CARLIER Guillaume

The objective of this thesis is to propose variational methods for the mathematical and numerical analysis of a class of HJ equations. The metric character of these equations allows to characterize the set of sub-solutions, namely, they are 1-Lipschitz with respect to the Finslerian distance associated to the Hamiltonian. Equivalently, this is equivalent to saying that the gradient of these functions belongs to a certain Finslerian ball. The solution we are looking for is the maximal subsolution, which can be described by a Hopf-Lax type formula, which solves a maximization problem with constraint on the gradient. We derive an associated dual problem involving the total Finslerian variation of vector measures with divergent constraint. We exploit the saddle-point structure to propose a numerical solution with the augmented Lagrangian method. This characterization of the HJ equation also shows the link with optimal transport problems to/from the edge. This link with optimal mass transport leads us to generalize the Evans-Gangbo approach. Indeed, we show that the maximal subsolution of the HJ equation is obtained by stretching p→∞ in a class of Finsler-type p-Laplacians with obstacles on the edge. This also allows us to construct the optimal flow for the associated Beckmann problem. Among the applications we look at is the Shape from Shading problem, which involves reconstructing the surface of a 3D object from a grayscale image of that object.

The aim of this short note is to give an elementary proof of linear convergence of the Sinkhorn algorithm for the entropic regularization of multi-marginal optimal transport. The proof simply relies on: i) the fact that Sinkhorn iterates are bounded, ii) strong convexity of the exponential on bounded intervals and iii) the convergence analysis of the coordinate descent (Gauss-Seidel) method of Beck and Tetruashvili [1].

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced in [5] as a regularization of Wasserstein barycenters [1]. After characterizing these barycenters in terms of a system of Monge-Ampère equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters.

We propose a (toy) MFG model for the evolution of residents and firms densities, coupled both by labour market equilibrium conditions and competition for land use (congestion). This results in a system of two Hamilton-Jacobi-Bellman and two Fokker-Planck equations with a new form of coupling related to optimal transport. This MFG has a convex potential which enables us to find weak solutions by a variational approach. In the case of quadratic Hamiltonians, the problem can be reformulated in Lagrangian terms and solved numerically by an IPFP/Sinkhorn-like scheme as in [4]. We present numerical results based on this approach, these simulations exhibit different behaviours with either agglomeration or segregation dominating depending on the initial conditions and parameters.

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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.

This thesis proposes models and methods to study risk control in large financial systems. In the first part, we propose a structural approach: we consider a financial system represented as a network of institutions connected to each other by strategic interactions that are sources of funding but also by interactions that expose them to default contagion risk. The novelty of our approach lies in the fact that these two types of interactions interfere. We propose new notions of equilibrium for these systems and study the optimal connectivity of the network and the associated systemic risk. In a second part, we introduce systemic risk measures defined by backward stochastic differential equations directed by mean-field operators and study associated optimal stopping problems. The last part deals with optimal portfolio liquidation issues.

Motivated by some variational problems subject to a convexity constraint, we consider an approximation using the logarithm of the Hessian determinant as a barrier for the constraint. We show that the minimizer of this penalization can be approached by solving a second boundary value problem for Abreu's equation which is a well-posed nonlinear fourth-order elliptic problem. More interestingly, a similar approximation result holds for the initial constrained variational problem.

Taking advantage of the Benamou-Brenier dynamic formulation of optimal transport, we propose a convex formulation for each step of the JKO scheme for Wasserstein gradient flows which can be attacked by an augmented Lagrangian method which we call the ALG2-JKO scheme. We test the algorithm in particular on the porous medium equation. We also consider a semi implicit variant which enables us to treat nonlocal interactions as well as systems of interacting species. Regarding systems, we can also use the ALG2-JKO scheme for the simulation of crowd motion models with several species.

We introduce a new, and elementary, approximation method for bilevel optimization problems motivated by Stackelberg leader-follower games. Our technique is based on the notion of two-scale Gibbs measures. The first scale corresponds to the cost function of the follower and the second scale to that of the leader. We explain how to choose the weights corresponding to these two scales under very general assumptions and establish rigorous Γ-convergence results. An advantage of our method is that it is applicable both to optimistic and to pessimistic bilevel problems.

We propose an entropy minimization viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We carefully analyze 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.

