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Filippo Santambrogio

Publications and source records attributed to Filippo Santambrogio.

72 records · Page 4Linked to original sources
arXiv2013

A Dacorogna-Moser approach to flow decomposition and minimal flow problems

The papers describes an easy approach, based on a classical construction by Dacorogna and Moser, to prove that optimal vector fields in some minimal flow problem linked to optimal transport models (congested traffic, branched transport, Beckmann's problem...) are induced by a probability measure on the space of paths. This gives a new, easier, proof of a classical result by Smirnov, and allows handling optimal flows without taking care of the presence of cycles.

math.OC↗
arXiv2013

Regularity in Monge's mass transfer problem

In this paper, we study the regularity of optimal mappings in Monge's mass transfer problem. Using the approximation to Monge's cost function given by the Euclidean distance c(x,y)=dist(x,y) through the costs c_\eps(x,y)=(\eps^2+dist(x,y)^2)^{1/2}, we consider the optimal mappings T_\eps for these costs, and we prove that the eigenvalues of the Jacobian matrix DT_\eps, which are all positive, are locally uniformly bounded. By an example we prove that T_\eps is in general not uniformly Lipschitz continuous as \eps-0, even if the mass distributions are positive and smooth, and the domains are c-convex.

math.AP↗
arXiv2012

Optimal transportation with an oscillation-type cost: the one-dimensional case

The main result of this paper is the existence of an optimal transport map $T$ between two given measures $μ$ and $ν$, for a cost which considers the maximal oscillation of $T$ at scale $δ$, given by $ω_δ(T):=\sup_{|x-y|<δ}|T(x)-T(y)|$. The minimization of this criterion finds applications in the field of privacy-respectful data transmission. The existence proof unfortunately only works in dimension one and is based on some monotonicity considerations.

math.OC↗
arXiv2011

A Modest Proposal for MFG with Density Constraints

We consider a typical problem in Mean Field Games: the congestion case, where in the cost that agents optimize there is a penalization for passing through zones with high density of agents, in a deterministic framework. This equilibrium problem is known to be equivalent to the optimization of a global functional including an $L^p$ norm of the density. The question arises as to produce a similar model replacing the $L^p$ penalization with an $L^\infty$ constraint, but the simplest approaches do not give meaningful definitions. Taking into account recent works about crowd motion, where the density constraint $ρ\leq 1$ was treated in terms of projections of the velocity field onto the set of admissible velocity (with a constraint on the divergence) and a pressure field was introduced, we propose a definition and write a system of PDEs including the usual Hamilton-Jacobi equation coupled with the continuity equation. For this system, we analyze an example and propose some open problems.

math.AP↗
arXiv2011

Optimal transportation for a quadratic cost with convex constraints and applications

We prove existence of an optimal transport map in the Monge-Kantorovich problem associated to a cost $c(x,y)$ which is not finite everywhere, but coincides with $|x-y|^2$ if the displacement $y-x$ belongs to a given convex set $C$ and it is $+\infty$ otherwise. The result is proven for $C$ satisfying some technical assumptions allowing any convex body in $\R^2$ and any convex polyhedron in $\R^d$, $d>2$. The tools are inspired by the recent Champion-DePascale-Juutinen technique. Their idea, based on density points and avoiding disintegrations and dual formulations, allowed to deal with $L^\infty$ problems and, later on, with the Monge problem for arbitrary norms.

math.OC↗
arXiv2011

A sharp inequality for transport maps in W^{1,p}(R) via approximation

For $f$ convex and increasing, we prove the inequality $ \int f(|U'|) \geq \int f(nT')$, every time that $U$ is a Sobolev function of one variable and $T$ is the non-decreasing map defined on the same interval with the same image measure as $U$, and the function $n(x)$ takes into account the number of pre-images of $U$ at each point. This may be applied to some variational problems in a mass-transport framework or under volume constraints.

math.AP↗
arXiv2010

Introduction to Optimal Transport Theory

These notes constitute a sort of Crash Course in Optimal Transport Theory. The different features of the problem of Monge-Kantorovitch are treated, starting from convex duality issues. The main properties of space of probability measures endowed with the distances $W_p$ induced by optimal transport are detailed. The key tools to put in relation optimal transport and PDEs are provided.

math.CA↗
arXiv2010

Models and applications of Optimal Transport in Economics, Traffic and Urban Planning

