Networks and Heterogeneous Media
September 2013 , Volume 8 , Issue 3
Special issue on Mathematics of Traffic Flow Modeling, Estimation and Control
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This Special Issue gathers contributions, most of which were presented at the Workshop ``Mathematics of Traffic Flow Modeling, Estimation and Control", organized at the Institute for Pure and Applied Mathematics of the University of California Los Angeles on December 7--9 2011.
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This paper is concerned with a conservation law model of traffic flow on a network of roads, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. The model includes various groups of drivers, with different origins and destinations and having different cost functions. Under a natural set of assumptions, two main results are proved: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In the case of Nash solutions, all departure rates are uniformly bounded and have compact support.
This paper deals with coupling conditions between the classical microscopic Follow The Leader model and a phase transition (PT) model. We propose a solution at the interface between the two models. We describe the solution to the Riemann problem.
The aim of this paper is to build a dynamical traffic model in a dense urban area. The main contribution of this article is to take into account the four possible directions of traffic flows with flow vectors of dimension $4$ and not $2$ as in fluid mechanic on a plan. Traffic flows are viewed as confrontation results between users demands and a travel supply of the network. The model gathers elements of intersection theory and two-dimensional continuum networks.
We discuss a numerical discretization of Hamilton--Jacobi equations on networks. The latter arise for example as reformulation of the Lighthill--Whitham--Richards traffic flow model. We present coupling conditions for the Hamilton--Jacobi equations and derive a suitable numerical algorithm. Numerical computations of travel times in a round-about are given.
A few applications of the viability theory to the solution to the Hamilton-Jacobi-Moskowitz problems are presented. In the considered problem the Hamiltonian (fundamental diagram) depends on time, position and/or some regulation parameters. We study such a problem in its equivalent variational formulation. In this case, the corresponding lagrangian depends on the state of the characteristic dynamical system. As the Lax-Hopf formulae that give the solution in a semi-explicit form for an homogeneous lagrangian do not hold, a capture basin algorithm is proposed to compute the Moskowitz function as a viability solution of the Hamilton-Jacobi-Moskowitz problem with general conditions (including initial, boundary and internal conditions). We present two examples of applications to traffic regulation problems.
A solution of the initial-boundary value problem on the strip $(0,\infty) \times [0,1]$ for scalar conservation laws with strictly convex flux can be obtained by considering gradients of the unique solution $V$ to an associated Hamilton-Jacobi equation (with appropriately defined initial and boundary conditions). It was shown in Frankowska (2010) that $V$ can be expressed as the minimum of three value functions arising in calculus of variations problems that, in turn, can be obtained from the Lax formulae. Moreover the traces of the gradients $V_x$ satisfy generalized boundary conditions (as in LeFloch (1988)). In this work we illustrate this approach in the case of the Burgers equation and provide numerical approximation of its solutions.
Fundamental diagrams of vehicular traffic flow are generally multi-valued in the congested flow regime. We show that such set-valued fundamental diagrams can be constructed systematically from simple second order macroscopic traffic models, such as the classical Payne-Whitham model or the inhomogeneous Aw-Rascle-Zhang model. These second order models possess nonlinear traveling wave solutions, called jamitons, and the multi-valued parts in the fundamental diagram correspond precisely to jamiton-dominated solutions. This study shows that transitions from function-valued to set-valued parts in a fundamental diagram arise naturally in well-known second order models. As a particular consequence, these models intrinsically reproduce traffic phases.
We review our previous results on partial differential equation(PDE) models of traffic flow. These models include the first order PDE models, a nonlocal PDE traffic flow model with Arrhenius look-ahead dynamics, and the second order PDE models, a discrete model which captures the essential features of traffic jams and chaotic behavior. We study the well-posedness of such PDE problems, finite time blow-up, front propagation, pattern formation and asymptotic behavior of solutions including the stability of the traveling fronts. Traveling wave solutions are wave front solutions propagating with a constant speed and propagating against traffic.
Traffic sensing systems rely more and more on user generated (insecure) data, which can pose a security risk whenever the data is used for traffic flow control. In this article, we propose a new formulation for detecting malicious data injection in traffic flow monitoring systems by using the underlying traffic flow model. The state of traffic is modeled by the Lighthill-Whitham-Richards traffic flow model, which is a first order scalar conservation law with concave flux function. Given a set of traffic flow data generated by multiple sensors of different types, we show that the constraints resulting from this partial differential equation are mixed integer linear inequalities for a specific decision variable. We use this fact to pose the problem of detecting spoofing cyber attacks in probe-based traffic flow information systems as mixed integer linear feasibility problem. The resulting framework can be used to detect spoofing attacks in real time, or to evaluate the worst-case effects of an attack offline. A numerical implementation is performed on a cyber attack scenario involving experimental data from the Mobile Century experiment and the Mobile Millennium system currently operational in Northern California.
In large scale deployments of traffic flow models, estimation of the model parameters is a critical but cumbersome task. A poorly calibrated model leads to erroneous estimates in data--poor environments, and limited forecasting ability. In this article we present a method for calibrating flow model parameters for a discretized scalar conservation law using only velocity measurements. The method is based on a Markov Chain Monte Carlo technique, which is used to approximate statistics of the posterior distribution of the model parameters. Numerical experiments highlight the difficulty in estimating jam densities and provide a new approach to improve performance of the sampling through re-parameterization of the model. Supporting source code for the numerical experiments is available for download at https://github.com/dbwork/MCMC-based-inverse-modeling-of-traffic.
The probability hypothesis density (PHD) methodology is widely used by the research community for the purposes of multiple object tracking. This problem consists in the recursive state estimation of several targets by using the information coming from an observation process. The purpose of this paper is to investigate the potential of the PHD filters for real-time traffic state estimation. This investigation is based on a Cell Transmission Model (CTM) coupled with the PHD filter. It brings a novel tool to the state estimation problem and allows to estimate the densities in traffic networks in the presence of measurement origin uncertainty, detection uncertainty and noises. In this work, we compare the PHD filter performance with a particle filter (PF), both taking into account the measurement origin uncertainty and show that they can provide high accuracy in a traffic setting and real-time computational costs. The PHD filtering framework opens new research avenues and has the abilities to solve challenging problems of vehicular networks.
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