2.1 Introduction

vvvv The last decade has witnessed a remarkable convergence between several sub-domains of the calculus of variations, namely optimal transport (and its many generalizations), infinite dimensional geometry of diffeomorphisms groups and inverse problems in imaging (in particular sparsity-based regularization). This convergence is due to (i) the mathematical objects manipulated in these problems, namely sparse measures (e.g. coupling in transport, edge location in imaging, displacement fields for diffeomorphisms) and (ii) the use of similar numerical tools from non-smooth optimization and geometric discretization schemes. Optimal Transportation, diffeomorphisms and sparsity-based methods are powerful modeling tools, that impact a rapidly expanding list of scientific applications and call for efficient numerical strategies. Our research program shows the important part played by the team members in the development of these numerical methods and their application to challenging problems.

2.2 Static Optimal Transport and Generalizations

Generalizations of OT.

In recent years, the classical Optimal Transportation problem has been extended in several directions. First, different ground costs measuring the “physical" displacement have been considered. In particular, well posedness for a large class of convex and concave costs has been established by McCann and Gangbo  94. Optimal Transportation techniques have been applied for example to a Coulomb ground cost in Quantum chemistry in relation with Density Functional theory  79. Given the densities of electrons Optimal Transportation models the potential energy and their relative positions. For more than more than 2 electrons (and therefore more than 2 densities) the natural extension of Optimal Transportation is the so called Multi-marginal Optimal Transport (see  123 and the references therein). Another instance of multi-marginal Optimal Transportation arises in the so-called Wasserstein barycenter problem between an arbitrary number of densities 20. An interesting overview of this emerging new field of optimal transport and its applications can be found in the recent survey of Ghoussoub and Pass  122.

2.3 Diffeomorphisms and Dynamical Transport

Dynamical transport.

While the optimal transport problem, in its original formulation, is a static problem (no time evolution is considered), it makes sense in many applications to rather consider time evolution. This is relevant for instance in applications to fluid dynamics or in medical images to perform registration of organs and model tumor growth.

In this perspective, the optimal transport in Euclidean space corresponds to an evolution where each particule of mass evolves in straight line. This interpretation corresponds to the Computational Fluid Dynamic (CFD) formulation proposed by Brenier and Benamou in  28. These solutions are time curves in the space of densities and geodesics for the Wasserstein distance. The CFD formulation relaxes the non-linear mass conservation constraint into a time dependent continuity equation, the cost function remains convex but is highly non smooth. A remarkable feature of this dynamical formulation is that it can be re-cast as a convex but non smooth optimization problem. This convex dynamical formulation finds many non-trivial extensions and applications, see for instance  30. The CFD formulation also appears to be a limit case of Mean Fields games (MFGs), a large class of economic models introduced by Lasry and Lions  104 leading to a system coupling an Hamilton-Jacobi with a Fokker-Planck equation. In contrast, the Monge case where the ground cost is the euclidan distance leads to a static system of PDEs  45.

2.4 Sparsity in Imaging

Regularization over measure spaces.

However, the theoretical analysis of sparse reconstructions involving real-life acquisition operators (such as those found in seismic imaging, neuro-imaging, astro-physical imaging, etc.) is still mostly an open problem. A recent research direction, triggered by a paper of Candès and Fernandez-Granda  62, is to study directly the infinite dimensional problem of reconstruction of sparse measures (i.e. sum of Dirac masses) using the total variation of measures (not to be mistaken for the total variation of 2-D functions). Several works  61, 87, 84 have used this framework to provide theoretical performance guarantees by basically studying how the distance between neighboring spikes impacts noise stability.

Segmentation input	output	Zooming input	output

2.5 Mokaplan unified point of view

The dynamical formulation of optimal transport creates a link between optimal transport and geodesics on diffeomorphisms groups. This formal link has at least two strong implications that Mokaplan will elaborate on: (i) the development of novel models that bridge the gap between these two fields ; (ii) the introduction of novel fast numerical solvers based on ideas from both non-smooth optimization techniques and Bregman metrics, as highlighted in Section 3.2.3.

In a similar line of ideas, we believe a unified approach is needed to tackle both sparse regularization in imaging and various generalized OT problems. Both require to solve related non-smooth and large scale optimization problems. Ideas from proximal optimization has proved crucial to address problems in both fields (see for instance  28, 129). Transportation metrics are also the correct way to compare and regularize variational problems that arise in image processing (see for instance the Radon inversion method proposed in  32) and machine learning (see  82). This unity in term of numerical methods is once again at the core of Section 3.2.3.

