Path Functionals over Wasserstein Spaces

Brancolini, Alessio; Buttazzo, Giuseppe; Santambrogio, Filippo

doi:10.4171/jems/61

Cited by 45 publications

(66 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Notice that if one wants to keep the usual transport interpretation given by a "dynamic cost" to be minimized along the solution of the continuity equation, one can simply introduce the velocity vector fieldṽ t := ρ Therefore, in this model the usual p-energy R d ρ t |ṽ t | p dx of the moving masses ρ t with velocityṽ t results locally modified by a factor f (ρ t ) depending on the local density of the mass occupied at the time t. Different non-local models have been considered in [8,4].…”

Section: 4)mentioning

confidence: 99%

A new class of transport distances between measures

2008

View full text Add to dashboard Cite

We introduce a new class of distances between nonnegative Radon measures in R d . They are modeled on the dynamical characterization of the Kantorovich-RubinsteinWasserstein distances proposed by BENAMOU-BRENIER [7] and provide a wide family interpolating between the Wasserstein and the homogeneous WFrom the point of view of optimal transport theory, these distances minimize a dynamical cost to move a given initial distribution of mass to a final configuration. An important difference with the classical setting in mass transport theory is that the cost not only depends on the velocity of the moving particles but also on the densities of the intermediate configurations with respect to a given reference measure γ. We study the topological and geometric properties of these new distances, comparing them with the notion of weak convergence of measures and the well established KantorovichRubinstein-Wasserstein theory. An example of possible applications to the geometric theory of gradient flows is also given.

show abstract

Section: 4)mentioning

confidence: 99%

A new class of transport distances between measures

2008

View full text Add to dashboard Cite

show abstract

“…This suggests a connection between the "multiplicative" model studied here and in [2], and the "additive" model…”

Section: Introductionmentioning

confidence: 99%

“…This choice of L is in fact a local functional on measures which, among probability measures, favours the most concentrated ones. In [2], as a natural counterpart, the case of local functionals L which prefer spread measures is considered as well and the two problems sound somehow specular. The aim of the present paper is in fact to consider this second problem and to find out optimality conditions in the form of PDEs.…”

Section: Introductionmentioning

confidence: 99%

“…In (1.1), |µ ′ |(t) is the rate of change of W p also called metric derivative, along the curve µ, see (2.1). This problem has been introduced in [2], where the main goal was to choose a factor L favouring atomic measures in order to give a timedependent approach to some branched transport problems which may be applied to the study of river networks, pipe systems, blood vessels, tree structures. .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Necessary optimality conditions for geodesics in weighted Wasserstein spaces

Ambrosio¹,

Santambrogio²

2007

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

Self Cite

View full text Add to dashboard Cite

Abstract: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.

show abstract

“…Equivalence of the different models and formulations are instead the topic of [MS09], [MS13]. Branched transport can also be modelled with curves in the Wasserstein space as in [BBS06], [BB10], [BS11b].…”

Section: Introductionmentioning

confidence: 99%

Equivalent formulations for the branched transport and urban planning problems

Brancolini

Wirth

2016

Journal de Mathématiques Pures et Appliquées

Self Cite

View full text Add to dashboard Cite

We consider two variational models for transport networks, an urban planning and a branched transport model, in both of which there is a preference for networks that collect and transport lots of mass together rather than transporting all mass particles independently. The strength of this preference determines the ramification patterns and the degree of complexity of optimal networks. Traditionally, the models are formulated in very different ways, via cost functionals of the network in case of urban planning or via cost functionals of irrigation patterns or of mass fluxes in case of branched transport. We show here that actually both models can be described by all three types of formulations; in particular, the urban planning can be cast into a Eulerian (flux-based ) or a Lagrangian (pattern-based ) framework.

show abstract

Path Functionals over Wasserstein Spaces

Cited by 45 publications

References 14 publications

A new class of transport distances between measures

A new class of transport distances between measures

Necessary optimality conditions for geodesics in weighted Wasserstein spaces

Equivalent formulations for the branched transport and urban planning problems

Contact Info

Product

Resources

About