This thesis deals with the study of the long time behavior of potential mean field games (MFG), independently of the convexity of the associated minimization problem. For the finite dimensional Hamiltonian system, similar problems have been treated by the weak KAM theory. We transpose many results of this theory to the context of potential mean field games. First, we characterize by ergodic approximation the boundary value associated with finite horizon MFG systems. We provide explicit examples in which this value is strictly greater than the energy level of the stationary solutions of the ergodic MFG system. This implies that the optimal trajectories of finite horizon MFG systems cannot converge to stationary configurations. Then, we prove the convergence of the minimization problem associated with finite horizon MFG to a solution of the critical Hamilton-Jacobi equation in the space of probability measures. Moreover, we show a mean field limit for the ergodic constant associated to the corresponding finite dimensional Hamilton-Jacobi equation. In the last part, we characterize the limit of the infinite horizon minimization problem that we used for the ergodic approximation in the first part of the manuscript.

We deal with a severe ill posed problem, namely the reconstruction process of an image during tomography acquisition with (very) few views. We present different methods that we have been investigated during the past decade. They are based on variational analysis. This is a survey paper and we refer to the quoted papers for more details. Mathematics Subject Classification (2010). 49K40, 45Q05,65M32.

Starting from Brenier's relaxed formulation of the incompress-ible 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 regular-ization with the so-called Bredinger entropic interpolation problem (see [1]). Numerical results in dimension one and two illustrate the feasibility of the method.

Motivated by some variational problems subject to a convexity constraint, we consider an approximation using the logarithm of the Hessian determinant as a barrier for the constraint. We show that the minimizer of this penalization can be approached by solving a second boundary value problem for Abreu's equation which is a well-posed nonlinear fourth-order elliptic problem. More interestingly, a similar approximation result holds for the initial constrained variational problem.

We develop an elementary and self-contained differential approach, in an L ∞ setting, for well-posedness (existence, uniqueness and smooth dependence with respect to the data) for the multi-marginal Schrödinger system which arises in the entropic regularization of optimal transport problems.

We study the Wasserstein distance between two measures µ, ν which are mutually singular. In particular, we are interested in minimization problems of the form W (µ, A) = inf W (µ, ν) : ν ∈ A where µ is a given probability and A is contained in the class µ ⊥ of probabilities that are singular with respect to µ. Several cases for A are considered. in particular, when A consists of L 1 densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problem min P (B) + kW (A, B) : |A ∩ B| = 0, |A| = |B| = 1 , where k > 0 is a fixed constant, P (A) is the perimeter of A, and both sets A, B may vary.

Starting from Brenier's relaxed formulation of the incompress-ible 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 regular-ization with the so-called Bredinger entropic interpolation problem (see [1]). Numerical results in dimension one and two illustrate the feasibility of the method.

We deal with a severe ill posed problem, namely the reconstruction process of an image during tomography acquisition with (very) few views. We present different methods that we have been investigated during the past decade. They are based on variational analysis. This is a survey paper and we refer to the quoted papers for more details. Mathematics Subject Classification (2010). 49K40, 45Q05,65M32.

In this thesis we study various aspects of martingale optimal transport in dimension greater than one, from duality to local structure, and finally propose methods of numerical approximation.We first prove the existence of irreducible components intrinsic to martingale transports between two given measures, and the canonicity of these components. We then prove a duality result for the optimal martingale transport in any dimension, the point by point duality is no longer true but a form of quasi-safe duality is proved. This duality allows us to prove the possibility of decomposing the quasi-safe optimal transport into a series of subproblems of point by point optimal transports on each irreducible component. We finally use this duality to prove a martingale monotonicity principle, analogous to the famous monotonicity principle of classical optimal transport. We then study the local structure of optimal transports, deduced from differential considerations. We obtain a characterization of this structure using real algebraic geometry tools. We deduce the structure of martingale optimal transports in the case of Euclidean norm power costs, thus solving a conjecture dating back to 2015. Finally, we compare existing numerical methods and propose a new method that is shown to be more efficient and to deal with an intrinsic problem of the martingale constraint that is the convex order defect. We also give techniques to handle the numerical problems in practice.

We study equations for the anisotropic p-Laplacian when some components of the p-vector are equal to 1.

We consider a bilevel optimization framework corresponding to a monopoly spatial pricing problem: the price for a set of given facilities maximizes the profit (upper level problem) taking into account that the demand is determined by consumers' cost minimization (lower level problem). In our model, both transportation costs and congestion costs are considered, and the lower level problem is solved via partial transport mass theory. The partial transport aspect of the problem comes from the fact that each consumer has the possibility to remain out of the market. We also generalize the model and our variational analysis to the stochastic case where utility involves a random term.

