Over the last twenty years, Optimal Mass Transportation has played a major role in PDEs, geometry, functional inequalities as well as in modelling and applied fields such as fluid mechanics, image processing and economics. This trend shows no sign of slowing and the field is still extremely active. However, the numerics remain underdeveloped, but recent progress in this new field of numerical Optimal Mass Transportation raise hope for significant advances in numerical simulations.

Mokaplan objectives are to design, develop and implement these new algorithms with and emphasis on economic applications.

*Optimal Mass Transportation* is a mathematical research topic which started two centuries ago with Monge's work on “des remblais et déblais". This engineering problem consists in minimizing the transport cost between two given mass densities.
In the 40's, Kantorovich solved the dual problem and interpreted it as an economic equilibrium. The *Monge-Kantorovich* problem became a specialized research topic in optimization and Kantorovich obtained the 1975 Nobel prize in economics for his contributions to resource allocations problems.
Following the seminal discoveries of Brenier in the 90's , Optimal Transportation has received renewed attention from mathematical analysts and the Fields Medal awarded in 2010 to C. Villani, who gave important contributions to Optimal Transportation and wrote
the modern reference monograph , arrived at a culminating moment for this theory.
Optimal Mass Transportation is today a mature area of mathematical analysis with a constantly growing range of applications
(see below).

In the modern Optimal Mass Transportation problem, we are given two probability measures or "mass" densities :
*transportation cost*,
*ground cost*,
over all *volume preserving maps* *Jacobian equation*

In the Euclidean distance squared ground cost,
the problem is well posed and in the seminal work of Brenier , the optimal map is characterized as the gradient of a convex
potential *second boundary value* condition *Brenier solutions*, may have discontinuous gradients when the
target density support *Wasserstein distance*
*Computational Fluid Dynamic* formulation proposed by Brenier and Benamou in
introduces a time extension of the domain and leads to a convex but non smooth optimization problem :
*Mean Fields games* , a large class of economic models introduced by Lasry and Lions.
The Wasserstein distance and its connection to Optimal Mass Transportation also appears in the construction of
semi-discrete Gradient Flows. This notion known as *JKO gradient flows* after its authors in is a popular tool
to study non-linear diffusion equations : the
implicit Euler scheme *Ma-Trudinger-Wang* condition
which gives necessary condition on

Our focus is on numerical methods in Optimal Mass Transportation and applications. The simplest way to build a numerical method is to consider sum of dirac masses
*assigment problem* between the points

For general densities data, the original optimization problem is not tractable because of the volume preserving
constraint on the map. Kantorovich dual formulation is a linear program but with a large number of constraints
set over the product of the source and target space

When interested in slightly more regular solutions which correspond to the assumption that the target support is convex, the recent
*wide stencil* monotone finite difference scheme for the Monge-Ampère equation can be adapted to the Optimal Mass Transportation problem.
This is the topic of . This approach is extremely fast as a Newton algorithm can be used to solve the discrete
system. Numerical studies confirm this with a linear empirical complexity.

For other costs, JKO schemes, multi marginal extensions, partial transport ... efficient numerical methods are to be invented.

As already mentioned the CFD formulation is a limit case of simple variational Mean-Field Games (MFG) . MFG is a new branch of game theory recently developed by J-M. Lasry and P-L. Lions. MFG models aim at describing the limiting behavior of stochastic differential games when the number of players tends to infinity. They are specifically designed to model economic problems where a large number of similar interacting agents try to maximize/minimize a utility/cost function which takes into account global but partial information on the game. The players in these models are individually insignificant but they collectively have a significant impact on the cost of the other players. Dynamic MFG models often lead to a system of PDEs which consists of a backward Hamilton-Jacobi Bellman equation for a value function coupled with a forward Fokker-Planck equation describing the space-time evolution of the density of agents.

In microeconomics, the *principal-agent problem* with adverse selection
plays a distinguished role in the literature on asymmetric information and
contract theory (with important contributions from several Nobel prizes
such as Mirrlees, Myerson, Spence or Tirole) and it has many important
applications in optimal taxation, insurance, nonlinear pricing.
The problem can be reduced to the maximization of
an integral functional subject to a convexity constraint
This is an unusual calculus of variations problem and the optimal price can only be
computed numerically. Recently, following a reformulation of Carlier , convexity/well-posedness results of McCann, Figalli and
Kim , connected to optimal transport theory, showed that there is some hope
to numerically solve the problem for general utility functions.

