# Gradient flows in Wasserstein spaces and applications to crowd movement

Filippo Santambrogio^{[1]}

- [1] Laboratoire de Mathématiques d’Orsay Faculté des Sciences Université Paris-Sud XI 91405 Orsay cedex France

Séminaire Équations aux dérivées partielles (2009-2010)

- Volume: 2009-2010, page 1-16

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topSantambrogio, Filippo. "Gradient flows in Wasserstein spaces and applications to crowd movement." Séminaire Équations aux dérivées partielles 2009-2010 (2009-2010): 1-16. <http://eudml.org/doc/251158>.

@article{Santambrogio2009-2010,

abstract = {Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in $\mathbb\{R\}^n$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step are obtained by looking at perturbation of the form $\rho _\varepsilon =(1-\varepsilon )\rho +\varepsilon \tilde\{\rho \}$ instead of $\rho _\varepsilon =(id+\varepsilon \xi )_\#\rho $. The ideas to make this approach rigorous are presented in the case of a Fokker-Planck equation, possibly with an interaction term, and then the paper is concluded by a section, where this method is applied to the original problem of crowd motion (referring to a recent paper in collaboration with B. Maury and A. Roudneff-Chupin for the details).},

affiliation = {Laboratoire de Mathématiques d’Orsay Faculté des Sciences Université Paris-Sud XI 91405 Orsay cedex France},

author = {Santambrogio, Filippo},

journal = {Séminaire Équations aux dérivées partielles},

keywords = {gradient flows; Wasserstein spaces; optimization; crowd motion; Fokker-Planck equation},

language = {eng},

pages = {1-16},

publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},

title = {Gradient flows in Wasserstein spaces and applications to crowd movement},

url = {http://eudml.org/doc/251158},

volume = {2009-2010},

year = {2009-2010},

}

TY - JOUR

AU - Santambrogio, Filippo

TI - Gradient flows in Wasserstein spaces and applications to crowd movement

JO - Séminaire Équations aux dérivées partielles

PY - 2009-2010

PB - Centre de mathématiques Laurent Schwartz, École polytechnique

VL - 2009-2010

SP - 1

EP - 16

AB - Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in $\mathbb{R}^n$, then in metric spaces, and finally in the space of probability measures endowed with the Wasserstein distance (induced by the quadratic transport cost). Differently from the usual theory by Jordan-Kinderlehrer-Otto and Ambrosio-Gigli-Savaré, we propose an approach where the optimality conditions for the minimizers of the optimization problems that one solves at every time step are obtained by looking at perturbation of the form $\rho _\varepsilon =(1-\varepsilon )\rho +\varepsilon \tilde{\rho }$ instead of $\rho _\varepsilon =(id+\varepsilon \xi )_\#\rho $. The ideas to make this approach rigorous are presented in the case of a Fokker-Planck equation, possibly with an interaction term, and then the paper is concluded by a section, where this method is applied to the original problem of crowd motion (referring to a recent paper in collaboration with B. Maury and A. Roudneff-Chupin for the details).

LA - eng

KW - gradient flows; Wasserstein spaces; optimization; crowd motion; Fokker-Planck equation

UR - http://eudml.org/doc/251158

ER -

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