arXiv subjects

Guillaume Carlier

Publications and source records attributed to Guillaume Carlier.

50 records · Page 3Linked to original sources

arXiv2014

A $Γ$-Convergence Result for the Upper Bound Limit Analysis of Plates

Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have proposed to use various finite elements discretizations. We provide in this paper a mathematical analysis which ensures the convergence of the finite element method, even with finite elements with discontinuous derivatives such as the quadratic 6 node Lagrange triangles and the cubic Hermite triangles. More precisely, we prove the $Γ$-convergence of the discretized problems towards the continuous limit analysis problem. Numerical results illustrate the relevance of this analysis for the yield design of both homogeneous and non-homogeneous materials.

math.NA↗

arXiv2014

Discretization of functionals involving the Monge-Ampère operator

Gradient flows in the Wasserstein space have become a powerful tool in the analysis of diffusion equations, following the seminal work of Jordan, Kinderlehrer and Otto (JKO). The numerical applications of this formulation have been limited by the difficulty to compute the Wasserstein distance in dimension >= 2. One step of the JKO scheme is equivalent to a variational problem on the space of convex functions, which involves the Monge-Ampère operator. Convexity constraints are notably difficult to handle numerically, but in our setting the internal energy plays the role of a barrier for these constraints. This enables us to introduce a consistent discretization, which inherits convexity properties of the continuous variational problem. We show the effectiveness of our approach on nonlinear diffusion and crowd-motion models.

math.NA↗

arXiv2014

Remarks on existence and uniqueness of Cournot-Nash equilibria in the non-potential case

This article is devoted to various methods (optimal transport, fixed-point, ordinary differential equations) to obtain existence and/or uniqueness of Cournot-Nash equilibria for games with a continuum of players with both attractive and repulsive effects. We mainly address separable situations but for which the game does not have a potential. We also present several numerical simulations which illustrate the applicability of our approach to compute Cournot-Nash equilibria.

math.OC↗

arXiv2014

From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem

The notion of Nash equilibria plays a key role in the analysis of strategic interactions in the framework of $N$ player games. Analysis of Nash equilibria is however a complex issue when the number of players is large. In this article we emphasize the role of optimal transport theory in: 1) the passage from Nash to Cournot-Nash equilibria as the number of players tends to infinity, 2) the analysis of Cournot-Nash equilibria.

math.AP↗

arXiv2014

Optimal regions for congested transport

We consider a given region $Ω$ where the traffic flows according to two regimes: in a region $C$ we have a low congestion, where in the remaining part $Ω\setminus C$ the congestion is higher. The two congestion functions $H_1$ and $H_2$ are given, but the region $C$ has to be determined in an optimal way in order to minimize the total transportation cost. Various penalization terms on $C$ are considered and some numerical computations are shown.

math.OC↗

arXiv2013

Optimal spatial pricing strategies with transportation costs

We consider an optimization problem in a given region $Q$ where an agent has to decide the price $p(x)$ of a product for every $x\in Q$. The customers know the pricing pattern $p$ and may shop at any place $y$, paying the cost $p(y)$ and additionally a transportation cost $c(x,y)$ for a given transportation cost function $c$. We will study two models: the first one where the agent operates everywhere on $Q$ and a second one where the agent operates only in a subregion. For both models we discuss the mathematical framework and we obtain an existence result for a pricing strategy which maximizes the total profit of the agent. We also present some particular cases where more detailed computations can be made, as the case of concave costs, the case of quadratic cost, and the onedimensional case. Finally we discuss possible extensions and developments, as for instance the case of Nash equilibria when more agents operate on the same market.

math.OC↗

arXiv2012

Optimal transport and Cournot-Nash equilibria

We study a class of games with a continuum of players for which Cournot-Nash equilibria can be obtained by the minimisation of some cost, related to optimal transport. This cost is not convex in the usual sense in general but it turns out to have hidden strict convexity properties in many relevant cases. This enables us to obtain new uniqueness results and a characterisation of equilibria in terms of some partial differential equations, a simple numerical scheme in dimension one as well as an analysis of the inefficiency of equilibria.

math.AP↗

arXiv2012

Geodesics for a class of distances in the space of probability measures

In this paper, we study the characterization of geodesics for a class of distances between probability measures introduced by Dolbeault, Nazaret and Savar e. We first prove the existence of a potential function and then give necessary and suffi cient optimality conditions that take the form of a coupled system of PDEs somehow similar to the Mean-Field-Games system of Lasry and Lions. We also consider an equivalent formulation posed in a set of probability measures over curves.

math.OC↗

arXiv2011

Pareto efficiency for the concave order and multivariate comonotonicity

In this paper, we focus on efficient risk-sharing rules for the concave dominance order. For a univariate risk, it follows from a comonotone dominance principle, due to Landsberger and Meilijson [25], that efficiency is characterized by a comonotonicity condition. The goal of this paper is to generalize the comonotone dominance principle as well as the equivalence between efficiency and comonotonicity to the multi-dimensional case. The multivariate setting is more involved (in particular because there is no immediate extension of the notion of comonotonicity) and we address it using techniques from convex duality and optimal transportation.

math.OC↗

arXiv2011

Exponential convergence for a convexifying equation and a non-autonomous gradient flow for global minimization

We consider an evolution equation similar to that introduced by Vese and whose solution converges in large time to the convex envelope of the initial datum. We give a stochastic control representation for the solution from which we deduce, under quite general assumptions that the convergence in the Lipschitz norm is in fact exponential in time. We then introduce a non-autonomous gradient flow and prove that its trajectories all converge to minimizers of the convex envelope.

math.AP↗

arXiv2009

A strategy for non-strictly convex transport costs and the example of ||x-y||p in R2

This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non strictly convex cost. We give a decomposition strategy to address this issue. As part of our strategy, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. As an illustration of our strategy, we prove existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form h(||x-y||) with h strictly convex increasing and ||. || an arbitrary norm in \R2.

math.CA↗

arXiv2009

Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory

This article is devoted to the optimal control of state equations with memory of the form: ?[x(t) = F(x(t),u(t), \int_0^{+\infty} A(s) x(t-s) ds), t>0, with initial conditions x(0)=x, x(-s)=z(s), s>0.]Denoting by $y_{x,z,u}$ the solution of the previous Cauchy problem and: \[v(x,z):=\inf_{u\in V} \{\int_0^{+\infty} e^{-λs} L(y_{x,z,u}(s), u(s))ds \}\] where $V$ is a class of admissible controls, we prove that $v$ is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form: \[λv(x,z)+H(x,z,\nabla_x v(x,z))+D_z v(x,z), \dot{z} >=0\] in the sense of the theory of viscosity solutions in infinite-dimensions of M. Crandall and P.-L. Lions.

math.OC↗

arXiv2008

From Knothe's transport to Brenier's map and a continuation method for optimal transport

A simple procedure to map two probability measures in $\mathbb{R}^d$ is the so-called \emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem.

math.OC↗

arXiv2006

Optimal transportation for the determinant

Among $\R^3$-valued triples of random vectors $(X,Y,Z)$ having fixed marginal probability laws, what is the best way to jointly draw $(X,Y,Z)$ in such a way that the simplex generated by $(X,Y,Z)$ has maximal average volume? Motivated by this simple question, we study optimal transportation problems with several marginals when the objective function is the determinant or its absolute value.

math.OC↗