We study the Wasserstein distance between two measures µ, ν which are mutually singular. In particular, we are interested in minimization problems of the form W (µ, A) = inf W (µ, ν) : ν ∈ A where µ is a given probability and A is contained in the class µ ⊥ of probabilities that are singular with respect to µ. Several cases for A are considered. in particular, when A consists of L 1 densities bounded by a constant, the optimal solution is given by the characteristic function of a domain. Some regularity properties of these optimal domains are also studied. Some numerical simulations are included, as well as the double minimization problem min P (B) + kW (A, B) : |A ∩ B| = 0, |A| = |B| = 1 , where k > 0 is a fixed constant, P (A) is the perimeter of A, and both sets A, B may vary.

We study the JKO scheme for the total variation, characterize the optimizers, prove some of their qualitative properties (in particular a sort of maximum principle and the regularity of level sets). We study in detail the case of step functions. Finally, in dimension one, we establish convergence as the time step goes to zero to a solution of a fourth-order nonlinear evolution equation.

We consider a class of doubly nonlinear constrained evolution equations which may be viewed as a nonlinear extension of the growing sandpile model of [15]. We prove existence of weak solutions for quite irregular sources by a semi-implicit scheme in the spirit of the seminal works of [13] and [14] but with the 1-Wasserstein distance instead of the quadratic one. We also prove an L 1-contraction result when the source is L 1 and deduce uniqueness and stability in this case.

During the last 20 years, optimal transport theory has proven to be an efficient tool to study the asymptotic behavior of diffusion equations, to prove functional inequalities and to extend geometric properties in extremely general spaces such as measured metric spaces, etc. The curvature-dimensional condition of the Bakry-Emery theory appears to be the cornerstone of these applications. The curvature-dimension condition of the Bakry-Emery theory appears as the cornerstone of these applications. It suffices to think of the simplest and most important case of the Wasserstein quadratic distance W2 : the contraction of the heat flow in W2 characterizes the uniform lower bounds for the Ricci curvature . the Talagrand inequality of the transport, comparing W2 to the relative entropy is implied and implies, by the HWI inequality, the log-Sobolev inequality . McCann geodesics in Wasserstein space (P2(Rn),W2) allow to prove important functional properties such as convexity, and standard functional inequalities such as isoperymmetry, measure concentration properties, Prekopa-Leindler inequality and so on. Nevertheless, the lack of regularity of the minimization schemes requires non-smooth analysis arguments. Schrödinger's problem is an entropy minimization problem with fixed marginal constraints and reference process. From large deviation theory, when the reference process is Brownian motion, its minimum value A converges to W2 when the temperature is zero. Entropic interpolations, solutions of the Schrödinger problem, are characterized in terms of Markov semigroups, which naturally implies Γ2 calculations and the curvature-dimension condition. Dating back to the 1930s and neglected for decades, the Schrodinger problem has been gaining popularity in recent years in various fields, thanks to its relation to optimal transport, the regularity of its solutions, and other powerful properties in numerical calculations. The aim of this work is twofold. First, we study some analogies between the Schrödinger problem and the optimal transport providing new proofs of the dual Kantorovich formulation and the dynamic Benamou-Brenier formulation for the entropy cost A. Then, as an application of these connections, we derive some functional properties and inequalities under curvature-dimension conditions. In particular, we prove the concavity of the exponential entropy along the entropy interpolations under the curvature-dimension condition CD(0, n) and the regularity of the entropy cost along the heat flow. We also give various proofs of the evolutionary variational inequality for A and the contraction of the heat flow in A, recovering as a limiting case, the classical results in W2 under CD(κ,∞) and CD(0, n). Finally, we propose a simple proof of the Gaussian concentration property via the Schrödinger problem as an alternative to classical arguments such as the Marton argument based on optimal transport.

No summary available.

Barycenters in Wasserstein space are a natural way to interpolate between several probability measures, useful in various applied domains such as image processing or sta-tistic learning. We conjecture that these barycenters obey a central limit theorem which we prove in some (very) particular cases.

Replacing positivity constraints by an entropy barrier is popular to approximate solutions of linear programs. In the special case of the optimal transport problem, this technique dates back to the early work of Schr\"odinger. This approach has recently been used successfully to solve optimal transport related problems in several applied fields such as imaging sciences, machine learning and social sciences. The main reason for this success is that, in contrast to linear programming solvers, the resulting algorithms are highly parallelizable and take advantage of the geometry of the computational grid (e.g. an image or a triangulated mesh). The first contribution of this article is the proof of the Γ-convergence of the entropic regularized optimal transport problem towards the Monge-Kantorovich problem for the squared Euclidean norm cost function. This implies in particular the convergence of the optimal entropic regularized transport plan towards an optimal transport plan as the entropy vanishes. Optimal transport distances are also useful to define gradient flows as a limit of implicit Euler steps according to the transportation distance. Our second contribution is a proof that implicit steps according to the entropic regularized distance converge towards the original gradient flow when both the step size and the entropic penalty vanish (in some controlled way).

