Section: Research Program

Managing the system (via probabilistic modeling)

Participants : Guy Fayolle, Cyril Furtlehner, Arnaud de La Fortelle, Jean-Marc Lasgouttes, Victorin Martin.

The research on the management of the transportation system is a natural continuation of the research of the Preval team, which joined IMARA in 2007. For many years, the members of this team (and of its ancestor Meval) have been working on understanding random systems of various origins, mainly through the definition and solution of mathematical models. The traffic modeling field is very fertile in difficult problems, and it has been part of the activities of the members of Preval since the times of the Praxitèle project.

Following this tradition, the roadmap of the group is to pursue basic research on probabilistic modeling with a clear slant on applications related to LaRA activities. A particular effort is made to publicize our results among the traffic analysis community, and to implement our algorithms whenever it makes sense to use them in traffic management. Of course, as aforementioned, these activities in no way preclude the continuation of the methodological work achieved in the group for many years in various fields: random walks in $Z_{+}^n$ ([1] , [2] , [5] ), large deviations, birth and death processes on trees, particle systems.

In practice, the group explores the links between large random systems and statistical physics, since this approach proves very powerful, both for macroscopic (fleet management [4] ) and microscopic (car-level description of traffic, formation of jams) analysis. The general setting is mathematical modeling of large systems (mostly stochastic), without any a priori restriction: networks [3] , random graphs or even objects coming from biology. When the size or the volume of those structures grows (this corresponds to the so-called thermodynamical limit), one aims at establishing a classification based on criteria of a twofold nature: quantitative (performance, throughput, etc) and qualitative (stability, asymptotic behavior, phase transition, complexity).

Exclusion processes

One of the simplest basic (but non trivial) probabilistic models for road traffic is the exclusion process. It lends itself to a number of extensions allowing to tackle some particular features of traffic flows: variable speed of particles, synchronized move of consecutive particles (platooning), use of geometries more complex than plain 1D (cross roads or even fully connected networks), formation and stability of vehicle clusters (vehicles that are close enough to establish an ad-hoc communication system), two-lane roads with overtaking.

Most of these generalizations lead to models that are obviously difficult to solve and require upstream theoretical studies. Some of these models have already been investigated by members of the group, and they are part of wide ongoing researches.

Message passing algorithms

Large random systems are a natural part of macroscopic studies of traffic, where several models from statistical physics can be fruitfully employed. One example is fleet management, where one main issue is to find optimal ways of reallocating unused vehicles: it has been shown that Coulombian potentials might be an efficient tool to drive the flow of vehicles. Another case deals with the prediction of traffic conditions, when the data comes from probe vehicles instead of static sensors. Using the Ising model, together with the Belief Propagation (BP) algorithm very popular in the computer science community, we have been able to show how real-time data can be used for traffic prediction and reconstruction (in the space-time domain).

This new use of BP algorithm raises some theoretical questions about the properties of the Bethe approximation of Ising models:

• find the best way to inject real-valued data in an Ising model with binary variables;

• build macroscopic variables that measure the overall state of the underlying graph, in order to improve the local propagation of information;

• make the underlying model as sparse as possible, in order to improve BP convergence and quality.

Statistical physics and hydrodynamic limits

These last years, having in mind a global project concerning the analysis of complex systems, we did focus on the interplay between discrete and continuous description: in some cases, this recurrent question can be addressed quite rigorously via probabilistic methods (see e.g.  [52] ).

To describe the systems of interest, which are in touch with many application domains, we started from paradigmatic elements, namely discrete curves subject to stochastic deformations. Up to some convenient mappings, it appears that most models can be set in terms of interacting exclusion processes, the ultimate goal being to derive hydrodynamic limits after proper scalings.

The key ideas can be found in [53] , where the basic ASEP system on the torus is the toy model. In this case, the usual sequence of empirical measures, converges in probability to a deterministic measure, which is the unique weak solution of a Cauchy problem.

The Gordian knot is indeed the analysis of a family of specific partial differential operators in infinite dimension. Indeed, the values of functions at given points play here the role of usual variables, their number becoming infinite. The method presents some new theoretical features, involving path integrals, promeasures (as introduced by Bourbaki), variational calculus, and the construction of generalized measures. In [53] , we present a detailed analysis of the asep system on the torus $\mathbb{Z}/N\mathbb{Z}$. Then, we claim that most of the arguments a priori for multi-type exclusion processes, and should lead to systems of coupled partial differential equations of Burgers' type. At the moment, this claim is being proved for the famous ABC model, reformulated in terms of the dynamics of a random walk on the triangular lattice.