IRIS Università Commerciale Luigi Bocconihttps://iris.unibocconi.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Tue, 25 Jan 2022 01:05:30 GMT2022-01-25T01:05:30Z1081Time-convexity of the entropy in the multiphasic formulation of the incompressible Euler equationhttp://hdl.handle.net/11565/4032305Titolo: Time-convexity of the entropy in the multiphasic formulation of the incompressible Euler equation
Abstract: We study the multiphasic formulation of the incompressible Euler equation introduced by Brenier: infinitely many phases evolve according to the compressible Euler equation and are coupled through a global incompressibility constraint. In a convex domain, we are able to prove that the entropy, when averaged over all phases, is a convex function of time, a result that was conjectured by Brenier. The novelty in our approach consists in introducing a time-discretization that allows us to import a flow interchange inequality previously used by Matthes, McCann and Savaré to study first order in time PDE, namely the JKO scheme associated with non-linear parabolic equations.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/11565/40323052017-01-01T00:00:00ZOptimal density evolution with congestion: L# bounds via flow interchange techniques and applications to variational Mean Field Gameshttp://hdl.handle.net/11565/4032309Titolo: Optimal density evolution with congestion: L# bounds via flow interchange techniques and applications to variational Mean Field Games
Abstract: We consider minimization problems for curves of measure, with kinetic and potential energy and a congestion penalization, as in the functionals that appear in Mean Field Games (MFG) with a variational structure. We prove L ∞ regularity results for the optimal density, which can be applied to the rigorous derivations of equilibrium conditions at the level of each agent’s trajectory, via time-discretization arguments, displacement convexity, and suitable Moser iterations. Similar L ∞ results have already been found by P.-L. Lions in his course on MFG, using a proof based on the use of a (very degenerate) elliptic equation on the dual potential (the value function) φ, in the case where the initial and final density were prescribed (planning problem). Here the strategy is highly different, and allows for instance to prove local-in-time estimates without assumptions on the initial and final data, and to insert a potential in the dynamics.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11565/40323092018-01-01T00:00:00ZTotal variation isoperimetric profileshttp://hdl.handle.net/11565/4032303Titolo: Total variation isoperimetric profiles
Abstract: Applications such as political redistricting demand quantitative measures of geometric compactness to distinguish between simple and contorted shapes. While the isoperimetric quotient, or ratio of area to perimeter squared, is commonly used in practice, it is sensitive to noisy data and irrelevant geographic features like coastline. These issues are addressed in theory by the isoperimetric profile, which plots the minimum perimeter needed to inscribe regions of different prescribed areas within the boundary of a shape. Efficient algorithms for computing this profile, however, are not known in practice. Hence, in this paper, we propose a convex Eulerian relaxation of the isoperimetric profile using total variation. We prove theoretical properties of our relaxation, showing that it still satisfies an isoperimetric inequality and yields a convex function of the prescribed area. Furthermore, we provide a discretization of the problem, an optimization technique, and experiments demonstrating the value of our relaxation.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11565/40323032019-01-01T00:00:00ZHidden convexity in a problem of nonlinear elasticityhttp://hdl.handle.net/11565/4036710Titolo: Hidden convexity in a problem of nonlinear elasticity
Abstract: We study compressible and incompressible nonlinear elasticity variational problems in a general context. Our main result gives a sufficient condition for an equilibrium to be a global energy minimizer, in terms of convexity properties of the pressure in the deformed configuration. We also provide a convex relaxation of the problem together with its dual formulation, based on measure-valued mappings, which coincides with the original problem under our condition.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/11565/40367102021-01-01T00:00:00ZNew estimates on the regularity of the pressure in density-constrained mean field gameshttp://hdl.handle.net/11565/4032311Titolo: New estimates on the regularity of the pressure in density-constrained mean field games
Abstract: We consider variational mean field games (MFGs) endowed with a constraint on the maximal density of the distribution of players. Minimizers of the variational formulation are equilibria for a game where both the running cost and the final cost of each player are augmented by a pressure effect, that is, a positive cost concentrated on the set where the density saturates the constraint. Yet, this pressure is a priori only a measure and regularity is needed to give a precise meaning to its integral on trajectories. We improve, in the limited case where the Hamiltonian is quadratic, which allows to use optimal transport techniques after time-discretization, the results obtained in a paper of the second author with Cardaliaguet and Mészáros. We prove (Formula presented.) and (Formula presented.) regularity under very mild assumptions on the data, and explain the consequences for the MFG, in terms of the value function and of the Lagrangian equilibrium formulation.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11565/40323112019-01-01T00:00:00ZDynamical optimal transport on discrete surfaceshttp://hdl.handle.net/11565/4032317Titolo: Dynamical optimal transport on discrete surfaces
Abstract: We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finitedimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between
distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows.
Mon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/11565/40323172018-01-01T00:00:00ZUnconditional convergence for discretizations of dynamical optimal transporthttp://hdl.handle.net/11565/4034608Titolo: Unconditional convergence for discretizations of dynamical optimal transport
Abstract: The dynamical formulation of optimal transport, also known as Benamou–Brenier formulation or computational fluid dynamics formulation, amounts to writing the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several discretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space. In this paper, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional problem for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by Gladbach, Kopfer, and Maas, as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien, and Solomon, fit in this framework.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/11565/40346082021-01-01T00:00:00ZHarmonic mappings valued in the Wasserstein spacehttp://hdl.handle.net/11565/4032313Titolo: Harmonic mappings valued in the Wasserstein space
Abstract: We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings μ:Ω→(P(D),W2) defined over a subset Ω of Rp and valued in the space P(D) of probability measures on a compact convex subset D of Rq endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Korevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetnyak of Sobolev spaces valued in metric spaces. We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary ∂Ω are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for Ω a segment of R. As the Wasserstein space (P(D),W2) is positively curved in the sense of Alexandrov we cannot apply the theory of Korevaar, Schoen and Jost and we use instead arguments based on optimal transport. We manage to get existence of harmonic mappings provided that the boundary values are Lipschitz on ∂Ω, uniqueness is an open question. If Ω is a segment of R, it is known that a curve valued in the Wasserstein space P(D) can be seen as a superposition of curves valued in D. We show that it is no longer the case in higher dimensions: a generic mapping Ω→P(D) cannot be represented as the superposition of mappings Ω→D. We are able to show the validity of a maximum principle: the composition F∘μ of a function F:P(D)→R convex along generalized geodesics and a harmonic mapping μ:Ω→P(D) is a subharmonic real-valued function. We also study the special case where we restrict ourselves to a given family of elliptically contoured distributions (a finite-dimensional and geodesically convex submanifold of (P(D),W2) which generalizes the case of Gaussian measures) and show that it boils down to harmonic mappings valued in the Riemannian manifold of symmetric matrices endowed with the distance coming from optimal transport.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/11565/40323132019-01-01T00:00:00Z