This paper is a brief presentation of those Mean Field Games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochastic framework (i.e. with diffusion) is also presented both in the stationary and dynamic case. The variational problems relevant for MFG are described via Eulerian and Lagrangian languages, and the connection with equilibria is explained by means of convex duality and of optimality conditions. The convex structure of the problem also allows for efficient numerical treatment, based on Augmented Lagrangian Algorithms, and some new simulations are shown at the end of the paper.

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No summary available.

We prove an existence result for nonlinear diffusion equations in the presence of a nonlocal density-dependent drift which is not necessarily potential. The proof is constructive and based on the Helmholtz decomposition of the drift and a splitting scheme. The splitting scheme combines transport steps by the divergence-free part of the drift and semi-implicit minimization steps à la Jordan-Kinderlherer Otto to deal with the potential part.

No summary available.

This paper is a brief presentation of those Mean Field Games with congestion penalization which have a variational structure, starting from the deterministic dynamical framework. The stochastic framework (i.e. with diffusion) is also presented both in the stationary and dynamic case. The variational problems relevant for MFG are described via Eulerian and Lagrangian languages, and the connection with equilibria is explained by means of convex duality and of optimality conditions. The convex structure of the problem also allows for efficient numerical treatment, based on Augmented Lagrangian Algorithms, and some new simulations are shown at the end of the paper.

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 $\Gamma$-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.

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.

Monge's problem with a Finsler cost is intimately related to an optimal flow problem. Discretization of this problem and its dual leads to a well-posed finite-dimensional saddle-point problem which can be solved numerically relatively easily by an augmented Lagrangian approach in the same spirit as the Benamou-Brenier method for the optimal transport problem with quadratic cost. Numerical results validate the method. We also emphasize that the algorithm only requires elementary operations and in particular never involves evaluation of the Finsler distance or of geodesics.

No summary available.

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.

In this chapter, we present a numerical method, based on iterative Bregman projections, to solve the optimal transport problem with Coulomb cost. This problem 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.

Barycenters in Wasserstein space are a natural way to interpolate between several probability measures, useful in various applied domains such as image processing or sta-tistic learning. We conjecture that these barycenters obey a central limit theorem which we prove in some (very) particular cases.

This thesis presents three main research topics, the first two being independent and the last one indicating the relation of the first two problems in a concrete case.In the first part we focus on the martingale optimal transport problem in Skorokhod space, whose first goal is to study systematically the tension of martingale transport schemes. We first focus on the upper semicontinuity of the primal problem with respect to the marginal distributions. Using the S-topology introduced by Jakubowski, we derive the upper semicontinuity and show the first duality. We also give two dual problems concerning the robust overcoverage of an exotic option, and we establish the corresponding dualities, by adapting the principle of dynamic programming and the discretization argument initiated by Dolinsky and Soner.The second part of this thesis deals with the optimal Skorokhod folding problem. We first formulate this optimization problem in terms of probability measures on an extended space and its dual problems. Using the classical duality. convex approach and the optimal stopping theory, we obtain the duality results. We also relate these results to martingale optimal transport in the space of continuous functions, from which the corresponding dualities are derived for a particular class of payment functions. Next, we provide an alternative proof of the monotonicity principle established by Beiglbock, Cox and Huesmann, which allows us to characterize optimizers by their geometric support. We show at the end a stability result which contains two parts: the stability of the optimization problem with respect to the target marginals and the connection with another problem of the optimal folding.The last part concerns the application of stochastic control to the martingale optimal transport with the local time dependent payoff function, and to the Skorokhod folding. For the case of one marginal, we find the optimizers for the primal and dual problems via the Vallois solutions, and consequently show the optimality of the Vallois solutions, which includes the optimal martingale transport and the optimal Skorokhod folding. For the case of two marginals, we obtain a generalization of the Vallois solution. Finally, a special case of several marginals is studied, where the stopping times given by Vallois are well ordered.

Since the seminal paper by Jordan, Kinderlehrer and Otto in 1998, it is well known that a large class of parabolic equations can be seen as gradient flows in Wasserstein space. The aim of this thesis is to extend this theory to some equations and systems which do not have exactly a gradient flow structure. The interactions studied are of different natures. The first chapter deals with systems with non local interactions in the drift. We then study cross-diffusion systems that apply to congestion models for several populations. Another model studied is the one where the coupling is in the reaction term like prey-predator systems with diffusion or tumor growth models. Finally, we will study new types of systems where the interaction is given by a multi-margin transport problem. A large part of these problems is illustrated by numerical simulations.

