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Luigi De Pascale

Publications and source records attributed to Luigi De Pascale.

At least 19 recordsLinked to original sources
arXiv2026

A variational formulation of a Multi-Population Mean Field Games with non-local interactions

We propose a MFG model with quadratic Hamiltonian involving $N$ populations. This results in a system of $N$ Hamilton-Jacobi-Bellman and $N$ Fokker-Planck equations with non-local interactions. As in the classical case we introduce an Eulerian variational formulation which, despite the non convexity of the interaction, still gives a weak solution to the MFG model. The problem can be reformulated in Lagrangian terms and solved numerically by a Sinkhorn-like scheme. We present numerical results based on this approach, these simulations exhibit different behaviours depending on the nature (repulsive or attractive) of the non-local interaction.

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arXiv2026

Some reverse inequality in optimal mass transportation

Controlling the $\mathcal W_\infty$ Wasserstein distance by the $\mathcal W_p$ Wasserstein distance is interesting both for theorical and numerical applications. A first paper on this problem was written several years ago [3]. Some year later [14] framed it in the same inequality for more general costs which increase with the distance. In this paper, we prove this type of inequality for optimal transport problems with pointwise cost which is a decreasing function of the distance. We show, in particular, that there is a general framework that encompasses all the cases above.

math.OC↗
arXiv2025

A regularized transportation cost stemming from entropic approximation

We study the entropic regularizations of optimal transport problems under suitable summability assumptions on the point-wise transport cost. These summability assumptions already appear in the literature. However, we show that the weakest compactness conditions that can be derived are already enough to obtain the convergence of the regularized functionals. This approach allows us to characterize the variational limit of the regularization even when it does not converge to the original problem. The results apply also to problems with more than two marginals.

math.OC↗
arXiv2025

A Rockafellar Theorem for cyclically quasi-monotone maps: the regular non-vanishing case

We study the connection between cyclic quasi-monotonicity and quasi-convexity, focusing on whether every cyclically quasi-monotone (possibly multivalued) map is included in the normal cone operator of a quasi-convex function, in analogy with Rockafellar's theorem for convex functions. We provide a positive answer for $\mathscr{C}^1$-regular, non-vanishing maps in any dimension, as well as for general multi-maps in dimension $1$. We further discuss connections to revealed preference theory in economics and to $L^\infty$ optimal transport. Finally, we present explicit constructions and examples, highlighting the main challenges that arise in the general case.

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arXiv2025

Dissociation limits in Density Functional Theory

In this paper we consider the {\it Density Functional Theory} (DFT) framework, where a functional of the form $$F_\eps(ρ)=\eps T(ρ)+bC(ρ)-U(ρ)$$ has to be minimized in the class of non-negative measures $ρ$ which have a prescribed total mass $m$ (the total electronic charge). The parameter $\eps$ is small and the terms $T$, $C$, $U$ respectively represent the kinetic energy, the electronic repulsive correlation, the potential interaction term between electrons and nuclei. Several expressions for the above terms have been considered in the literature and our framework is general enough to include most of them. It is known that in general, when the positive charge of the nuclei is small, the so-called {\it ionization phenomenon} may occur, consisting in the fact that the minimizers of $F_\eps$ can have a total mass lower than $m$; this physically means that some of the electrons may escape to infinity when the attraction of the nuclei is not strong enough. Our main goal, continuing the research we started in \cite{bbcd18}, is to study the asymptotic behavior of the minimizers of $F_\eps$ as $\eps\to0$. We show that the $Γ$-limit functional is defined on sums of Dirac masses and has an explicit expression that depends on the terms $T$, $C$, $U$ that the model takes into account. Some explicit examples illustrate how the electrons are distributed around the nuclei according to the model used.

math-ph↗
arXiv2023

Entropic approximation of $\infty$-optimal transport problems

We propose an entropic approximation approach for optimal transportation problems with a supremal cost. We establish $Γ$-convergence for suitably chosen parameters for the entropic penalization and that this procedure selects $\infty$-cyclically monotone plans at the limit. We also present some numerical illustrations performed with Sinkhorn's algorithm.

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arXiv2023

$L^\infty$-optimal transport for a class of quasiconvex cost functions

We consider the $L^\infty$-optimal mass transportation problem \[ \min_{Π(μ, ν)} γ-\mathrm{ess\,sup\,} c(x,y), \] for a new class of costs $c(x,y)$ for which we introduce a tentative notion of twist condition. In particular we study the conditions under which the infinitely-motonone minimizers are induced by a transportation map. We also state a uniqueness result for infinitely cyclically monotone Monge minimizers that corresponds to this class of cost functions. We compare the results to previous works.