3.1 Modeling and Analysis

The first layer of methodological tools developed by our team is a set of theoretical continuous models that aim at formalizing the problems studied in the applications. These theoretical findings will also pave the way to efficient numerical solvers that are detailed in Section 3.2.

3.1.1 Static Optimal Transport and Generalizations

Convexity constraint and Principal Agent problem in Economics.

(Participants: G. Carlier, J-D. Benamou, V. Duval, Xavier Dupuis (LUISS Guido Carli University, Roma)) The principal agent problem plays a distinguished role in the literature on asymmetric information and contract theory (with important contributions from several Nobel prizes such as Mirrlees, Myerson or Spence) and it has many important applications in optimal taxation, insurance, nonlinear pricing. The typical problem consists in finding a cost minimizing strategy for a monopolist facing a population of agents who have an unobservable characteristic, the principal therefore has to take into account the so-called incentive compatibilty constraint which is very similar to the cyclical monotonicity condition which characterizes optimal transport plans. In a special case, Rochet and Choné 130 reformulated the problem as a variational problem subject to a convexity constraint. For more general models, and using ideas from Optimal Transportation, Carlier  65 considered the more general $c$-convexity constraint and proved a general existence result. Using the formulation of   65 McCann, Figalli and Kim  88 gave conditions under which the principal agent problem can be written as an infinite dimensional convex variational problem. The important results of   88 are intimately connected to the regularity theory for optimal transport and showed that there is some hope to numerically solve the principal-agent problem for general utility functions.

Our expertise: We have already contributed to the numerical resolution of the Principal Agent problem in the case of the convexity constraint, see 70, 117, 119.

Goals: So far, the mathematical PA model can be numerically solved for simple utility functions. A Bregman approach inspired by 32 is currently being developed 68 for more general functions. It would be extremely useful as a complement to the theoretical analysis. A new semi-Discrete Geometric approach is also investigated where the method reduces to non-convex polynomial optimization.

Optimal transport and conditional constraints in statistics and finance.

(Participants: G. Carlier, J-D. Benamou, G. Peyré) A challenging branch of emerging generalizations of Optimal Transportation arising in economics, statistics and finance concerns Optimal Transportation with conditional constraints. The martingale optimal transport  26, 93 which appears naturally in mathematical finance aims at computing robust bounds on option prices as the value of an optimal transport problem where not only the marginals are fixed but the coupling should be the law of a martingale, since it represents the prices of the underlying asset under the risk-neutral probability at the different dates. Note that as soon as more than two dates are involved, we are facing a multimarginal problem.

Our expertise: Our team has a deep expertise on the topic of OT and its generalization, including many already existing collaboration between its members, see for instance  32, 37, 30 for some representative recent collaborative publications.

Goals: This is a non trivial extension of Optimal Transportation theory and Mokaplan will develop numerical methods (in the spirit of entropic regularization) to address it. A popular problem in statistics is the so-called quantile regression problem, recently Carlier, Chernozhukov and Galichon  66 used an Optimal Transportation approach to extend quantile regression to several dimensions. In this approach again, not only fixed marginals constraints are present but also constraints on conditional means. As in the martingale Optimal Transportation problem, one has to deal with an extra conditional constraint. The duality approach usually breaks down under such constraints and characterization of optimal couplings is a challenging task both from a theoretical and numerical viewpoint.

JKO gradient flows.

(Participants: G. Carlier, J-D. Benamou, M. Laborde, Q. Mérigot, V. Duval) The connection between the static and dynamic transportation problems (see Section 2.3) opens the door to many extensions, most notably by leveraging the use of gradient flows in metric spaces. The flow with respect to the transportation distance has been introduced by Jordan-Kindelherer-Otto (JKO)  101 and provides a variational formulation of many linear and non-linear diffusion equations. The prototypical example is the Fokker Planck equation. We will explore this formalism to study new variational problems over probability spaces, and also to derive innovative numerical solvers. The JKO scheme has been very successfully used to study evolution equations that have the structure of a gradient flow in the Wasserstein space. Indeed many important PDEs have this structure: the Fokker-Planck equation (as was first considered by  101), the porous medium equations, the granular media equation, just to give a few examples. It also finds application in image processing  54. Figure 4 shows examples of gradient flows.

