunswick, April 5th 2010
From vehicular traffic to
motion of intelligent groups
Benedetto Piccoli
Rutgers University - Camden
Supply chains
Car Traffic
Irrigation Channels
Gas pipelines
Tlc and data networks
Air traffic management
Blood circulation
Quantum nanotubes
Traffic lights and Viale del Muro Torto
Data reconstruction
error: 9% free phase, 17% congested
0
2 phase
1
Dynamics at junctions
Rule (A) :
Out. Fluxes Vector = A · Inc. Fluxes Vector
Traffic distribution matrix A = (α ji ) , 0<α ji<1, Σ j α ji =1
Rule (B) :
Max
║Inc. Fluxes Vector║
Rule (B) is an “entropy” type rule : maximize velocity
Numerics and FSF scheme
Network with 5000 roads parametrized by [0,1],
h space mesh size, T real time
1. Use simplified flux function with two characteristic speeds
f
2. Make use of theoretical results to bound the number of
regime changes
Free phase
Congested phase
G = Godunov, FG = Fast Godunov,
K3V = 3-velocities Kinetic, FSF = Fast Shock Fitting
3. Track
exactly
the empty
regime
change
(generalized
characteristic)
Lemma.
If we
start from
network,
then
each road presents
at most ρone
σ
regime
change for
every
time
and
use upwind
for
each
zone
ρ
max
Real data
Problems :
1. Dimensionality: big networks
2. Data: measurements and elaboration
Manual counting
Satellite data
Radars
1500 arcs network
Videocameras
Plates reading
NETWORK of SALERNO
Model for data networks
First attempt
d
c
d
b
b
a
Second attempt
Processor with queue model
(Goettlich-Herty-Klar)
Mixed ODE-PDE model
Queue
Processor j
Queue buffer occupancy change is given by
the difference between incoming and outgoing flux
Motions of intelligent groups
Gas particles:
Intelligent particles:
•
•
•
•
•
•
•
•
•
•
Robust
Isotropic
Blind
Local interaction
Energy balance
Fragile
Anisotropic
Vision + decision
Non-local interaction
No energy balance
Vehicular traffic vs Pedestrians, robots and animals:
1. Dimensionality : 1-D vs 2-D or 3-D
2. Non locality : one car ahead vs many neighbors
As a result: Entropy vs Self-organization
Time-evolving measures model
Measure μ : (t,E) → μ(t,E) number of pedestrians in the region E
Flow map ɣ : x → x + v(x,μ) Δt move each point with given velocity
At next time step is given by μ(t+Δt ,E) = μ(t,ɣ⁻¹ (E))
The velocity v is the sum of desired velocity v d
ɣ⁻¹ (E)
E
and interaction term v i (μ)
ɣ⁻¹
vd
E
ɣ
v i (μ)
Simultaneous microscopic and macroscopic approach
Metro exit and opposite flows
Initial condition
Desired velocity field
Exiting the metro : real movie
Exiting the metro : simulation
Animal groups
Attraction
Repulsion
Visual field
Logic variables activating the forces -> networks of interacting animals
Animal groups : structure of flocks
A>>C, total vision
A>>R, front vision
A=R, front repulsion
Thank you for your attention!
Benedetto Piccoli
Math Dept and CCIB, Rutgers-Camden
Phone: 856 225 6356
Email: piccoli@camden.rutgers.edu
Finsler metric on L^1
L1
Piecewise constant functions
u
u  PC  L1
v
2
1
Perturbations:
u, u'  PC
v
L1
2
1
u
u'
ui
v
Family of piecewise smooth curves in
PC connecting u and u':
 : [0, 1]  PC
x1
(0) = u, (1) = u'
x2
the length of each of these curves as
||(v, )|| = ||v||Define
L1 + |i| ui
i
1
∫
L() = ||(v, )(s)|| ds
0
Lipschitz continuous dependence
u'(t)
u'(0)
(v, )(0)
t(s)
0(s)
(v, )(t)
u(0)
u(t)
Lemma:||(v, )(t)||  ||(v, )(0)||
d(u(t), u'(t)) = inf L()  inf L(t)
 : u(t)  u'(t)
t : u(t)  u'(t)
Lemma
 inf L(0) = d(u(0), u'(0))
0 : u(0)  u'(0)
Lemma is true for data networks and GHK supply chain model
Car trajectory on loaded network
Godunov type scheme : first density then car position
Wave front tracking schemes
Simulation result
Thank you for your attention!
