# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

All Issues

## Discrete & Continuous Dynamical Systems - S

2014 , Volume 7 , Issue 3

Issue on traffic modeling and management: Trends and perspectives

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2014, 7(3): i-ii doi: 10.3934/dcdss.2014.7.3i +[Abstract](80) +[PDF](97.6KB)
Abstract:
The present issue of Discrete and Continuous Dynamical Systems -- Series S is devoted to Traffic Modeling and Management. This subject dramatically developed in recent years. On one hand, the successes of the analytical theory of conservation laws have provided new tools to traffic researchers while, on the other hand, the requirements coming from the applications have grown dramatically. Remarkably, two of the papers that opened the way to this decades long development date the same year. In 1995 The Unique Limit of the Glimm Scheme'' by A. Bressan (Archive for Rational Mechanics and Analysis, 130, 3, 205--230) gave a basis for several well posedness results for 1D systems of conservation laws. In the same year, Requiem for High-Order Fluid Approximations of Traffic Flow'' by C. Daganzo (Transportation Research Part B: Methodological, 29B, 4, 277--287) posed serious criticisms to models studied at that time and started to fix minimal requirements for a traffic model to be seriously considered.

2014, 7(3): 363-377 doi: 10.3934/dcdss.2014.7.363 +[Abstract](115) +[PDF](7424.0KB)
Abstract:
In the present paper we discuss the coupling of traffic flow with pedestrian motion. First we review the coupling of the Lighthill-Whitham model for road traffic and the Hughes pedestrian model as presented in [5]. In different numerical examples we investigate the mutual interaction of both dynamics. A special focus is given on the possible placement of crosswalks to facilitate the passage for the pedestrians over a crowded street.
2014, 7(3): 379-394 doi: 10.3934/dcdss.2014.7.379 +[Abstract](94) +[PDF](375.6KB)
Abstract:
In this paper we propose a Godunov-based discretization of a hyperbolic system of conservation laws with discontinuous flux, modeling vehicular flow on a network. Each equation describes the density evolution of vehicles having a common path along the network. We show that the algorithm selects automatically an admissible solution at junctions, hence ad hoc external procedures (e.g., maximization of the flux via a linear programming method) usually employed in classical approaches are no needed. Since users have not to deal explicitly with vehicle dynamics at junction, the numerical code can be implemented in minutes. We perform a detailed numerical comparison with a Godunov-based scheme coming from the classical theory of traffic flow on networks which maximizes the flux at junctions.
2014, 7(3): 395-409 doi: 10.3934/dcdss.2014.7.395 +[Abstract](80) +[PDF](552.7KB)
Abstract:
We review numerical and analytical results that have been obtained for a general model of intersecting flows of two types of particles propagating towards east ($\varepsilon$) and north ($\mathcal{N}$) on a bidimensional $M \times M$ square lattice. The behaviour of this model can also be reproduced by a system of mean field equations. The low density behaviour of both models is studied, with a focus on pattern formation. Using periodic boundary conditions, particles self-organize into a pattern of alternating diagonal stripes, which corresponds to an instability in the mean-field equations. With open boundary conditions, translational symmetry is broken. One then observes an asymmetry between the organization of the two types of particles, leading to tilted diagonals whose angle of inclination differs from $45^\circ$, both for the particle system and the equations. The angle of inclination of the stripes is measured using two different numerical methods. Finally, simplified theoretical arguments based on a particular mode of propagation give a quantitative estimate for the angle of the stripes. A complementary understanding of the phenomenon in terms of effective interactions between particles is presented.
2014, 7(3): 411-433 doi: 10.3934/dcdss.2014.7.411 +[Abstract](92) +[PDF](623.4KB)
Abstract:
In this paper, we consider a numerical scheme to solve first order Hamilton-Jacobi (HJ) equations posed on a junction. The main mathematical properties of the scheme are first recalled and then we give a traffic flow interpretation of the key elements. The scheme formulation is also adapted to compute the vehicles densities on a junction. The equivalent scheme for densities recovers the well-known Godunov scheme outside the junction point. We give two numerical illustrations for a merge and a diverge which are the two main types of traffic junctions. Some extensions to the junction model are finally discussed.
2014, 7(3): 435-447 doi: 10.3934/dcdss.2014.7.435 +[Abstract](115) +[PDF](627.3KB)
Abstract:
In this paper we introduce a numerical method for tracking a bus trajectory on a road network. The mathematical model taken into consideration is a strongly coupled PDE-ODE system: the PDE is a scalar hyperbolic conservation law describing the traffic flow while the ODE, that describes the bus trajectory, needs to be intended in a Carathéodory sense. The moving constraint is given by an inequality on the flux which accounts for the bottleneck created by the bus on the road. The finite volume algorithm uses a locally non-uniform moving mesh which tracks the bus position. Some numerical tests are shown to describe the behavior of the solution.
2014, 7(3): 449-462 doi: 10.3934/dcdss.2014.7.449 +[Abstract](101) +[PDF](524.5KB)
Abstract:
In this paper we investigate the ability of some recently introduced discrete kinetic models of vehicular traffic to catch, in their large time behavior, typical features of theoretical fundamental diagrams. Specifically, we address the so-called spatially homogeneous problem'' and, in the representative case of an exploratory model, we study the qualitative properties of its solutions for a generic number of discrete microscopic states. This includes, in particular, asymptotic trends and equilibria, whence fundamental diagrams originate.