We consider a bilevel optimization framework corresponding to a monopoly spatial pricing problem: the price for a set of given facilities maximizes the profit (upper level problem) taking into account that the demand is determined by consumers' cost minimization (lower level problem). In our model, both transportation costs and congestion costs are considered, and the lower level problem is solved via partial transport mass theory. The partial transport aspect of the problem comes from the fact that each consumer has the possibility to remain out of the market. We also generalize the model and our variational analysis to the stochastic case where utility involves a random term.

In this thesis, our goal is to give a general numerical framework to approximate the solutions of optimal transport (OT) problems. The general idea is to introduce an entropy regularization of the initial problem. The regularized problem corresponds to minimize a relative entropy with respect to a given reference measure. Indeed, this is equivalent to finding the projection of a coupling with respect to the Kullback-Leibler divergence. This allows us to use the Bregman/Dykstra algorithm and to solve several variational problems related to TO. We are particularly interested in the solution of multi-marginal optimal transport (MMOT) problems that arise in the context of fluid dynamics (incompressible Euler equations à la Brenier) and quantum physics (the density functional theory). In these cases, we show that entropy regularization plays a more important role than simple numerical stabilization. Moreover, we give results concerning the existence of optimal transports (e.g. fractal transports) for the TOMM problem.

The principal-agent problem in economics leads to variational problems subject to global constraints of $b$-convexity on the admissible functions, capturing the so-called incentive-compatibility constraints. Typical examples are minimization problems subject to a convexity constraint. In a recent pathbreaking article, Figalli et al. (J Econ Theory 146(2):454–478, 2011) identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over $b$-convex functions and show how they can be solved numerically using Dykstra’s iterated projection algorithm to handle the $b$-convexity constraint in the framework of (Figalli et al. in J Econ Theory 146(2):454–478, 2011). Our method also turns out to be simple for convex envelope computations.

No summary available.

We prove an existence result for nonlinear diffusion equations in the presence of a nonlocal density-dependent drift which is not necessarily potential. The proof is constructive and based on the Helmholtz decomposition of the drift and a splitting scheme. The splitting scheme combines transport steps by the divergence-free part of the drift and semi-implicit minimization steps à la Jordan-Kinderlherer Otto to deal with the potential part.

We consider a one-dimensional kinetic model of granular media in the case where the interaction potential is quadratic. Taking advantage of a simple first integral, we can use a reformulation (equivalent to the initial kinetic model for classical solutions) which allows measure solutions. This reformulation has a Wasserstein gradient flow structure (on a possibly infinite product of spaces of measures) for a convex energy which enables us to prove global in time well-posedness.

Taking advantage of the Benamou-Brenier dynamic formulation of optimal transport, we propose a convex formulation for each step of the JKO scheme for Wasserstein gradient flows which can be attacked by an augmented Lagrangian method which we call the ALG2-JKO scheme. We test the algorithm in particular on the porous medium equation. We also consider a semi implicit variant which enables us to treat nonlocal interactions as well as systems of interacting species. Regarding systems, we can also use the ALG2-JKO scheme for the simulation of crowd motion models with several species.

This paper is devoted to existence and uniqueness results for classes of nonlinear diffusion equations (or systems) which may be viewed as regular perturbations of Wasserstein gradient flows. First, in the case where the drift is a gradient (in the physical space), we obtain existence by a semi-implicit Jordan-Kinderlehrer-Otto scheme. Then, in the nonpotential case, we derive existence from a regularization procedure and parabolic energy estimates. We also address the uniqueness issue by a displacement convexity argument.

The principal-agent problem in economics leads to variational problems subject to global constraints of b-convexity on the admissible functions, capturing the so-called incentive-compatibility constraints. Typical examples are minimization problems subject to a convexity constraint. In a recent pathbreaking article, Fi-galli, Kim and McCann [19] identified conditions which ensure convexity of the principal-agent problem and thus raised hope on the development of numerical methods. We consider special instances of projections problems over b-convex functions and show how they can be solved numerically using Dykstra's iterated projection algorithm to handle the b-convexity constraint in the framework of [19]. Our method also turns out to be simple for convex envelope computations.

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.

