Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in , 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...
Starting from a motivation in the modeling of crowd movement, the paper presents the topics of gradient flows, first in , 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...
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.
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.
Given a metric space we consider a general class of functionals which measure the cost of a path in joining two given points and , providing abstract existence results for
optimal paths. The results are then applied to the case when is aWasserstein space of probabilities
on a given set and the cost of a path depends on the value of classical functionals over measures. Conditions for linking arbitrary extremal measures and by means of finite cost paths are given.
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.
Given the probability measure over the given region , we consider the optimal location of a set composed by points in in order to minimize the average distance (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...
Given the probability measure over the given region
, we consider the optimal location of a set
composed by points in in order to minimize the
average distance (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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