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G. Bretti, A. Cutolo, B. Piccoli: On calibration of traffic data, to appear on Applied and
Computational Mathematics.
Bretti G., D'Apice C., Manzo R., Piccoli B., A continuum-discrete model for supply chains
dynamics, Networks and Heterogeneous Media 2 (2007), 661–694.
G. Bretti, R. Natalini and B. Piccoli, Numerical Approximations of Traffic Flow Models on
Networks, Networks and Heterogeneous Media, vol. 1 n. 1, (2006), 57-84.
G. Bretti, R. Natalini and B. Piccoli, Fast algorithms for the approximation of a traffic flow
model on networks, Discr. Cont. Dyn. Systems – B 6 (2006), 427-448.
G. Bretti, R. Natalini, B. Piccoli: A Fluid-Dynamic Traffic Model on Road Networks, Arch.
Comput Methods Eng., 14 (2007), 139-172.
G. Bretti, R. Natalini, B. Piccoli: Numerical algorithms for simulation of a traffic model on road
networks, Journal of Computational and Applied Mathematics, 201 (2007), 71-77.
G. Bretti and B. Piccoli, A tracking algorithm for a single car moving in a road network,
SIAM Appl. Dyn Syst. 7 (2008), 71-77.
A. Cascone, A. Marigo, B. Piccoli, L.Rarità, Decentralized optimal routing for packets flow on
data networks, Discrete and Continuous Dynamical Systems - Series B, submitted.
A. Cutolo, C. D’Apice, B. Piccoli, L. Rarità, Numerical approaches for supply chains, working
paper.
C. D’Apice, R. Manzo, B. Piccoli, Packets flow on telecommunication networks, SIAM J. Math.
Anal. 37 (2006), 717-740.
D’Apice C., Manzo R., Piccoli B., Modelling supply networks with partial differential equations,
to appear on Quarterly Appl. Math. 2009.
D'apice C., Manzo R., Piccoli B. A fluid dynamic model for telecommunication networks with
sources and destinations, SIAM J. Appl. Math. (2008), 981-1003.
D'apice C., Manzo R., Piccoli B., Existence of solutions to Cauchy problems for a mixed
continuum-discrete model for supply chains and networks, submitted to JMAA.
M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics,
vol. 1, American Institute of Mathematical Sciences, 2006, ISBN-13: 978-1-60133-000-0.
M. Herty, A. Klar, B. Piccoli, Existence of solutions for supply chain models based on partial
differential equations, SIAM J. Math. Anal. 39 (2007), 160-173.
Numerical schemes : Godunov
Approximation schemes (explicit schemes):
Godunov’s scheme (first order), Kinetic scheme with 3 vel. of first and second order
Solve Riemann problems
Apply Gauss-Green
Use flows on vertical lines
Godunov scheme reads:
Assume f concave
with unique maximum σ
then numerical flux :
Numerical (2) : Wave Front Tracking
1. Approximate initial datum by a piecewise constant function
2. Solve RPs, replace rarefactions by rarefaction shocks fans:
initially waves evolve independently of one another
3. At time t* > 0 a first interaction between two of such
discontinuities occurs (two shocks collide in this example)
4. Then we solve a new Riemann problem and so on
t*
u1
u2
u3
u4
u5
…
Tens, hundreds, thousands
of pedestrians
Helbing et al., microscopic
Colombo-Rosini, macroscopic 1D
Maury-Venel, microscopic
Bellomo-Dogbé, macroscopic