We consider a one-dimensional kinetic model of granular media in the case where the interaction potential is quadratic. Taking advantage of a simple first integral, we can use a reformulation (equivalent to the initial kinetic model for classical solutions) which allows measure solutions. This reformulation has a Wasserstein gradient flow structure (on a possibly infinite product of spaces of measures) for a convex energy which enables us to prove global in time well-posedness.

Equilibrium multi-population matching (matching for teams) is a problem from mathematical economics which is related to multi-marginal optimal transport. A special but important case is the Wasserstein barycenter problem, which has applications in image processing and statistics. Two algorithms are presented: a linear programming algorithm and an efficient nonsmooth optimization algorithm, which applies in the case of the Wasserstein barycenters. The measures are approximated by discrete measures: convergence of the approximation is proved. Numerical results are presented which illustrate the efficiency of the algorithms.

No summary available.

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∖C the congestion is higher. The two congestion functions H1 and H2 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.

Equilibrium multi-population matching (matching for teams) is a problem from mathematical economics which is related to multi-marginal optimal transport. A special but important case is the Wasserstein barycenter problem, which has applications in image processing and statistics. Two algorithms are presented: a linear programming algorithm and an efficient nonsmooth optimization algorithm, which applies in the case of the Wasserstein barycenters. The measures are approximated by discrete measures: convergence of the approximation is proved. Numerical results are presented which illustrate the efficiency of the algorithms.

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.

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.

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.

Motivated by questions posed by F. Santambrogio, this thesis is dedicated to the study of mean-field games and models involving optimal transport with density constraints. In order to study second order MFG models in the spirit of F. Santambrogio's work, we introduce as an elementary brick a diffusive model of crowd motion with density constraints (generalizing in a sense the work of Maury et al.). The model is described by the evolution of the density of the crowd, which can be seen as a curve in Wasserstein space. From a PDE point of view, it corresponds to a modified Fokker-Planck equation, with an additional term, the gradient of a pressure (only in the saturated zone) in the drift. By going through the dual equation and using well known parabolic estimates, we prove the uniqueness of the density and pressure pair. Motivated initially by the splitting algorithm (used in the existence result above), we study fine properties of the Wasserstein projection below a given threshold. Integrating this question into a larger class of problems involving optimal transport, we demonstrate BV estimates for optimizers. Other possible applications (in partial transport, shape optimization and degenerate parabolic problems) of these BV estimates are also discussed. In this sense, MFG systems are obtained as first order optimality conditions for two convex problems in duality. In these systems an additional term appears, interpreted as a price to be paid when agents pass through saturated areas. First, taking advantage of the elliptic regularity results, we show the existence and characterization of second order stationary MFG solutions with density constraints. As an additional result, we characterize the subdifferential of a functional introduced by Benamou-Brenier to give a dynamic formulation of the optimal transport problem. Second, (based on a penalty technique) we show that a class of first order MFG systems with density constraints is well posed. An unexpected connection with the incompressible Euler equations à la Brenier is also given.

No summary available.

Many problems from mass transport can be reformulated as variational problems under a prescribed divergence constraint (static problems) or subject to a time-dependent continuity equation, which again can be formulated as a divergence constraint but in time and space. The variational class of mean field games, introduced by Lasry and Lions, may also be interpreted as a generalization of the time-dependent optimal transport problem. Following Benamou and Brenier, we show that augmented Lagrangian methods are well suited to treat such convex but non-smooth problems. They include in particular Monge historic optimal transport problem. A finite-element discretization and implementation of the method are used to provide numerical simulations and a convergence study.

We propose a notion of conditional vector quantile function and a vector quantile regression. A conditional vector quantile function (CVQF) of a random vector Y, taking values in ℝd given covariates Z=z, taking values in ℝk, is a map u↦QY∣Z(u,z), which is monotone, in the sense of being a gradient of a convex function, and such that given that vector U follows a reference non-atomic distribution FU, for instance uniform distribution on a unit cube in ℝd, the random vector QY∣Z(U,z) has the distribution of Y conditional on Z=z. Moreover, we have a strong representation, Y=QY∣Z(U,Z) almost surely, for some version of U. The vector quantile regression (VQR) is a linear model for CVQF of Y given Z. Under correct specification, the notion produces strong representation, Y=β(U)⊤f(Z), for f(Z) denoting a known set of transformations of Z, where u↦β(u)⊤f(Z) is a monotone map, the gradient of a convex function, and the quantile regression coefficients u↦β(u) have the interpretations analogous to that of the standard scalar quantile regression. As f(Z) becomes a richer class of transformations of Z, the model becomes nonparametric, as in series modelling. A key property of VQR is the embedding of the classical Monge-Kantorovich's optimal transportation problem at its core as a special case. In the classical case, where Y is scalar, VQR reduces to a version of the classical QR, and CVQF reduces to the scalar conditional quantile function. Several applications to diverse problems such as multiple Engel curve estimation, and measurement of financial risk, are considered.

