arXiv subjects
Publications and source records attributed to Luca Nenna.
This note is devoted to study a class of games with a continuum of players for which Cournot-Nash equilibria can be obtained by minimisation of some cost related to Optimal Transport. In particular we focus on the case of an Optimal Transport term between unequal dimension. We also present several numerical simulations.
We develop the theory of a metric, which we call the $ν$-based Wasserstein metric and denote by $W_ν$, on the set of probability measures $\mathcal P(X)$ on a domain $X \subseteq \mathbb{R}^m$. This metric is based on a slight refinement of the notion of generalized geodesics with respect to a base measure $ν$ and is relevant in particular for the case when $ν$ is singular with respect to $m$-dimensional Lebesgue measure; it is also closely related to the concept of linearized optimal transport. The $ν$-based Wasserstein metric is defined in terms of an iterated variational problem involving optimal transport to $ν$; we also characterize it in terms of integrations of classical Wasserstein distance between the conditional probabilities and through limits of certain multi-marginal optimal transport problems. As we vary the base measure $ν$, the $ν$-based Wasserstein metric interpolates between the usual quadratic Wasserstein distance and a metric associated with the uniquely defined generalized geodesics obtained when $ν$ is sufficiently regular. When $ν$ concentrates on a lower dimensional submanifold of $\mathbb{R}^m$, we prove that the variational problem in the definition of the $ν$-based Wasserstein distance has a unique solution. We establish geodesic convexity of the usual class of functionals and of the set of source measures $μ$ such that optimal transport between $μ$ and $ν$ satisfies a strengthening of the generalized nestedness condition introduced in \cite{McCannPass20}.We finally introduce a slight variant of the dual metric mentioned above in order to prove convergence of an iterative scheme to solve a variational problem arising in game theory.
This paper is devoted to variational problems on the set of probability measures which involve optimal transport between unequal dimensional spaces. In particular, we study the minimization of a functional consisting of the sum of a term reflecting the cost of (unequal dimensional) optimal transport between one fixed and one free marginal, and another functional of the free marginal (of various forms). Motivating applications include Cournot-Nash equilibria where the strategy space is lower dimensional than the space of agent types. For a variety of different forms of the term described above, we show that a nestedness condition, which is known to yield much improved tractability of the optimal transport problem, holds for any minimizer. Depending on the exact form of the functional, we exploit this to find local differential equations characterizing solutions, prove convergence of an iterative scheme to compute the solution, and prove regularity results.
We propose a new viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We show the equivalence of such mean-field games with a relative entropy minimization at the level of probabilities on curves. We also address the time-discretization of such problems, establish $Γ$-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm.
Starting from Brenier's relaxed formulation of the incompressible Euler equation in terms of geodesics in the group of measure-preserving diffeomorphisms, we propose a numerical method based on Sinkhorn's algorithm for the entropic regularization of optimal transport. We also make a detailed comparison of this entropic regularization with the so-called Bredinger entropic interpolation problem. Numerical results in dimension one and two illustrate the feasibility of the method.
The strong-interaction limit of the Hohenberg-Kohn functional defines a multimarginal optimal transport problem with Coulomb cost. From physical arguments, the solution of this limit is expected to yield strictly-correlated particle positions, related to each other by co-motion functions (or optimal maps), but the existence of such a deterministic solution in the general three-dimensional case is still an open question. A conjecture for the co-motion functions for radially symmetric densities was presented in Phys.~Rev.~A {\bf 75}, 042511 (2007), and later used to build approximate exchange-correlation functionals for electrons confined in low-density quantum dots. Colombo and Stra [Math.~Models Methods Appl.~Sci., {\bf 26} 1025 (2016)] have recently shown that these conjectured maps are not always optimal. Here we revisit the whole issue both from the formal and numerical point of view, finding that even if the conjectured maps are not always optimal, they still yield an interaction energy (cost) that is numerically very close to the true minimum. We also prove that the functional built from the conjectured maps has the expected functional derivative also when they are not optimal.
We consider a class of games with continuum of players where equilibria can be obtained by the minimization of a certain functional related to optimal transport as emphasized in [7]. We then use the powerful entropic regularization technique to approximate the problem and solve it numerically in various cases. We also consider the extension to some models with several populations of players.
This paper intents to present the state of art and recent developments of the optimal transportation theory with many marginals for a class of repulsive cost functions. We introduce some aspects of the Density Functional Theory (DFT) from a mathematical point of view, and revisit the theory of optimal transport from its perspective. Moreover, in the last three sections, we describe some recent and new theoretical and numerical results obtained for the Coulomb cost, the repulsive harmonic cost and the determinant cost.
In this paper, we present a numerical method, based on iterative Bregman projections, to solve the optimal transport problem with Coulomb cost. This is related to the strong interaction limit of Density Functional Theory. The first idea is to introduce an entropic regularization of the Kantorovich formulation of the Optimal Transport problem. The regularized problem then corresponds to the projection of a vector on the intersection of the constraints with respect to the Kullback-Leibler distance. Iterative Bregman projections on each marginal constraint are explicit which enables us to approximate the optimal transport plan. We validate the numerical method against analytical test cases.
This article details a general numerical framework to approximate so-lutions to linear programs related to optimal transport. The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback-Leibler Bregman di-vergence projection of a vector (representing some initial joint distribu-tion) on the polytope of constraints. We show that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form. This allows us to make use of iterative Bregman projections (when there are only equality constraints) or more generally Bregman-Dykstra iterations (when inequality constraints are in-volved). We illustrate the usefulness of this approach to several variational problems related to optimal transport: barycenters for the optimal trans-port metric, tomographic reconstruction, multi-marginal optimal trans-port and in particular its application to Brenier's relaxed solutions of in-compressible Euler equations, partial un-balanced optimal transport and optimal transport with capacity constraints.