Many relevant markets are markets of indivisible goods characterized by a certain quality: houses, jobs, marriages... On the theoretical side, recent papers by Ekeland, McCann, Chiappori showed that finding equilibria in such markets is equivalent to solving a certain optimal transport problem (where the cost function depends on the sellers and buyers preferences). On the empirical side, this allows for trying to recover information on the preferences from observed matching; this is an inverse problem as in a recent work of Galichon and Salanié Interestingly, these problems naturally lead to numerically challenging variants of the Monge-Kantorovich problem: the multi-marginal OT problem and the entropic approximation of the Monge-Kantorovich problem (which is actually due to Schrödinger in the early 30's).

The Skorohod embedding problem (SEP) consists in
finding a martingale interpolation between two probability measures.
When a particular stochastic ordering between
the two measures is given, Galichon et al have shown
that a very natural variational formulation could be given to a class of
problems that includes the SEP. This formulation is related to the
CFD formulation of the OT problem
and has applications to *model-free bounds of derivative prices in Finance*.
It can also be interpreted as a a multi marginal Optimal Mass Transportation with infinitely many marginals .

The volume preserving property appears naturally in this context where motion is constrained by the density of player.

Optimal Mass Transportation and MFG theories can be an extremely powerful tool to attack some of these problems arising from spatial economics or to design new ones. For instance, various urban/traffic planning models have been proposed by Buttazzo, Santambrogio, Carlier ( ) in recent years.

Many models from PDEs and fluid mechanics have been used to give a description of *people or vehicles moving in a congested environment*.
These models have to be classified according to the dimension (1D model are mostly used for cars on traffic networks, while 2D models are most suitable for pedestrians), to the congestion effects (“soft” congestion standing for the phenomenon where high densities slow down the movement, “hard” congestion for the sudden effects when contacts occur, or a certain threshold is attained), and to the possible rationality of the agents
Maury et al recently developed a theory for 2D hard congestion models without rationality, first in a discrete and then in a continuous framework.
This model produces a PDE that is difficult to attack with usual PDE methods, but has been successfully studied via Optimal Mass Transportation techniques again related to the JKO
gradient flow paradigm.

In and , the authors show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales – a few Mpc – where multi-streaming becomes significant. Above 6 Mpc/h we propose and implement an effective Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation. After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than cubic time complexity in N and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60% of exactly reconstructed points.

The Wasserstein distance between densities is the value function of the Optimal Mass Transportation problem. This distance may be considered to have "orthogonal" properties to the widely used least square distance. It is for instance quadratic with respect to dilations and translation. On the other hand it is not very sensitive to rigid transformations, is an attempt at generalizing the CFD formulation in this context. The Wasserstein distance is an interesting tool for applications where distances between signals and in particular oscillatory signals need to to computed, this is assuming one understands how to transform the information into positive densities.

Tannenbaum and co-authors have designed several variants of the CFD numerical method and applied it to warping, morphing and registration (using the Optimal Mass Transportation map) problems in medical imaging.

Gabriel Peyre and co-authors have proposed an easier to compute relaxation of the Wasserstein distance (the sliced Wasserstein distance) and applied it to two image processing problems: color transfer and texture mixing.

In, Brenier reviews in a unified framework the connection between optimal transport theory and classical convection theory for geophysical flows. Inspired by the numerical model proposed in , the starting point is a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, he relates different variants of the NSB equations (in particular what he calls the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport and the related Monge-Ampère equation. This includes the 2D semi-geostrophic equations and some fully nonlinear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel system for chemotaxis .

The necessity to preserve areas/volumes is a intrinsic feature of mesh deformations more generally Lagrangian numerical methods. Numerical method of Optimal Mass Transportation which preserve some notions of convexity and as a consequence the monotonicity of the computed transport maps can play a role in this context, see for instance .

The precise modeling of electron correlations continues to constitute the major obstacle in developing high-accuracy, low-cost methods for electronic structure computations in molecules and solids. The article sheds a new light on the longstanding problem of how to accurately incorporate electron correlation into DFT, by deriving and analyzing the semiclassical limit of the exact Hohenberg-Kohn functional with the single-particle density

A generalisation of the ALG2 algorithm corresponding to
the paper ha been implemented in FreeFem++.
The scripts and numerical simulations are available at
https://

We still plan to implement a parallel version on Rocquencourt Inria cluster. We are waiting for FreeFem to be installed on the cluster.

