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37
Modeling MicroMacro Pedestrian Counterflow
 in Heterogeneous Domains, Nonlinear Phenomena in Complex Systems, Volume 14, Number 1
, 2011
"... Abstract We present a micromacro strategy able to describe the dynamics of crowds in heterogeneous media. Herein we focus on the example of pedestrian counterflow. The main working tools include the use of mass and porosity measures together with their transport as well as suitable application of ..."
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Abstract We present a micromacro strategy able to describe the dynamics of crowds in heterogeneous media. Herein we focus on the example of pedestrian counterflow. The main working tools include the use of mass and porosity measures together with their transport as well as suitable application of a version of RadonNikodym Theorem formulated for finite measures. Finally, we illustrate numerically our microscopic model and emphasize the effects produced by an implicitly defined social velocity.
Existence and uniqueness of measure solutions for a system of continuity equations with nonlocal flow
 NODEA NONLINEAR DIFFERENTIAL EQUATIONS APPL
, 2011
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A hierarchy of heuristicbased models of crowd dynamics,”
 Journal of Statistical Physics,
, 2013
"... Abstract We derive a hierarchy of kinetic and macroscopic models from a noisy variant of the heuristic behavioral IndividualBased Model of Acknowledgments: This work has been supported by the French 'Agence Nationale pour la Recherche (ANR)' in the frame of the contracts 'Pedigree& ..."
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Cited by 6 (1 self)
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Abstract We derive a hierarchy of kinetic and macroscopic models from a noisy variant of the heuristic behavioral IndividualBased Model of Acknowledgments: This work has been supported by the French 'Agence Nationale pour la Recherche (ANR)' in the frame of the contracts 'Pedigree' (ANR08SYSC01501) and 'CBDifFr' (ANR08BLAN033301)
Sparse Stabilization and Control of the CuckerSmale Model
, 2013
"... From a mathematical point of view selforganization can be described as patterns to which certain dynamical systems modeling social dynamics tend spontaneously to be attracted. In this paper we explore situations beyond selforganization, in particular how to externally control such dynamical system ..."
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Cited by 4 (4 self)
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From a mathematical point of view selforganization can be described as patterns to which certain dynamical systems modeling social dynamics tend spontaneously to be attracted. In this paper we explore situations beyond selforganization, in particular how to externally control such dynamical systems in order to eventually enforce pattern formation also in those situations where this wished phenomenon does not result from spontaneous convergence. Our focus is on dynamical systems of CuckerSmale type, modeling consensus emergence, and we question the existence of stabilization and optimal control strategies which require the minimal amount of external intervention for nevertheless inducing consensus in a group of interacting agents. First we follow a greedy approach, by designing instantaneous feedback controls with two different sparsity properties: componentwise sparsity, meaning that the controls have at most one nonzero component at every instant of time and their implementation is based on a variational criterion involving ℓ1norm penalization terms; time sparsity, meaning that the number of switchings is bounded on every compact interval of time, and such controls are realized by means of a sampleandhold procedure. Controls sharing these two sparsity features are very realistic and convenient for practical issues. Moreover we show
On properties of the generalized Wasserstein distance
, 2013
"... In this article, we continue the investigation of the generalized Wasserstein distance W a,bp, that we introduced in [12]. We first prove that the particular choice W 1,11 coincides with the socalled flat metric. This provides a dual formulation for the flat metric, in the spirit of the Kantorovich ..."
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Cited by 4 (2 self)
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In this article, we continue the investigation of the generalized Wasserstein distance W a,bp, that we introduced in [12]. We first prove that the particular choice W 1,11 coincides with the socalled flat metric. This provides a dual formulation for the flat metric, in the spirit of the KantorovichRubinstein theorem. We then prove another duality formula for the caseW a,b2. We prove that the square of this Wasserstein distance is indeed the minimizer of an action functional related to the transport equation with sources. This generalizes the BenamouBrenier formula for the standard Wasserstein distance. We finally show that, under some standard regularity hypotheses, one has existence and uniqueness of the solution of a transport equation with source. The proof is based on a time discretization and the use of Gronwall estimates for the generalized Wasserstein distance.
An adaptive finitevolume method for a model of twophase pedestrian flow, Networks and Heterogeneous
 Media
"... Abstract. A flow composed of two populations of pedestrians moving in different directions is modeled by a twodimensional system of convectiondiffusion equations. An efficient simulation of the twodimensional model is obtained by a finitevolume scheme combined with a fully adaptive multiresolut ..."
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Abstract. A flow composed of two populations of pedestrians moving in different directions is modeled by a twodimensional system of convectiondiffusion equations. An efficient simulation of the twodimensional model is obtained by a finitevolume scheme combined with a fully adaptive multiresolution strategy. Numerical tests show the flow behavior in various settings of initial and boundary conditions, where different species move in countercurrent or perpendicular directions. The equations are characterized as hyperbolicelliptic degenerate, with an elliptic region in the phase space, which in one space dimension is known to produce oscillation waves. When the initial data are chosen inside the elliptic region, a spatial segregation of the populations leads to pattern formation. The entries of the diffusionmatrix determine the stability of the model and the shape of the patterns. 1. Introduction. In
Visionbased macroscopic pedestrian models
, 2013
"... We propose a hierarchy of kinetic and macroscopic models for a system consisting of a large number of interacting pedestrians. The basic interaction rules are derived from [44] where the dangerousness level of an interaction with another pedestrian is measured in terms of the derivative of the beari ..."
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Cited by 3 (0 self)
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We propose a hierarchy of kinetic and macroscopic models for a system consisting of a large number of interacting pedestrians. The basic interaction rules are derived from [44] where the dangerousness level of an interaction with another pedestrian is measured in terms of the derivative of the bearing angle (angle between the walking direction and the line connecting the two subjects) and of the timetointeraction (time before reaching the closest distance between the two subjects). A meanfield kinetic model is derived. Then, three different macroscopic continuum models are proposed. The first two ones rely on two different closure assumptions of the kinetic model, respectively based on a monokinetic and a von MisesFisher distribution. The third one is derived through a hydrodynamic limit. In each case, we discuss the relevance of the model for practical simulations of pedestrian crowds.