A MACROSCOPIC CROWD MOTION MODEL OF GRADIENT FLOW TYPE

By Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio. 2010

275 citations recorded by the source corpus.

A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U(x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities. Instead of looking at a microscopic setting (where individuals are represented by rigid discs), here the macroscopic approach is investigated, where the unknwon is the evolution of the density . If a gradient structure is given, say U is the opposite of the gradient of D where D is, for instance, the distance to the exit door, the problem is presented as a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density), nor geodesically convex, which requires for an ad-hoc study of the convergence of a discrete scheme.

Subfields: Applied Mathematics, Control and Systems Engineering, Ocean Engineering

View Paper: https://www.semanticscholar.org/paper/99faf58521e001e864a28233681eb9e9923708c7

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Citations in the corpus

Quantitative stability of optimal transport maps under variations of the target measure (2021, unknown): “This corollary will be a consequence of the following lemma from [27], which we will use as a replacement of the Kantorovich-Rubinstein inequality.”
Scaling Algorithms for Unbalanced Transport Problems (2016, background): “Another use of these implicit steps is to construct minimizing flows of non-smooth functionals, for instance to model crowd motions [50].”
A CLASS OF NONLOCAL MODELS FOR PEDESTRIAN TRAFFIC (2011, background): “To underline the variety of analytical techniques with which crowd dynamics has been tackled, we recall the further approaches: optimal transport in [5], the mean field limit in [15], the functional analytic one in [26, 27] and the nonclassical shocks used in [6, 10, 11].”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, methodology): “In [225] these difficulties have been overtaken in the case u D rD (where D W ̋ ! R is a given Lipschitz function), and the existence of a solution (with numerical simulations) is proven via a gradient flow method.”
{Euclidean, metric, and Wasserstein} gradient flows: an overview (2016, methodology): “The main axiom of the model (presented, for instance, in [67]) is that the density will be advected by a vector field vt which is not u but its projection, at each time t , on adm( t ).”
Generalized Wasserstein Distance and its Application to Transport Equations with Source (2012, methodology): “This equation is intensively used for the modelling of crowd dynamics, where μt represents a pedestrian density (see for example [6,8,12–14,16–18]).”
Refining Deep Generative Models via Discriminator Gradient Flow (2020, methodology): “For example, they have been applied to the study of the porous-medium equation (Otto, 2001), crowd modeling (Maury et al., 2010; 2011), and mean-field games (Almulla et al., 2017).”
Computational Optimal Transport (2018, methodology): “used to model crowd motion [Maury et al., 2010] (see also the review paper [Santambrogio, 2018]).”
A variational approach to second order mean field games with density constraints: the stationary case (2015, background): “have been already studied in the literature (see for instance [24, 25]).”