Following the pioneering work of Caffarelli and Oliker , Wang has shown that the inverse problem of freeforming a *convex* reflector which sends a prescribed source to a target intensity is a particular instance of Optimal Mass Transportation. The method developed in has been used by researchers of TU Eindhoven in collaboration with Philips Lightning Labs to compute reflectors in a simplified setting. The industrial motivation is the automatic design of reflector given prescribed source and target illuminance.
From the mathematical point of view there is a hierarchy of Optimal Mass Transportation reflector and lenses problems and only the simplest "far field" one can be solved with state of the art
Monge-Ampère solvers.
We will adapt the Monge-Ampère solvers and also attempt to build real optimized reflector prototypes.
We plan on investigating the more complicated near field models and design numerical methods.
Finally Monge-Ampère based Optimal Mass Transportation solvers will be made available. This could be used for example in
Mesh adaptation.

The web site is under construction https://

This ADT (Simon Legrand) on the numerical free forming of specular reflectors started in december. We implement different types of MA solvers
in collaboration with Quentin Mérigot (CEREMADE), Boris Thibert (LJK Grenoble) and Vincent Duval.
See https://

All of the new results below are important break through and most of them non-incremental research.

Mokaplan has extended its collaborations to several researchers at Ceremade and is under review to become a project team.

*Benamou, Jean-David and Carlier, Guillaume and Cuturi, Marco and Nenna, Luca and Peyré, Gabriel*

We provide a general numerical framework to approximate solutions 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 distribution) 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 involved). 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 unbalanced optimal transport and optimal transport with capacity constraints.

The extension of the method to the Principal Agent problem, Density Functional theory and Transport under martingal constraint is under way.

*Benamou, Jean-David and Froese, Brittany D. *

We consider the Monge-Kantorovich optimal transportation problem between two measures, one of which is a weighted sum of Diracs. This problem is traditionally solved using expensive geometric methods. It can also be reformulated as an elliptic partial differential equation known as the Monge-Ampere equation. However, existing numerical methods for this non-linear PDE require the measures to have finite density. We introduce a new formulation that couples the viscosity and Aleksandrov solution definitions and show that it is equivalent to the original problem. Moreover, we describe a local reformulation of the subgradient measure at the Diracs, which makes use of one-sided directional derivatives. This leads to a consistent, monotone discretisation of the equation. Computational results demonstrate the correctness of this scheme when methods designed for conventional viscosity solutions fail.

The method offers a new insight into the duality between Aleksandrov and Brenier solutions of the Monge Ampère equations. We still work on the viscosity existence/uniqueness convergence of sheme theory.

*Benamou, Jean-David and Carlier, Guillaume and Mérigot, Quentin and Oudet, Edouard*

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 larger than 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.

*Benamou, Jean-David and Carlier, Guillaume*

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.

We have good hopes to use this method to many non-linear diffusion equations through the use of JKO gradient schemes.

*Benamou, Jean-David and Collino, Francis and Mirebeau, Jean-Marie*

We introduce a novel discretization of the Monge-Ampere operator, simultaneously consistent and degenerate elliptic, hence accurate and robust in applications. These properties are achieved by exploiting the arithmetic structure of the discrete domain, assumed to be a two dimensional cartesian grid. The construction of our scheme is simple, but its analysis relies on original tools seldom encountered in numerical analysis, such as the geometry of two dimensional lattices, and an arithmetic structure called the Stern-Brocot tree. Numerical experiments illustrate the method's efficiency.

*Bleyer, Jérémy and Carlier, Guillaume and Duval, Vincent and Mirebeau, Jean-Marie and Peyré, Gabriel*

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.

*Carlier, Guillaume and Blanchet, Adrien*

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. It is therefore natural to investigate the continuous limit as N tends to infinity and to investigate whether it corresponds to the notion of Cournot-Nash equilibria. In , this kind of convergence result is studied in a Wasserstein framework. In [BC1], we go one step further by giving a class of games with a continnum of players for which equilibria may be found as minimizers as a functional on measures which is very similar to the one-step JKO case, uniqueness results are the obtained from displacement convexity arguments. Finally, in some situations which are non variational are considered and existence is obtained by methods combining fixed point arguments and optimal transport.

*Carlier, Guillaume, Benamou, Jean-David and Dupuis Xavier*

The numerical resolution of principal Agent for a bilinear utility has been attacked and solved successfully in a series of recent papers see and references therein.