Our expertise: There is an ongoing collaboration between the team members on the theoretical and numerical analysis of gradient flows.

Goals: We apply and extend our research on JKO numerical methods to treat various extensions:

• Wasserstein gradient flows with a non displacement convex energy (as in the parabolic-elliptic Keller-Segel chemotaxis model 72)
• systems of evolution equations which can be written as gradient flows of some energy on a product space (possibly mixing the Wasserstein and $L^2$ structures) : multi-species models or the parabolic-parabolic Keller-Segel model  41
• perturbation of gradient flows: multi-species or kinetic models are not gradient flows, but may be viewed as a perturbation of Wasserstein gradient flows, we shall therefore investigate convergence of splitting methods for such equations or systems.

From networks to continuum congestion models.

(Participants: G. Carlier, J-D. Benamou, G. Peyré) Congested transport theory in the discrete framework of networks has received a lot of attention since the 50's starting with the seminal work of Wardrop. A few years later, Beckmann proved that equilibria are characterized as solution of a convex minimization problem. However, this minimization problem involves one flow variable per path on the network, its dimension thus quickly becomes too large in practice. An alternative, is to consider continuous in space models of congested optimal transport as was done in  69 which leads to very degenerate PDEs  46.

Our expertise: MOKAPLAN members have contributed a lot to the analysis of congested transport problems and to optimization problems with respect to a metric which can be attacked numerically by fast marching methods  37.

Goals: The case of general networks/anisotropies is still not well understood, general $\Gamma$-convergence results will be investigated as well as a detailed analysis of the corresponding PDEs and numerical methods to solve them. Benamou and Carlier already studied numerically some of these PDEs by an augmented Lagrangian method see figure 5. Note that these class of problems share important similarities with metric learning problem in machine learning, detailed below.

3.1.2 Diffeomorphisms and Dynamical Transport

Growth Models for Dynamical Optimal Transport.

(Participants: F-X. Vialard, J-D. Benamou, G. Peyré, L. Chizat) A major issue with the standard dynamical formulation of OT is that it does not allow for variation of mass during the evolution, which is required when tackling medical imaging applications such as tumor growth modeling  57 or tracking elastic organ movements  133. Previous attempts  112, 127 to introduce a source term in the evolution typically lead to mass teleportation (propagation of mass with infinite speed), which is not always satisfactory.

Our expertise: Our team has already established key contributions both to connect OT to fluid dynamics  28 and to define geodesic metrics on the space of shapes and diffeomorphisms  76.

Goals: Lenaic Chizat's PhD thesis aims at bridging the gap between dynamical OT formulation, and LDDDM diffeomorphisms models (see Section 2.3). This will lead to biologically-plausible evolution models that are both more tractable numerically than LDDM competitors, and benefit from strong theoretical guarantees associated to properties of OT.

Mean-field games.

(Participants: G. Carlier, J-D. Benamou) The Optimal Transportation Computational Fluid Dynamics (CFD) formulation is a limit case of variational Mean-Field Games (MFGs), a new branch of game theory recently developed by J-M. Lasry and P-L. Lions  104 with an extremely wide range of potential applications  96. Non-smooth proximal optimization methods used successfully for the Optimal Transportation can be used in the case of deterministic MFGs with singular data and/or potentials  31. They provide a robust treatment of the positivity constraint on the density of players.

Our expertise: J.-D. Benamou has pioneered with Brenier the CFD approach to Optimal Transportation. Regarding MFGs, on the numerical side, our team has already worked on the use of augmented Lagrangian methods in MFGs 30 and on the analytical side 64 has explored rigorously the optimality system for a singular CFD problem similar to the MFG system.

Goals: We will work on the extension to stochastic MFGs. It leads to non-trivial numerical difficulties already pointed out in  19.

Macroscopic Crowd motion, congestion and equilibria.

(Participants: G. Carlier, J-D. Benamou, Q. Mérigot, F. Santambrogio (U. Paris-Sud), Y. Achdou (Univ. Paris 7), R. Andreev (Univ. Paris 7)) 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 2-D 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  115 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 Transportation techniques again related to the JKO gradient flow paradigm. Another possibility to model crowd motion is to use the mean field game approach of Lions and Lasry which limits of Nash equilibria when the number of players is large. This also gives macroscopic models where congestion may appear but this time a global equilibrium strategy is modelled rather than local optimisation by players like in the JKO approach. Numerical methods are starting to be available, see for instance  19, 53.