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arXiv2022

Sufficiency of $c$-cyclical monotonicity in a class of multimarginal optimal transport problems

$c$-cyclical monotonicity is the most important optimality condition for an optimal transport plan. While the proof of necessity is relatively easy, the proof of sufficiency is often more difficult or even elusive. We present here a new approach and we show how known results are derived in this new framework and how this approach allows to prove sufficiency in situations previously not treatable.

math.OC↗
arXiv2021

On deterministic solutions for multi-marginal optimal transport with Coulomb cost

In this paper we study the three-marginal optimal mass transportation problem for the Coulomb cost on the plane $\R^2$. The key question is the optimality of the so-called Seidl map, first disproved by Colombo and Stra. We generalize the partial positive result obtained by Colombo and Stra and give a necessary and sufficient condition for the radial Coulomb cost to coincide with a much simpler cost that corresponds to the situation where all three particles are aligned. Moreover, we produce an infinite class of regular counterexamples to the optimality of this family of maps.

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arXiv2019

Relaxed multi-marginal costs and quantization effects

We propose a duality theory for multi-marginal repulsive cost that appear in optimal transport problems arising in Density Functional Theory. The related optimization problems involve probabilities on the entire space and, as minimizing sequences may lose mass at infinity, it is natural to expect relaxed solutions which are sub-probabilities. We first characterize the $N$-marginals relaxed cost in terms of a stratification formula which takes into account all $k$ interactions with $k\le N$. We then develop a duality framework involving continuous functions vanishing at infinity and deduce primal-dual necessary and sufficient optimality conditions Next we prove the existence and the regularity of an optimal dual potential under very mild assumptions. In the last part of the paper, we apply our results to a minimization problem involving a given continuous potential and we give evidence of a mass quantization effect for optimal solutions.

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arXiv2018

Dissociating limit in Density Functional Theory with Coulomb optimal transport cost

In the framework of Density Functional Theory with Strongly Correlated Electrons we consider the so called bond dissociating limit for the energy of an aggregate of atoms. We show that the multi-marginals optimal transport cost with Coulombian electron-electron repulsion may correctly describe the dissociation effect. The variational limit is completely calculated in the case of N=2 electrons. The theme of fractional number of electrons appears naturally and brings into play the question of optimal partial transport cost. A plan is outlined to complete the analysis which involves the study of the relaxation of optimal transport cost with respect to the weak* convergence of measures.

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arXiv2017

A study of the dual problem of the one-dimensional L-infinity optimal transport problem with applications

The Monge-Kantorovich problem for the infinite Wasserstein distance presents several peculiarities. Among them the lack of convexity and then of a direct duality. We study in dimension 1 the dual problem introduced by Barron, Bocea and Jensen. We construct a couple of Kantorovich potentials which is "as less trivial as possible". More precisely, we build a potential which is non constant around any point that the plan which is locally optimal moves at maximal distance. As an application, we show that the set of points which are displaced to maximal distance by a locally optimal transport plan is minimal.

math.OC↗
arXiv2017

Optimal transport with Coulomb cost and the semiclassical limit of Density Functional Theory

We present some progress in the direction of determining the semiclassical limit of the Hoenberg-Kohn universal functional in Density Functional Theory for Coulomb systems. In particular we give a proof of the fact that for Bosonic systems with an arbitrary number of particles the limit is the multimarginal optimal transport problem with Coulomb cost and that the same holds for Fermionic systems with 2 or 3 particles. Comparisons with previous results are reported . The approach is based on some techniques from the optimal transportation theory.

math.AP↗
arXiv2017

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

We investigate the approximation of the Monge problem (minimizing \int\_$Ω$ |T (x) -- x| d$μ$(x) among the vector-valued maps T with prescribed image measure T \# $μ$) by adding a vanishing Dirichlet energy, namely $ε$ \int\_$Ω$ |DT |^2. We study the $Γ$-convergence as $ε$ $\rightarrow$ 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 H ^1 map, we study the selected limit map, which is a new "special" Monge transport, possibly different from the monotone one, and we find the precise asymptotics of the optimal cost depending on $ε$, where the leading term is of order $ε$| log $ε$|.

math.OC↗
arXiv2016

Continuity and estimates for multimarginal optimal transportation problems with singular costs

We consider some repulsive multimarginal optimal transportation problems which include, as a particular case, the Coulomb cost. We prove a regularity property of the minimizers (optimal transportation plan) from which we deduce existence and some basic regularity of a maximizer for the dual problem (Kantorovich potential). This is then applied to obtain some estimates of the cost and to the study of continuity properties.

math.OC↗
arXiv2015

Duality theory and optimal transport for sand piles growing in a silos

We prove existence and uniqueness of solutions for a system of PDEs which describes the growth of a sandpile in a silos with flat bottom under the action of a vertical, measure source. The tools we use are a discrete approximation of the source and the duality theory for optimal transport (or Monge-Kantorovich) problems.

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