Asymptotics of an optimal compliance-location problem

We consider the problem of placing n small balls of given radius in a certain domain subject to a force f in order to minimize the compliance of the configuration. Then we let n tend to infinity and look at the asymptotics of the minimization problem, after properly scaling the functionals involved, and to the limit distribution of the centres of the balls. This problem is both linked to optimal location and shape optimization problems.Comment: 20 pages with 2 figures; final accepted version (minor changes, some extra details on the positivity assumption on $f$

Optimal transportation with traffic congestion and Wardrop equilibria

In the classical Monge-Kantorovich problem, the transportation cost only depends on the amount of mass sent from sources to destinations and not on the paths followed by this mass. Thus, it does not allow for congestion effects. Using the notion of traffic intensity, we propose a variant taking into account congestion. This leads to an optimization problem posed on a set of probability measures on a suitable paths space. We establish existence of minimizers and give a characterization. As an application, we obtain existence and variational characterization of equilibria of Wardrop type in a continuous space setting

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

Given the probability measure $\nu$ over the given region $\Omega\subset \R^n$, we consider the optimal location of a set $\Sigma$ composed by $n$ points \Om in order to minimize the average distance \Sigma\mapsto \int_\Om \dist(x,\Sigma) d\nu (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 the configuration built at previous steps. We show that the respective optimization problems exhibit qualitatively different asymptotic behavior as $n\to\infty$, although the optimization costs in both cases have the same asymptotic orders of vanishing.Comment: for more pictures and some movies as well, see http://www.sissa.it/~brancoli

Handling congestion in crowd motion modeling

We address here the issue of congestion in the modeling of crowd motion, in the non-smooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in terms of behavior, to focus on the particular problems raised by the non-smooth character of the models. We consider that individuals tend to move according to a desired, or spontanous, velocity. We account for congestion by assuming that the evolution realizes at each time an instantaneous balance between individual tendencies and global constraints (overlapping is forbidden): the actual velocity is defined as the closest to the desired velocity among all admissible ones, in a least square sense. We develop those principles in the microscopic and macroscopic settings, and we present how the framework of Wasserstein distance between measures allows to recover the sweeping process nature of the problem on the macroscopic level, which makes it possible to obtain existence results in spite of the non-smooth character of the evolution process. Micro and macro approaches are compared, and we investigate the similarities together with deep differences of those two levels of description

A Benamou-Brenier approach to branched transport

The problem of branched transportation aims to describe the movement of masses when, due to concavity effects, they have the interest to travel together as much as possible, because the cost for a path of length $\ell$ covered by a mass $m$ is proportional to $m^\alpha\ell$ with $0<\alpha<1$. The optimization of this criterion let branched structures appear and is suitable to applications like road systems, blood vessels, river networks\dots Several models have been employed in the literature to present this transport problem, and the present paper looks at a dynamical one, similar to the celebrated Benamou-Brenier formulation of Kantorovitch optimal transport. The movement is represented by a path $\rho_t$ of probabilities, connecting an initial state $\mu_0$ to a final state $\mu_1$, satisfying the continuity equation \partial_t\rho+\dive_xq=0 together with a velocity field $v$ (with $q=\rho v$ being the momentum). The transportation cost to be minimized is non-convex and finite on atomic measures: $\int_0^1\big(\int_\Omega\rho^{\alpha-1}|q|\,d\#(x)\big)\,dt$

Derivatives with respect to metrics and applications: subgradient marching algorithm

This paper introduces a subgradient descent algorithm to compute a Riemannian metric that minimizes an energy involving geodesic distances. The heart of the method is the Subgradient Marching Algorithm to compute the derivative of the geodesic distance with respect to the metric. The geodesic distance being a concave function of the metric, this algorithm computes an element of the subgradient in O(N 2 log(N)) operations on a discrete grid of N points. It performs a front propagation that computes a subgradient of a discrete geodesic distance. We show applications to landscape modeling and to traffic congestion. Both applications require the maximization of geodesic distances under convex constraints, and are solved by subgradient descent computed with our Subgradient Marching. We also show application to the inversion of travel time tomography, where the recovered metric is the local minimum of a non-convex variational problem involving geodesic distance

Finite-size effects in the superconformal beta-deformed N=4 SYM

We study finite size effects for composite operators in the SU(2) sector of the superconformal beta-deformed N=4 SYM theory. In particular we concentrate on the spectrum of one single magnon. Since in this theory one-impurity states are non BPS we compute their anomalous dimensions including wrapping contributions up to four loops and discuss higher order effects.Comment: LaTeX, mpost, feynmf, 20 pages, 4 figures, 5 tables; v2: references added, equations (4.13) and (4.17) correcte

The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile

We investigate the approximation of the Monge problem (minimizing ?????|T(x)???x|d??(x) among the vector-valued maps T with prescribed image measure T_\\#\mu) by adding a vanishing Dirichlet energy, namely ???????|DT|2, where ?????0. We study the ??-convergence as ?????0, proving a density result for Sobolev (or Lipschitz) transport maps in the class of transport plans. In a certain two-dimensional framework that we analyze in details, when no optimal plan is induced by an H1 map, we study the selected limit map, which is a new "special" Monge transport, different from the monotone one, and we find the precise asymptotics of the optimal cost depending on ??, where the leading term is of order ??|log??