Our expertise: We have developed numerical methods to tackle both the JKO approach and the MFG approach. The Augmented Lagrangian (proximal) numerical method can actually be applied to both models 30, JKO and deterministic MFGs.

Goals: We want to extend our numerical approach to more realistic congestion model where the speed of agents depends on the density, see Figure 6 for preliminary results. Comparison with different numerical approaches will also be performed inside the ANR ISOTACE. Extension of the Augmented Lagrangian approach to Stochastic MFG will be studied.

Diffeomorphic image matching.

(Participants: F-X. Vialard, G. Peyré, B. Schmitzer, L. Chizat) Diffeomorphic image registration is widely used in medical image analysis. This class of problems can be seen as the computation of a generalized optimal transport, where the optimal path is a geodesic on a group of diffeomorphisms. The major difference between the two approaches being that optimal transport leads to non smooth optimal maps in general, which is however compulsory in diffeomorphic image matching. In contrast, optimal transport enjoys a convex variational formulation whereas in LDDMM the minimization problem is non convex.

Our expertise: F-X. Vialard is an expert of diffeomorphic image matching (LDDMM) 139, 52, 137. Our team has already studied flows and geodesics over non-Riemannian shape spaces, which allows for piecewise smooth deformations  76.

Goals: Our aim consists in bridging the gap between standard optimal transport and diffeomorphic methods by building new diffeomorphic matching variational formulations that are convex (geometric obstructions might however appear). A related perspective is the development of new registration/transport models in a Lagrangian framework, in the spirit of  134, 133 to obtain more meaningful statistics on longitudinal studies.

Diffeomorphic matching consists in the minimization of a functional that is a sum of a deformation cost and a similarity measure. The choice of the similarity measure is as important as the deformation cost. It is often chosen as a norm on a Hilbert space such as functions, currents or varifolds. From a Bayesian perspective, these similarity measures are related to the noise model on the observed data which is of geometric nature and it is not taken into account when using Hilbert norms. Optimal transport fidelity have been used in the context of signal and image denoising  106, and it is an important question to extends these approach to registration problems. Therefore, we propose to develop similarity measures that are geometric and computationally very efficient using entropic regularization of optimal transport.

Our approach is to use a regularized optimal transport to design new similarity measures on all of those Hilbert spaces. Understanding the precise connections between the evolution of shapes and probability distributions will be investigated to cross-fertilize both fields by developing novel transportation metrics and diffeomorphic shape flows.

The corresponding numerical schemes are however computationally very costly. Leveraging our understanding of the dynamic optimal transport problem and its numerical resolution, we propose to develop new algorithms. These algorithms will use the smoothness of the Riemannian metric to improve both accuracy and speed, using for instance higher order minimization algorithm on (infinite dimensional) manifolds.

Metric learning and parallel transport for statistical applications.

(Participants: F-X. Vialard, G. Peyré, B. Schmitzer, L. Chizat) The LDDMM framework has been advocated to enable statistics on the space of shapes or images that benefit from the estimation of the deformation. The statistical results of it strongly depend on the choice of the Riemannian metric. A possible direction consists in learning the right invariant Riemannian metric as done in 140 where a correlation matrix (Figure 7) is learnt which represents the covariance matrix of the deformation fields for a given population of shapes. In the same direction, a question of emerging interest in medical imaging is the analysis of time sequence of shapes (called longitudinal analysis) for early diagnosis of disease, for instance 89. A key question is the inter subject comparison of the organ evolution which is usually done by transport of the time evolution in a common coordinate system via parallel transport or other more basic methods. Once again, the statistical results (Figure 8) strongly depend on the choice of the metric or more generally on the connection that defines parallel transport.

Our expertise: Our team has already studied statistics on longitudinal evolutions in 89, 90.

Goals: Developing higher order numerical schemes for parallel transport (only low order schemes are available at the moment) and developing variational models to learn the metric or the connections for improving statistical results.

