arXiv subjects

Filippo Santambrogio

Publications and source records attributed to Filippo Santambrogio.

At least 37 records · Page 2Linked to original sources

arXiv2022

A cross-diffusion system obtained via (convex) relaxation in the JKO scheme

In this paper, we start from a very natural system of cross-diffusion equations, which can be seen as a gradient flow for the Wasserstein distance of a certain functional. Unfortunately, the cross-diffusion system is not well-posed, as a consequence of the fact that the underlying functional is not lower semi-continuous. We then consider the relaxation of the functional, and prove existence of a solution in a suitable sense for the gradient flow of (the relaxed functional). This gradient flow has also a cross-diffusion structure, but the mixture between two different regimes, that are determined by the relaxation, makes this study non-trivial.

math.AP↗

arXiv2022

Strong $L^2 H^2$ convergence of the JKO scheme for the Fokker-Planck equation

Following a celebrated paper by Jordan, Kinderleherer and Otto it is possible to discretize in time the Fokker-Planck equation $\partial_t\varrho=Δ\varrho+\nabla\cdot(ρ\nabla V)$ by solving a sequence of iterated variational problems in the Wasserstein space, and the sequence of piecewise constant curves obtained from the scheme is known to converge to the solution of the continuous PDE. This convergence is uniform in time valued in the Wasserstein space and also strong in $L^1$ in space-time. We prove in this paper, under some assumptions on the domain (a bounded and smooth convex domain) and on the initial datum (which is supposed to be bounded away from zero and infinity and belong to $W^{1,p}$ for an exponent $p$ larger than the dimension), that the convergence is actually strong in $L^2_tH^2_x$, hence strongly improving the previously known results in terms of the order of derivation in space. The technique is based on some inequalities, obtained with optimal transport techniques, that can be proven on the discrete sequence of approximate solutions, and that mimic the corresponding continuous computations.

math.OC↗

arXiv2022

Second order local minimal-time Mean Field Games

The paper considers a forward-backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain $Ω$ in minimal expected time. Agents are constrained by a bound on the drift depending on the density of other agents at their location. Existence for a finite time horizon $T$ is proven via a fixed point argument, but the natural setting for this problem is in infinite time horizon. Estimates are needed to treat the limit $T\to\infty$, and the asymptotic behavior of the solution obtained in this way is also studied. This passes through classical parabolic arguments and specific computations for MFGs. Both the Fokker--Planck equation on the density of agents and the Hamilton--Jacobi--Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach $\partialΩ$. The initial datum for the density is given, and the long-time limit of the value function is characterized as the solution of a stationary problem.

math.AP↗

arXiv2021

Non-asymptotic convergence bounds for Wasserstein approximation using point clouds

Several issues in machine learning and inverse problems require to generate discrete data, as if sampled from a model probability distribution. A common way to do so relies on the construction of a uniform probability distribution over a set of $N$ points which minimizes the Wasserstein distance to the model distribution. This minimization problem, where the unknowns are the positions of the atoms, is non-convex. Yet, in most cases, a suitably adjusted version of Lloyd's algorithm -- in which Voronoi cells are replaced by Power cells -- leads to configurations with small Wasserstein error. This is surprising because, again, of the non-convex nature of the problem, as well as the existence of spurious critical points. We provide explicit upper bounds for the convergence speed of this Lloyd-type algorithm, starting from a cloud of points sufficiently far from each other. This already works after one step of the iteration procedure, and similar bounds can be deduced, for the corresponding gradient descent. These bounds naturally lead to a modified Poliak-Lojasiewicz inequality for the Wasserstein distance cost, with an error term depending on the distances between Dirac masses in the discrete distribution.

math.OC↗

arXiv2021

Stein kernels for $q$-moment measures and new bounds for the rate of convergence in the central limit theorem

Given an isotropic probability measure $μ$ on ${\mathbb R}^d$ with ${\rm d}μ\left( x \right) = {\left( {\varrho \left( x \right)} \right)^{ - α}}{\rm d}x$, where $α> d + 1$ and $\varrho :{{\mathbb R}^d} \to \left( {0, + \infty } \right)$ is a continuous function and uniformly convex (${\nabla ^2}\varrho \ge {\varepsilon_0}{\rm {Id}}$). By using Stein kernels for $\left( {α- d} \right)$-moment measures, we prove that the rates of convergence in the central limit theorem with sequence of i.i.d. random variables ${X_1},{X_2},...,{X_n}$ of the law $μ$, to be of form $c_{{\varepsilon}_0}\,\sqrt {\dfrac{d}{n}} $. The general case (i.e., $\varrho$ is only convex and continuous) remains open.

math.PR↗

arXiv2020

A Cucker-Smale inspired deterministic mean field game with velocity interactions

We introduce a mean field game model for pedestrians moving in a given domain and choosing their trajectories so as to minimize a cost including a penalization on the difference between their own velocity and that of the other agents they meet. We prove existence of an equilibrium in a Lagrangian setting by using its variational structure, and then study its properties and regularity.