We consider a given region where the tra c ows according to two regimes: in a region C we have a low congestion, where in the remaining part n C the congestion is higher. The two congestion functions H1 and H2 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.

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-Ampere 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.

We obtain new a priori estimates for spatially inhomogeneous solutions of a kinetic equation for granular media, as first proposed in [3] and, more recently, studied in [1]. In particular, we show that a family of convex functionals on the phase space is non-increasing along the flow of such equations, and we deduce consequences on the asymptotic behaviour of solutions. Furthermore, using an additional assumption on the interaction kernel and a ``potential for interaction'', we prove a global entropy estimate in the one-dimensional case.

We consider a given region where the tra c ows according to two regimes: in a region C we have a low congestion, where in the remaining part n C the congestion is higher. The two congestion functions H1 and H2 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.

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.

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.

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Motivated by applications to congested optimal transport problems, we prove higher integrability results for the gradient of solutions to some anisotropic elliptic equations, exhibiting a wide range of degeneracy. The model case we have in mind is the following: \[ \partial_x \left[(|u_{x}|-\delta_1)_+^{q-1}\, \frac{u_{x}}{|u_{x}|}\right]+\partial_y \left[(|u_{y}|-\delta_2)_+^{q-1}\, \frac{u_{y}}{|u_{y}|}\right]=f, \] for $2\le q<\infty$ and some non negative parameters $\delta_1,\delta_2$. Here $(\,\cdot\,)_+$ stands for the positive part. We prove that if $f\in L^\infty_{loc}$, then $\nabla u\in L^r_{loc}$ for every $r\ge 1$.

Many problems from mass transport can be reformulated as variational problems under a prescribed divergence constraint (static problems) or subject to a time dependent continuity equation which again can also be formulated as a divergence constraint but in time and space. The variational class of Mean-Field Games introduced by Lasry and Lions may also be interpreted as a generalisation of the time-dependent optimal transport problem. Following Benamou and Brenier, we show that augmented Lagrangian methods are well-suited to treat convex but nonsmooth problems. It includes in particular Monge historic optimal transport problem. A Finite Element discretization and implementation of the method is used to provide numerical simulations and a convergence study.

This report presents the adaptation of the ALG2 algorithm [1] in order to numerically solve variational mean field games [5].

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.

An exchange economy in which agents have convex incomplete preferences defined by families of concave utility functions is considered. Sufficient conditions for the set of efficient allocations and equilibria to coincide with the set of efficient allocations and equilibria that result when each agent has a utility in her family are provided. Welfare theorems in an incomplete preferences framework therefore hold under these conditions and efficient allocations and equilibria are characterized by first order conditions.

We consider $H$ expected utility maximizers that have to share a risky aggregate multivariate endowment $X 2 R N$ and address the following two questions: does efficient risk-sharing imply restrictions on the form of individual consumptions as a function of $X$? Can one identify the individual utility functions from the observation of the risk-sharing? We show that when $H 2N N 1$ efficient risk sharings have to satisfy a system of nonlinear PDEs. Under an additional rank condition, we prove an identification theorem.

The research conducted in this thesis proposes a contribution to the analysis of risk in the French automobile insurance market. Three new axes are presented: the first axis is based on a theoretical framework of the automobile insurance market. An original model of double asymmetric information is presented. The main result is the existence of two kinds of equilibrium contracts: a separating contract and a mixing contract. The second point is related to the consideration of past claims experience in the study of the risk-coverage relationship. Bivariate and trivariate models are applied for this purpose. This shows that the hypothesis of information asymmetry is verified. Finally, the third issue raised in this thesis concerns the application of the surcharge to young drivers. We show through econometric modelling of claims experience that the legitimacy of insurers to almost systematically offer higher rates to young drivers than to experienced drivers is not always verified.

This thesis is dedicated to the study of some problems in the computation of variations of deviated argument functionals involved for instance in optimal control problems of deviated argument differential equations and in variational problems with deviated arguments. We use the direct method to show the existence of the deviated argument problem in dimension n > 1 in a Sobolev type functional space with weight related to the deviation. Then, we put forward the necessary conditions of optimality based on the area formula. We obtain an equivalence between a problem without deviation and a problem with deviation in a convex framework. We also show a form of Pontryagin's principle for a class of optimal control problems governed by an equation with a memory. Some application examples are considered. Infin we complete with some existence and uniqueness results for some nonlocal elliptic equations known as deviated argument, encountered in the literature.