Axial	Coronal	Sagittal

3.2 Numerical Tools

The above continuous models require a careful discretization, so that the fundamental properties of the models are transferred to the discrete setting. Our team aims at developing innovative discretization schemes as well as associated fast numerical solvers, that can deal with the geometric complexity of the variational problems studied in the applications. This will ensure that the discrete solution is correct and converges to the solution of the continuous model within a guaranteed precision. We give below examples for which a careful mathematical analysis of the continuous to discrete model is essential, and where dedicated non-smooth optimization solvers are required.

3.2.3 First Order Proximal Schemes

$L^2$ proximal methods.

(Participants: G. Peyré, J-D. Benamou, G. Carlier, Jalal Fadili (ENSICaen)) Both sparse regularization problems in imaging (see Section 2.4) and dynamical optimal transport (see Section 2.3) are instances of large scale, highly structured, non-smooth convex optimization problems. First order proximal splitting optimization algorithms have recently gained lots of interest for these applications because they are the only ones capable of scaling to giga-pixel discretizations of images and volumes and at the same time handling non-smooth objective functions. They have been successfully applied to optimal transport  28, 121, congested optimal transport  56 and to sparse regularizations (see for instance  129 and the references therein).

Our expertise: The pioneering work of our team has shown how these proximal solvers can be used to tackle the dynamical optimal transport problem  28, see also  121. We have also recently developed new proximal schemes that can cope with non-smooth composite objectives functions  129.

Goals: We aim at extending these solvers to a wider class of variational problems, most notably optimization under divergence constraints  30. Another subject we are investigating is the extension of these solvers to both non-smooth and non-convex objective functionals, which are mandatory to handle more general transportation problems and novel imaging regularization penalties.

Bregman proximal methods.

(Participants: G. Peyré G. Carlier, L. Nenna, J-D. Benamou, L. Nenna, Marco Cuturi (Kyoto Univ.)) The entropic regularization of the Kantorovich linear program for OT has been shown to be surprisingly simple and efficient, in particular for applications in machine learning  82. As shown in  32, this is a special instance of the general method of Bregman iterations, which is also a particular instance of first order proximal schemes according to the Kullback-Leibler divergence.

Our expertise: We have recently  32 shown how Bregman projections  47 and Dykstra algorithm  24 offer a generic optimization framework to solve a variety of generalized OT problems. Carlier and Dupuis  68 have designed a new method based on alternate Dykstra projections and applied it to the principal-agent problem in microeconomics. We have applied this method in computer graphics in a paper accepted in SIGGRAPH 2015  135. Figure 9 shows the potential of our approach to handle giga-voxel datasets: the input volumetric densities are discretized on a ${100}^3$ computational grid.

Goals: Following some recent works (see in particular  75) we first aim at studying primal-dual optimization schemes according to Bregman divergences (that would go much beyond gradient descent and iterative projections), in order to offer a versatile and very effective framework to solve variational problems involving OT terms. We then also aim at extending the scope of usage of this method to applications in quantum mechanics (Density Functional Theory, see  79) and fluid dynamics (Brenier's weak solutions of the incompressible Euler equation, see  50). The computational challenge is that realistic physical examples are of a huge size not only because of the space discretization of one marginal but also because of the large number of marginals involved (for incompressible Euler the number of marginals equals the number of time steps).

4.1 Natural Sciences

FreeForm Optics, Fluid Mechanics (Incompressible Euler, Semi-Geostrophic equations), Quantum Chemistry (Density Functional Theory), Statistical Physics (Schroedinger problem), Porous Media.

4.2 Signal Processing and inverse problems

Full Waveform Inversion (Geophysics), Super-resolution microscopy (Biology), Satellite imaging (Meteorology)

4.3 Social Sciences

Mean-field games, spatial economics, principal-agent models, taxation, nonlinear pricing.