math.AP↗

arXiv2020

q-Moment Measures and Applications: A New Approach via Optimal Transport

In 2017, Bo'az Klartag obtained a new result in differential geometry on the existence of affine hemisphere of elliptic type. In his approach, a surface is associated with every a convex function $Φ$ : R^n $\rightarrow$ (0, +$\infty$) and the condition for the surface to be an affine hemisphere involves the 2-moment measure of $Φ$ (a particular case of q-moment measures, i.e measures of the form ($\nabla$$Φ$) \# ($Φ$^{--(n+q)}) for q > 0). In Klartag's paper, q-moment measures are studied through a variational method requiring to minimize a functional among convex functions, which is studied using the Borell-Brascamp-Lieb inequality. In this paper, we attack the same problem through an optimal transport approach, since the convex function $Φ$ is a Kantorovich potential (as already done for moment measures in a previous paper). The variational problem in this new approach becomes the minimization of a local functional and a transport cost among probability measures and the optimizer turns out to be of the form $ρ$ = $Φ$^{--(n+q)}.

math.AP↗

arXiv2020

Lipschitz estimates on the JKO scheme for the Fokker-Plack equation on bounded convex domains

Given a semi-convex potential V on a convex and bounded domain $Ω$, we consider the Jordan-Kinderlehrer-Otto scheme for the Fokker-Planck equation with potential V, which defines, for fixed time step $τ$ > 0, a sequence of densities $ρ$ k $\in$ P($Ω$). Supposing that V is $α$-convex, i.e. D 2 V $\ge$ $α$I, we prove that the Lipschitz constant of log $ρ$ + V satisfies the following inequality: Lip(log($ρ$ k+1) + V)(1 + $α$$τ$) $\le$ Lip(log($ρ$ k) + V). This provides exponential decay if $α$ > 0, Lipschitz bounds on bounded intervals of time, which is coherent with the results on the continuous-time equation, and extends a previous analysis by Lee in the periodic case.

math.AP↗

arXiv2020

Lagrangian discretization of crowd motion and linear diffusion

We study a model of crowd motion following a gradient vector field, with possibly additional interaction terms such as attraction/repulsion, and we present a numerical scheme for its solution through a Lagrangian discretization. The density constraint of the resulting particles is enforced by means of a partial optimal transport problem at each time step. We prove the convergence of the discrete measures to a solution of the continuous PDE describing the crowd motion in dimension one. In a second part, we show how a similar approach can be used to construct a Lagrangian discretization of a linear advection-diffusion equation, interpreted as a gradient flow in Wasserstein space. We provide also a numerical implementation in 2D to demonstrate the feasibility of the computations.

math.NA↗

arXiv2019

JKO estimates in linear and non-linear Fokker-Planck equations, and Keller-Segel: L p and Sobolev bounds

We analyze some parabolic PDEs with different drift terms which are gradient flows in the Wasserstein space and consider the corresponding discrete-in-time JKO scheme. We prove with optimal transport techniques how to control the L p and L $\infty$ norms of the iterated solutions in terms of the previous norms, essentially recovering well-known results obtained on the continuous-in-time equations. Then we pass to higher order results, and in particulat to some specific BV and Sobolev estimates, where the JKO scheme together with the so-called "five gradients inequality" allows to recover some inequalities that can be deduced from the Bakry-Emery theory for diffusion operators, but also to obtain some novel ones, in particular for the Keller-Segel chemiotaxis model. 1 Short introduction The goal of this paper is to present some estimates on evolution PDEs in the space of probability densities which share two important features: they include a linear diffusion term, and they are gradient flows in the Wasserstein space W2. These PDEs will be of the form $\partial$t$ρ$ -- $Δ$$ρ$ -- $\nabla$ $\times$ ($ρ$$\nabla$u[$ρ$]) = 0, complemented with no-flux boundary conditions and an intial condition on $ρ$0. We will in particular concentrate on the Fokker-Plack case, where u[$ρ$] = V and V is a fixed function (with possible regularity assumptions) independent of $ρ$, on the case where u[$ρ$] = W * $ρ$ is obtained by convolution and models interaction between particles, and on the parabolic-elliptic Keller-Segel case where u[$ρ$] is related to $ρ$ via an elliptic equation. This last case models the evolution of a biological population $ρ$ subject to diffusion but attracted by the concentration of a chemo-attractant, a nutrient which is produced by the population itself, so that its distribution is ruled by a PDE where the density $ρ$ appears as a source term. Under the assumption that the production rate of this nutrient is much faster than the motion of the cells, we can assume that its distribution is ruled by a statical PDE with no explicit time-dependence, and gives rise to a system which is a gradient flow in the variable $ρ$ (the parabolic-parabolic case, where the time scale for the cells and for the nutrient are comparable, is also a gradient flow, in the product space W2 x L 2 , but we will not consider this case). Since we mainly concentrate on the case of bounded domains, in the Keller-Segel case the term u[$ρ$] cannot be expressed as a convoluton and requires ad-hoc computations. In all the paper, the estimates will be studied on a time-discretized version of these PDEs, consisting in the so-called JKO (Jordan-Kinderleherer-Otto) scheme, based on iterated optimization problems involving the Wasserstein distance W2. We will first present 0-order estimates, on the L p and L $\infty$ norms of the solution. This is just a translation into the JKO language of well-known properties of these equations. The main goal of this part is hence to

math.AP↗

arXiv2019

New estimates on the regularity of the pressure in density-constrained Mean Field Games

