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

Abstract

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Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in 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 ρ ε = ( 1 - ε ) ρ + ε ρ ˜ instead of ρ ε = ( i d + ε ξ ) # ρ . 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).

How to cite

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Santambrogio, 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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