*Duval, Vincent and Peyré, Gabriel*

We study sparse spikes deconvolution over the space of measures. We focus our attention to the recovery properties of the support of the measure, i.e. the location of the Dirac masses. For non-degenerate sums of Diracs, we show that, when the signal-to-noise ratio is large enough, total variation regularization (which is the natural extension of the L1 norm of vectors to the setting of measures) recovers the exact same number of Diracs. We also show that both the locations and the heights of these Diracs converge toward those of the input measure when the noise drops to zero. The exact speed of convergence is governed by a specific dual certificate, which can be computed by solving a linear system. We draw connections between the support of the recovered measure on a continuous domain and on a discretized grid. We show that when the signal-to-noise level is large enough, the solution of the discretized problem is supported on pairs of Diracs which are neighbors of the Diracs of the input measure. This gives a precise description of the convergence of the solution of the discretized problem toward the solution of the continuous grid-free problem, as the grid size tends to zero.

Jean-David Benamou is the coordinator of the ANR
ISOTACE (Interacting Systems and Optimal Transportation, Applications to Computational Economics) ANR-12-MONU-0013 (2012-2016). The consortium explores new numerical methods in Optimal Transportation AND Mean Field Game
theory with applications in Economics and congested crowd motion.
Four extended seminars have been organized/co-organized by Mokaplan. Check
https://

Christophe Duquesne (Aurigetech) is a software and mobility consultant hired on the ANR budget. He helps the consortium to develop its industrial partnerships.

Title: Numerical Optimal Transportation in (Mathematical) Economics

International Partner (Institution - Laboratory - Researcher):

McGill University (CANADA)

Duration: 2014 - 2016

See also: https://

The overall scientific goals is to develop numerical methods for large scale optimal transport and models based on optimal transport tools

see https://

A few additional applications were suggested at our annual workshop in october

https://

Adam Oberman (U. Mc Gill) visited Mokaplan in June.

Guillaume Carlier in on sabbatical for the academic year (délégation CNRS at the UMI-CNRS 3069 PIMS at UVIC, Victoria, British Columbia, Canada). He is taking advantage of this full-research year to work on optimal transport methods for kinetic models for granular media (with M. Agueh and Reinhard Illner), Wasserstein barycenters and to continue to develop joint projects on numerical optimal transport with J.D. Benamou's MOKAPLAN team.

Guillaume Carlier was organiser of the RICAM special semester on Calculus of Variation
http://

Guillaume carlier is in the scientific committee for SMAI 2015.

Vincent Duval has reviewed several contributions for the *Scale Space and Variational Methods* SSVM 2015 conference.

Guillaume carlier is member of the editorial Board of "Journal de l'Ecole Polytechnique" and co-editor of "Mathematics and Financial Economics".

Vincent Duval has reviewed several papers for the following jounals:

SIIMS (SIAM Journal on Imaging Sciences)

JMAA (Journal of Mathematical Analysis and Applications)

IPol (Image Processing Online)

JVCI (Journal of Visual Communication and Image Representation)

In 2014, Guillaume Carlier gave an advanced course on Mean-Field Games (M2 EDP-MAD and Masef) and a master course (M1) on dynamic programming at Dauphine

Master : Guillaume Carlier, Mean-Field Games, M2 EDP-MAD , U. Paris- Dauphine.

Master : Guillaume Carlier, Dynamic Programming , M1 , U. Paris- Dauphine.

PhD in progress: Quentin Denoyelle, “Analyse théorique et numérique de la super-résolution sans grille”, 2014, Gabriel Peyré and Vincent Duval.

PhD in progress : Roméo Hatchi , "Analyse mathématique de modèles de trafic congestionné", 2012, Guillaume Carlier.

PhD in progress : Maxime Laborde , " Dynamique des systèmes de particules en interaction, approche par flots de gradient et applications", 2013 , Guillaume Carlier

PhD in progress : Luca Nenna , "Méthodes numṕerique pour le transport optimal multimarge" , 2013, Jean-David Benamou et Guillaume Carlier.

PhD in progress: Quentin Denoyelle, “Analyse théorique et numérique de la super-résolution sans grille”, thèse commencée le 1er octobre 2014, supervised by Gabriel Peyré (main supervisor) and Vincent Duval (co-supervisor).

Guillaume carlier was in the Ph.D. committee of Serena Guarino (Pisa) and Miryana Grigorova (Paris 7).