Multiscale Modeling of Pedestrian Dynamics

In this chapter we begin the discussion about crowd dynamics from an informal phenomenological point of view. In particular, we put in evidence how simple interaction rules adopted independently by pedestrians generate, at a collective level, complex group behaviors featuring various forms of self-organization. Bearing in mind the ultimate goal of the book, which is mathematical modeling, we promote the idea that understanding such basic behavioral rules contributes to the modeling at all scales, also those not directly focused on single individuals. In the light of these arguments, we critically analyze the main scales of observation and representation which are typically used in mathematical modeling, namely the microscopic, the macroscopic, and the mesoscopic (or kinetic) scale. For each of them we discuss the advantages/drawbacks in catching/losing specific features of crowd dynamics, with a view also to the interplay with the available experimental knowledge about crowds. Finally we elucidate the role of the book in this cultural framework and we give reading directions through the various chapters targeted to a few different kinds of readerships.

In this chapter we give an informal introduction to the multiscale model and present some case studies of interest for applications, along with related numerical simulations. Results presented here are somehow complementary to those usually presented by physicists, engineers, and computer scientists. Indeed, we aim at showing how mathematical modeling can help in developing truthful pedestrian models, and at giving a sample of phenomena which can be simulated without the introduction of artificial or ad hoc effects.

In this chapter we review some of the most important models at microscopic, macroscopic, and mesoscopic scale, which, in our opinion, represent milestones in their respective fields or are of particular interest for this book. We also report some models for rational pedestrians, which make use of techniques from optimal control theory. For the sake of convenience, we present all models in two space dimensions.

This chapter is devoted to a multiscale approach to the modeling of crowd dynamics, which is the core topic of the book. We begin by presenting, in Sect. 5.1, a general measure-based modeling framework suitable to include the basic features of pedestrian kinematics at any scale. Specifically, we assume that pedestrian motion results from the interplay between the individual will to follow a preferred travel program and the necessity to face the rest of the crowd. We discuss in Sect. 5.2 how to properly model these behavioral aspects. In Sect. 5.3 we show how discrete (microscopic) and continuous (macroscopic) models can be obtained in the proposed framework, before focusing, in Sect. 5.4, on multiscale modeling issues. We also propose a detailed dimensional analysis, which highlights the role of a few significant parameters, and a numerical scheme for the approximate solution of the equations. The scheme is obtained in two steps in Sect. 5.5. First we derive a discrete-in-time model; next we discretize the space variable as well, obtaining an algorithm (cf. Appendix B) which can be implemented on a computer to produce simulations (cf. Chap. 2). Finally, in Sect. 5.6 we extend the previous modeling structures to the case of two interacting crowds.

This chapter is devoted to the mathematical foundations of the model introduced in Chap. 5. Contents go continuously back and forth between modeling and analysis, however with a more formal approach than that used in the previous chapter. The first three sections, from Sects. 6.1 to 6.3, discuss how the measure-based model can be derived from a particle description of pedestrians, thereby formalizing the link between individualities and collectivity which is at the basis of most of the complexity of crowd behaviors. In addition, in the light of such a derivation they propose a probabilistic reading of the measure-based model, which turns out to be particularly meaningful for applications. The central part of the chapter, encompassing Sects. 6.4–6.7, is concerned with the basic theory of well-posedness and numerical approximation of measure-valued Cauchy problems for first order models based on conservation laws, also in a multiscale perspective. Minimal generic assumptions are stated in order to achieve proofs, to be regarded possibly also as guidelines in the modeling approach. Finally, Sect. 6.8 resumes the discussion about the crowd model presented in Chap. 5 studying under which conditions it is in the scope of the theory set forth in the preceding sections.

In this chapter, we provide a fairly general mathematical setting for the nonlinear transport equation analyzed in Chap. 6 (namely Eqs. (5.1) and (6.6)). More precisely, we study the evolution of solutions in measures spaces endowed with the Wasserstein distance and its generalizations. Moreover, we illustrate the connections between the Wasserstein distance, the transport equation and optimal transportation problems in the sense of Monge-Kantorovich. We also deal with numerical schemes for the transport equation in measure spaces and prove convergence of a Lagrangian scheme to the unique solution, when the discretization parameters approach zero. Convergence of an Eulerian scheme is then achieved under more strict hypotheses. Both schemes are discretizations of the push-forward formula defined by the transport equation as in Chap. 6. As a by-product, we obtain existence and uniqueness of the solution under general assumptions. All the results of convergence are proved with respect to the Wasserstein distance. We first show that the total variation distance is not natural for such equations, since we lose uniqueness of the solution. Then transport equations with sources are considered. In this case the solution does not conserve its total mass, thus we can not directly use the classical Wasserstein distance. For this we introduce a generalized Wasserstein distance, which allows mass creation/destruction and has interest in itself as distance among measures with different total mass.

In this chapter we present some natural generalizations of the multiscale approach described in Chap. 5. In most of the cases, the following ideas are not yet fully developed. Nevertheless, they give some interesting directions for future research, from theoretical, numerical, and applied point of view.