5.1 Awards

I. Waldspurger received the Bronze medal from CNRS.

I. Waldspurger is the laureate of the “cours Peccot du collège de France”.

6.1 An Entropic Optimal Transport Numerical Approach to the Reflector Problem

Benamou, Jean-David and Rukhaia, Giorgi and IJzerman, Wilbert. In 6 : The point source far field reflector design problem is one of the main classic optimal transport problems with a non-euclidean displacement cost [Wang, 2004] [Glimm and Oliker, 2003]. This work describes the use of Entropic Optimal Transport and the associated Sinkhorn algorithm [Cuturi, 2013] to solve it numerically. As the reflector modelling is based on the Kantorovich potentials , several questions arise. First, on the convergence of the discrete entropic approximation and here we follow the recent work of [Berman, 2017] and in particular the imposed discretization requirements therein. Secondly, the correction of the Entropic bias induced by the Entropic OT, as discussed in particular in [Ramdas et al., 2017] [Genevay et al., 2018] [Feydy et al., 2018], is another important tool to achieve reasonable results. The paper reviews the necessary mathematical and numerical tools needed to produce and discuss the obtained numerical results. We find that Sinkhorn algorithm may be adapted, at least in simple academic cases, to the resolution of the far field reflector problem. Sinkhorn canonical extension to continuous potentials is needed to generate continuous reflector approximations. The use of Sinkhorn divergences [Feydy et al., 2018] is useful to mitigate the entropic bias.

In collaboration with G. Chazareix, We are currently applying this technique to a more realistic “finite source” extension of the point source reflector problem.

6.2 Capacity Constrained Entropic Optimal Transport, Sinkhorn Saturated Domain Out-Summation and Vanishing Temperature

Benamou, Jean-David and Martinet, Mélanie. In 7 : We propose a new method to reduce the computational cost of the Entropic Optimal Transport in the vanishing temperature (ε) limit. As in [Schmitzer, 2016], the method relies on a Sinkhorn continuation "ε-scaling" approach; but instead of truncating out the small values of the Kernel, we rely on the exact "out-summation" of saturated domains for a modified constrained Entropic Optimal problem. The constraint depends on an additional parameter λ. In pratice λ = ε also vanishes and the constraint disappear. Using [Berman, 2017], the convergence of the (ε, λ) continuation method based on this modified problem is established. We then show that the saturated domain can be over estimated from the previous larger (ε, λ). On the saturated zone the solution is constant and known and the domain can be "out-summed" (removed) from Sinkhorn algorithm. The computational cost and memory foot print is shown to be almost linear thanks again to an estimate given by [Berman, 2017]. This is confirmed on 1-D numerical experiments.

6.3 Stability of optimal traffic plans in the irrigation problem

Colombo, Maria and De Rosa, Antonio and Marchese, Andrea and Pegon, Paul and Prouff, Antoine. In 12 : We prove the stability of optimal traffic plans in the irrigation problem, namely the optimality of weak limits P of sequences of traffic plans which are optimal for marginals which weakly converge. This is achieved under a uniform bound on the α-energy of branched transport for any exponent α strictly between 0 and 1. The proof builds upon the Eulerian counterpart of this statement which has been recently established in [Colombo, De Rosa, Marchese, 2019], showing that the limit plan does not produce cancellations in the induced current, as a limit of optimal traffic plans. As a consequence, we prove that P is a good decomposition of its induced current and we deduce the optimality of P.

6.4 Rank optimality for the Burer-Monteiro formulation

Waldspurger, Irène and Waters, Alden. In 4 : When solving large scale semidefinite programs that admit a low-rank solution, a very efficient heuristic is the Burer-Monteiro factorization: Instead of optimizing over the full matrix, one optimizes over its low-rank factors. This strongly reduces the number of variables to optimize, but destroys the convexity of the problem, thus possibly introducing spurious second-order critical points which can prevent local optimization algorithms from finding the solution. Boumal, Voroninski, and Bandeira [2018] have recently shown that, when the size of the factors is of the order of the square root of the number of linear constraints, this does not happen: For almost any cost matrix, second-order critical points are global solutions. In this article, we show that this result is essentially tight: For smaller values of the size, second-order critical points are not generically optimal, even when considering only semidefinite programs with a rank 1 solution.

6.5 A mean field game model for the evolution of cities

Barilla, César and Carlier, Guillaume and Lasry, Jean-Michel In 5 : 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. 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.

6.6 Explicit constants in Harnack inequalities and regularity estimates, with an application to the fast diffusion equation

Bonforte, Matteo and Dolbeault, Jean and Nazaret, Bruno and Simonov, Nikita In 8 : This paper is devoted to the computation of various explicit constants in functional inequalities and regularity estimates for solutions of parabolic equations, which are not available from the literature. We provide new expressions and simplified proofs of the Harnack inequality and the corresponding Hölder continuity of the solution of a linear parabolic equation. We apply these results to the computation of a constructive estimate of a threshold time for the uniform convergence in relative error of the solution of the fast diffusion equation.

