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Gradient flows in Wasserstein spaces and applications to crowd movement

Filippo Santambrogio

Séminaire Équations aux dérivées partielles

Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in 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...

Gradient flows in Wasserstein spaces and applications to crowd movement

Filippo Santambrogio

Séminaire Équations aux dérivées partielles

Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in 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...

A variational model for urban planning with traffic congestion

Guillaume CarlierFilippo Santambrogio — 2005

ESAIM: Control, Optimisation and Calculus of Variations

We propose a variational model to describe the optimal distributions of residents and services in an urban area. The functional to be minimized involves an overall transportation cost taking into account congestion effects and two aditional terms which penalize concentration of residents and dispersion of services. We study regularity properties of the minimizers and treat in details some examples.

A variational model for urban planning with traffic congestion

Guillaume CarlierFilippo Santambrogio — 2010

ESAIM: Control, Optimisation and Calculus of Variations

We propose a variational model to describe the optimal distributions of residents and services in an urban area. The functional to be minimized involves an overall transportation cost taking into account congestion effects and two aditional terms which penalize concentration of residents and dispersion of services. We study regularity properties of the minimizers and treat in details some examples.

Path functionals over Wasserstein spaces

Alessio BrancoliniGiuseppe ButtazzoFilippo Santambrogio — 2006

Journal of the European Mathematical Society

Given a metric space X we consider a general class of functionals which measure the cost of a path in X joining two given points x 0 and x 1 , providing abstract existence results for optimal paths. The results are then applied to the case when X is aWasserstein space of probabilities on a given set Ω and the cost of a path depends on the value of classical functionals over measures. Conditions for linking arbitrary extremal measures μ 0 and μ 1 by means of finite cost paths are given.

Asymptotics of an optimal compliance-location problem

Giuseppe ButtazzoFilippo SantambrogioNicolas Varchon — 2006

ESAIM: Control, Optimisation and Calculus of Variations

We consider the problem of placing a Dirichlet region made by small balls of given radius in a given domain subject to a force in order to minimize the compliance of the configuration. Then we let tend to infinity and look for the Γ-limit of suitably scaled functionals, in order to get informations on the asymptotical distribution of the centres of the balls. This problem is both linked to optimal location and shape optimization problems.

Long-term planning versus short-term planning in the asymptotical location problem

Alessio BrancoliniGiuseppe ButtazzoFilippo SantambrogioEugene Stepanov — 2009

ESAIM: Control, Optimisation and Calculus of Variations

Given the probability measure ν over the given region Ω n , we consider the optimal location of a set Σ composed by n points in Ω in order to minimize the average distance Σ Ω dist ( x , Σ ) d ν (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving...

Long-term planning short-term planning in the asymptotical location problem

Alessio BrancoliniGiuseppe ButtazzoFilippo SantambrogioEugene Stepanov — 2008

ESAIM: Control, Optimisation and Calculus of Variations

Given the probability measure over the given region Ω n , we consider the optimal location of a set composed by points in in order to minimize the average distance Σ Ω dist ( x , Σ ) d ν (the classical optimal facility location problem). The paper compares two strategies to find optimal configurations: the long-term one which consists in placing all points at once in an optimal position, and the short-term one which consists in placing the points one by one adding at each step at most one point and preserving the...

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