Some optimization or equilibrium problems involving somehow the concept of optimal transport are presented in these notes, mainly devoted to applications to economic and game theory settings. A variant model of transport, taking into account traffic congestion effects is the first topic, and it shows various links with Monge-Kantorovich theory and PDEs. Then, two models for urban planning are introduced. The last section is devoted to two problems from economics and their translation in the language of optimal transport.

math.CA↗
arXiv2010

On the strategic use of risk and undesirable goods in multidimensional screening

A monopolist sells goods with possibly a characteristic consumers dislike (for instance, he sells random goods to risk averse agents), which does not affect the production costs. We investigate the question whether using undesirable goods is profitable to the seller. We prove that in general this may be the case, depending on the correlation between agents types and aversion. This is due to screening effects that outperform this aversion. We analyze, in a continuous framework, both 1D and multidimensional cases.

math.OC↗
arXiv2010

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

math.OC↗
arXiv2010

A macroscopic crowd motion model of gradient flow type

A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U(x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities. Instead of looking at a microscopic setting (where individuals are represented by rigid discs), here the macroscopic approach is investigated, where the unknwon is the evolution of the density . If a gradient structure is given, say U is the opposite of the gradient of D where D is, for instance, the distance to the exit door, the problem is presented as a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density), nor geodesically convex, which requires for an ad-hoc study of the convergence of a discrete scheme.

math.AP↗
arXiv2009

Least Squares estimation of two ordered monotone regression curves

In this paper, we consider the problem of finding the Least Squares estimators of two isotonic regression curves $g^\circ_1$ and $g^\circ_2$ under the additional constraint that they are ordered; e.g., $g^\circ_1 \le g^\circ_2$. Given two sets of $n$ data points $y_1, ..., y_n$ and $z_1, >...,z_n$ observed at (the same) design points, the estimates of the true curves are obtained by minimizing the weighted Least Squares criterion $L_2(a, b) = \sum_{j=1}^n (y_j - a_j)^2 w_{1,j}+ \sum_{j=1}^n (z_j - b_j)^2 w_{2,j}$ over the class of pairs of vectors $(a, b) \in \mathbb{R}^n \times \mathbb{R}^n $ such that $a_1 \le a_2 \le ...\le a_n $, $b_1 \le b_2 \le ...\le b_n $, and $a_i \le b_i, i=1, ...,n$. The characterization of the estimators is established. To compute these estimators, we use an iterative projected subgradient algorithm, where the projection is performed with a "generalized" pool-adjacent-violaters algorithm (PAVA), a byproduct of this work. Then, we apply the estimation method to real data from mechanical engineering.

stat.ME↗
arXiv2009

A Modica-Mortola approximation for branched transport

The M^αenergy which is usually minimized in branched transport problems among singular 1-dimensional rectifiable vector measures with prescribed divergence is approximated (and convergence is proved) by means of a sequence of elliptic energies, defined on more regular vector fields. The procedure recalls the Modica-Mortola one for approximating the perimeter, and the double-well potential is replaced by a concave power.

math.OC↗
arXiv2009

A strategy for non-strictly convex transport costs and the example of ||x-y||p in R2

This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non strictly convex cost. We give a decomposition strategy to address this issue. As part of our strategy, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. As an illustration of our strategy, we prove existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form h(||x-y||) with h strictly convex increasing and ||. || an arbitrary norm in \R2.

math.CA↗
arXiv2008

From Knothe's transport to Brenier's map and a continuation method for optimal transport

A simple procedure to map two probability measures in $\mathbb{R}^d$ is the so-called \emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem.

math.OC↗
arXiv2006

Necessary optimality conditions for geodesics in weighted Wasserstein spaces

The geodesic problem in Wasserstein spaces with a metric perturbed by a conformal factor is considered, and necessary optimality conditions are estabilished in a case where this conformal factor favours the spreading of the probability measure along the curve. These conditions have the form of a system of PDEs of the kind of the compressible Euler equations. Moreover, self-similar solutions to this system are discussed.

math.AP↗
arXiv2005

Transport and Concentration Problems with Interaction Effects

After presenting an overview about variational problems on probability measures for functionals involving transport costs and extra terms encouraging or discouraging concentration, we look for optimality conditions, regularity properties and explicit computations in the case where Wasserstein distances and interaction energies are considered.

math.OC↗