This thesis is dedicated to the study of optimal transportation problems, alternative to the Monge-Kantorovich problem: they appear naturally in practical applications, such as the design of optimal transportation networks or the modeling of urban traffic problems. In particular, we consider problems where the cost of transportation has a nonlinear dependence on the mass: typically in such problems, the cost of moving a mass m for a length ℓ is φ (m) ℓ, where φ is an assigned function, thus obtaining a total cost of type Σ φ (m) ℓ. Two important cases are discussed in detail in this work: the case where the function φ is subadditive (branched transport), so that the mass has an interest in traveling together, so as to reduce the total cost. the case where φ is superadditive (congested transport), where on the contrary, the mass tends to diffuse as much as possible. In the case of branched transport, we introduce two new models: in the first one, the transport is described by curves of probability measures that minimize a geodesic type functional (with a coefficient that penalizes the measures that are not atomic). The second one is more in the spirit of the formulation of Benamou and Brenier for the Wasserstein distances, in particular, the transport is described by pairs of ``measurement curves--velocity fields'', linked by the continuity equation, which minimize an adequate (non-convex) energy. For both models, we demonstrate the existence of minimal configurations and the equivalence with other existing formulations in the literature. For the case of congested transportation, we review two already existing models, in order to prove their equivalence: while the first of these models can be considered as a Lagrangian approach to the problem and it has interesting links with equilibrium issues for urban traffic, the second one is a convex optimization problem with divergence constraints by The proof of the equivalence between the two models constitutes the main body of the second part of this thesis and contains various elements of interest, including: the flow theory of sparsely regular vector fields (DiPerna-Lions), the Dacorogna and Moser construction for transport applications, and in particular the regularity results (which we prove here) for a highly degenerate elliptic equation, which does not seem to have been studied much.

In this thesis we explore the use of optimal control and mass transport for economic modeling. We thus take the opportunity to bring together several works involving these two tools, sometimes interacting with each other. We first briefly introduce the recent mean-field game theory introduced by Lasry and Lions and focus on the control aspect of the Fokker-Planck equation. We exploit this aspect both to obtain equilibrium existence results and to develop numerical solution methods. We test the algorithms in two complementary cases, namely the convex framework (crowd aversion, two population dynamics) and the concave framework (attraction, externalities and scale effects in a stylized model of technological transition). In a second step, we study a matching problem mixing optimal transport and optimal control. The planner seeks an optimal matching, fixed for a given period (commitment), given that the margins evolve (possibly randomly) in a controlled manner. Finally, we reformulate a risk sharing problem between d agents (for which we prove an existence result) into an optimal control problem with comonotonicity constraints. This allows us to obtain optimality conditions with which we construct a simple and convergent algorithm.

This thesis is devoted to some optimization problems arising in contract theory. We first consider some one-dimensional problems. In this simple case, the principal's program consists in minimizing a certain cost on the cone of increasing functions. We consider non-convex cases and use a variant of the Hardy-Littlewood inequality. We then consider variational problems under convexity constraints. After giving existence results for non-convex Lagrangians, we introduce a penalty to recover the Euler equation due to Pierre-Louis Lions. In a collaborative work with Thomas Lachand-Robert, we then establish C1 regularity results for minimizers in various situations (Dirichlet, Neumann, Choné and Rochet model. . . ). Finally, in a paper co-authored with Thomas Lachand-Robert and Bertrand Maury, we present a method for numerical approximation of quadratic problems under convexity constraints. The presented algorithm is, to our knowledge, the only one whose convergence is established. In the second part, we characterize incentive contracts in a general way by means of abstract convexity and subdifferentiability notions used in the theory of mass transportation and we prove the existence of optimal incentive contracts without any particular functional specification on the agents' preferences. We then use the link between the characterization of incentive contracts and a class of optimal transportation problems. First, existence, uniqueness, and duality results are established for mass transportation problems where the cost verifies a generalizing Spence-Mirrlees hypothesis, which is classical in the economic literature in dimension 1. This allows us, returning to incentive problems, to de��monstrate a reallocation principle: any measurable allocation profile can be uniquely rearranged into an implementable profile, via a suitable optimal transfer problem. The final section is devoted to two specific economic problems. The first one is motivated by insurance fraud issues that give rise to a non-convex problem and consists of a paper co-authored with Rose-Anne Dana and Maxime Renaudin. The second one is related to the design of labor contracts in the presence of two-dimensional adverse selection. It is a paper co-authored with Damien Gaumont.