We consider variational Mean Field Games 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 is augmented by a pressure effect, i.e. 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 M{é}sz{á}ros. We prove H1 and L infinity 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.

math.AP↗

arXiv2019

Minimal-time mean field games

This paper considers a mean field game model inspired by crowd motion where agents want to leave a given bounded domain through a part of its boundary in minimal time. Each agent is free to move in any direction, but their maximal speed is bounded in terms of the average density of agents around their position in order to take into account congestion phenomena. After a preliminary study of the corresponding minimal-time optimal control problem, we formulate the mean field game in a Lagrangian setting and prove existence of Lagrangian equilibria using a fixed point strategy. We provide a further study of equilibria under the assumption that agents may leave the domain through the whole boundary, in which case equilibria are described through a system of a continuity equation on the distribution of agents coupled with a Hamilton--Jacobi equation on the value function of the optimal control problem solved by each agent. This is possible thanks to the semiconcavity of the value function, which follows from some further regularity properties of optimal trajectories obtained through Pontryagin Maximum Principle. Simulations illustrate the behavior of equilibria in some particular situations.

math.OC↗

arXiv2018

Weighted ultrafast diffusion equations: from well-posedness to long-time behaviour

In this paper we devote our attention to a class of weighted ultrafast diffusion equations arising from the problem of quantisation for probability measures. These equations have a natural gradient flow structure in the space of probability measures endowed with the quadratic Wasserstein distance. Exploiting this structure, in particular through the so-called JKO scheme, we introduce a notion of weak solutions, prove existence, uniqueness, BV and H^1 estimates, L^1 weighted contractivity, Harnack inequalities, and exponential convergence to a steady state.

math.AP↗

arXiv2018

$L^p$ bounds for boundary-to-boundary transport densities, and $W^{1,p}$ bounds for the BV least gradient problem in 2D

The least gradient problem (minimizing the total variation with given boundary data) is equivalent, in the plane, to the Beckmann minimal-flow problem with source and target measures located on the boundary of the domain, which is in turn related to an optimal transport problem. Motivated by this fact, we prove L p summability results for the solution of the Beckmann problem in this setting, which improve upon previous results where the measures were themselves supposed to be L p. In the plane, we carry out all the analysis for general strictly convex norms, which requires to first introduce the corresponding optimal transport tools. We then obtain results about the W 1,p regularity of the solution of the anisotropic least gradient problem in uniformly convex domains.

math.OC↗

arXiv2017

A fractal shape optimization problem in branched transport

We investigate the following question: what is the set of unit volume which can be best irrigated starting from a single source at the origin, in the sense of branched transport? We may formulate this question as a shape optimization problem and prove existence of solutions, which can be considered as a sort of "unit ball" for branched transport. We establish some elementary properties of optimizers and describe these optimal sets A as sublevel sets of a so-called landscape function which is now classical in branched transport. We prove $β$-H{ö}lder regularity of the landscape function, allowing us to get an upper bound on the Minkowski dimension of the boundary: dim $\partial$A $\le$ d -- $β$ (where $β$ := d($α$ -- (1 -- 1/d)) $\in$ (0, 1) is a relevant exponent in branched transport, associated with the exponent $α$ > 1 -- 1/d appearing in the cost). We are not able to prove the upper bound, but we conjecture that $\partial$A is of non-integer dimension d -- $β$. Finally, we make an attempt to compute numerically an optimal shape, using an adaptation of the phase-field approximation of branched transport introduced some years ago by Oudet and the second author.

math.OC↗

arXiv2017

Splitting Schemes & Segregation In Reaction-(Cross-)Diffusion Systems

One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, i.e. population densities with disjoint supports. We analyse such a reaction cross-diffusion system. In order to prove existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum.

math.AP↗

arXiv2017

Spatial rational expectations equilibria in the Ramsey model of optimal growth

It is the aim of this work provide a rigorous treatment concerning the formation of spatial rational expectations equlibria in a general class of spatial economic models under the effect of externalities, using techniques from the calculus of variations. Using detailed estimates for a para-metric optimisation problem, the existence of spatial rational expectations equilibria is proved and they are characterised in terms of a nonlocal Euler-Lagrange equation.

math.OC↗

arXiv2017

L^$\infty$ estimates for the jko scheme in parabolic-elliptic keller-segel systems

We prove L^$\infty$ estimates on the densities that are obtained via the JKO scheme for a general form of a parabolic-elliptic Keller-Segel type system, with arbitrary diffusion, arbitrary mass, and in arbitrary dimension. Of course, such an estimate blows up in finite time, a time proportional to the inverse of the initial L^$\infty$ norm. This estimate can be used to prove short-time well-posedness for a number of equations of this form regardless of the mass of the initial data. The time of existence of the constructed solutions coincides with the maximal time of existence of Lagrangian solutions without the diffusive term by characteristic methods.

math.AP↗