2014, 7(3): 463-482 doi: 10.3934/dcdss.2014.7.463 +[Abstract](77) +[PDF](471.1KB)
Abstract:
We consider the Lighthill-Whitham-Richards traffic flow model on a network composed by a single junction $J$ with $n$ incoming roads, $m$ outgoing roads and $m$ buffers, one for each outgoing road. We introduce a concept solution at $J$, which is compared with that proposed in [14]. Finally we study the Cauchy problem and, in the special case of $n \le 2$ and $m \le 2$, we prove existence of solutions to the Cauchy problem, via the wave-front tracking method.
2014, 7(3): 483-501 doi: 10.3934/dcdss.2014.7.483 +[Abstract](73) +[PDF](652.4KB)
Abstract:
This article is devoted to traffic flow networks including traffic lights at intersections. Mathematically, we consider a nonlinear dynamical traffic model where traffic lights are modeled as piecewise constant functions for red and green signals. The involved control problem is to find stop and go configurations depending on the current traffic volume. We propose a numerical solution strategy and present computational results.
2014, 7(3): 503-523 doi: 10.3934/dcdss.2014.7.503 +[Abstract](87) +[PDF](770.0KB)
Abstract:
The LASSO is a widely used shrinkage and selection method for linear regression. We propose a generalization of the LASSO in which the $l_1$ penalty is applied on a linear transformation of the regression parameters, allowing to input prior information on the structure of the problem and to improve interpretability of the results. We also study time varying system with an $l_1$-penalty on the variations of the state, leading to estimates that exhibit few jumps''. We propose a homotopy algorithm that updates the solution as additional measurements are available. The algorithm takes advantage of the sparsity of the solution for computational efficiency and is promising for mining large datasets. The algorithm is implemented on three experimental data sets representing applications to traffic estimation from sparsely sampled probe vehicles, flow estimation in tidal channels and text analysis of on-line news.
2014, 7(3): 525-542 doi: 10.3934/dcdss.2014.7.525 +[Abstract](101) +[PDF](4378.1KB)
Abstract:
This article presents a new robust control framework for transportation problems in which the state is modeled by a first order scalar conservation law. Using an equivalent formulation based on a Hamilton-Jacobi equation, we pose the problem of controlling the state of the system on a network link, using initial density control and boundary flow control, as a Linear Program. We then show that this framework can be extended to arbitrary control problems involving the control of subsets of the initial and boundary conditions. Unlike many previously investigated transportation control schemes, this method yields a globally optimal solution and is capable of handling shocks (i.e. discontinuities in the state of the system). We also demonstrate that the same framework can handle robust control problems, in which the uncontrollable components of the initial and boundary conditions are encoded in intervals on the right hand side of inequalities in the linear program. The lower bound of the interval which defines the smallest feasible solution set is used to solve the robust LP/MILP. Since this framework leverages the intrinsic properties of the Hamilton-Jacobi equation used to model the state of the system, it is extremely fast. Several examples are given to demonstrate the performance of the robust control solution and the trade-off between the robustness and the optimality.
2014, 7(3): 543-556 doi: 10.3934/dcdss.2014.7.543 +[Abstract](83) +[PDF](630.5KB)
Abstract:
We present two frameworks for the description of traffic, both consisting in the coupling of systems of different types. First, we consider the Free--Congested model [7,11], where a scalar conservation law is coupled with a $2\times2$ system. Then, we present the coupling of a micro- and a macroscopic models, the former consisting in a system of ordinary differential equations and the latter in the usual LWR conservation law, see [10]. A comparison between the two different frameworks is also provided.
2014, 7(3): 557-578 doi: 10.3934/dcdss.2014.7.557 +[Abstract](81) +[PDF](2070.2KB)
Abstract:
This article experimentally assesses the influence of sensor data rates on travel time estimates computed from filtered traffic speed estimates. Using velocity data obtained from GPS smartphones and inductive loop detector data collected during the Mobile Century experiment near Berkeley, CA, and an evolution equation for average velocity along the roadway, an estimate of the traffic state is obtained via ensemble Kalman filtering. A large--scale batch of computations is run to produce estimates of traffic velocity with varying degrees of input data, and instantaneous and a posteriori dynamic travel times are compared to travel times recorded using license plate re-identification. We illustrate that dynamic travel time estimates can be computed with less than 10% error regardless of the data source, and that existing inductive loop detector data can significantly improve the accuracy of travel time estimates when GPS data is sparse.
2014, 7(3): 579-591 doi: 10.3934/dcdss.2014.7.579 +[Abstract](60) +[PDF](847.3KB)
Abstract:
We investigate the correlations between a macroscopic Lighthill--Whitham and Richards model and a microscopic follow-the-leader model for traffic flow. We prove that the microscopic model tends to the macroscopic one in a sort of kinetic limit, i.e. as the number of individuals tends to infinity, keeping the total mass fixed. Based on this convergence result, we approximately compute the solutions to a conservation law by means of the integration of an ordinary differential system. From the numerical point of view, the limiting procedure is then extended to the case of several populations, referring to the macroscopic model in [2] and to the natural multi--population analogue of the microscopic one.

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