6.7 Stability in Gagliardo-Nirenberg inequalities

Bonforte, Matteo and Dolbeault, Jean and Nazaret, Bruno and Simonov In 9 : The purpose of this paper is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg inequalities. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. In both cases, the tail behaviour plays a key role. The regularity properties of the parabolic flow allow us to connect an improved entropy-entropy production inequality during the initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms the distance to the manifold of optimal functions.

6.8 Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT

Carlier, Guillaume and Eichinger, Katharina and Kroshnin, Alexey In 10 : 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.

6.9 An Epigraphical Approach to the Representer Theorem

Duval, Vincent In 13 : Describing the solutions of inverse problems arising in signal or image processing is an important issue both for theoretical and numerical purposes. We propose a principle which describes the solutions to convex variational problems involving a finite number of measurements. We discuss its optimality on various problems concerning the recovery of Radon measures.

6.10 A mixed finite element discretization of dynamical optimal transport

Natale, Andrea and Todeschi, Gabriele In 14 :

In this paper we introduce a new class of finite element discretizations of the quadratic optimal transport problem based on its dynamical formulation. These generalize to the finite element setting the finite difference scheme proposed by 121. We solve the discrete problem using a proximal-splitting approach and we show how to modify this in the presence of regularization terms which are relevant for imaging applications.

6.11 TPFA Finite Volume Approximation of Wasserstein Gradient Flows

Natale, Andrea and Todeschi, Gabriele In 16 : Numerous infinite dimensional dynamical systems arising in different fields have been shown to exhibit a gradient flow structure in the Wasserstein space. We construct Two Point Flux Approximation Finite Volume schemes discretizing such problems which preserve the variational structure and have second order accuracy in space. We propose an interior point method to solve the discrete variational problem, providing an efficient and robust algorithm. We present two applications to test the scheme and show its order of convergence.

7.1 European Initiatives

7.2 National Initiatives

8.0.1 Journal

8.0.2 Invited Talks

• J-D. Benamou at Variational Methods for Evolution, MF Oberwolfach workshop 2038, September 13-19th.
• J-D. Benamou at INS colloquium at Shanghai Jiao Tong University, China. May 15th.
• V. Duval at One World Seminar Mathematical Methods for Arbitrary Data Sources (MADS), June 8th, online.
• V. Duval at International Traveling Workshop on Interactions between low complexity data models and Sensing techniques (iTwist), Decembert 2-4, online.
• G. Carlier: séminaire EDP du Collège de France, workshop ICODE à Paris 7, séminaire (en ligne) sur les MFGs, séminaire (en ligne) Erlangen, workshop Optimal Transport: Regularization and Applications (Columbia, NY, en ligne), Heriot-Watt stat. seminar (en ligne).
• I. Waldspurger at the Optimization for Machine Learning workshop, March 10th, in CIRM (Marseille).
• I. Waldspurger at the PDE and inverse problems machine learning workshop, April 22nd, organized online by the University of California, Los Angeles.
• I. Waldspurger at the séminaire signal-apprentissage of I2M / LIS in Marseille, June 19th, online.
• I. Waldspurger at the Virtual Maxwell Analysis Seminar of Edinburgh, October 23nd.
• I. Waldspurger at International Traveling Workshop on Interactions between low complexity data models and Sensing techniques (iTwist), Decembert 2-4, online.

8.0.3 Research Administration

V. Duval is a member of the Comité de Suivi Doctoral (CSD) and the Comité des Emplois Scientifiques (CES) of the Inria Paris research center.

8.1 Teaching - Supervision - Juries

8.2 Popularization

9.1 Publications of the year

9.2 Other

9.3 Cited publications

• 18 article IsabelleI. Abraham, RomainR. Abraham, MaïtineM. Bergounioux and GuillaumeG. Carlier. Tomographic reconstruction from a few views: a multi-marginal optimal transport approach Preprint Hal-01065981 2014
  back to text
• 19 articleY. Achdou and V. Perez. Iterative strategies for solving linearized discrete mean field games systemsNetw. Heterog. Media722012, 197--217URL: http://dx.doi.org/10.3934/nhm.2012.7.197
  DOIback to textback to text
• 20 articleM. Agueh and GuillaumeG. Carlier. Barycenters in the Wasserstein spaceSIAM J. Math. Anal.4322011, 904--924
  back to text
• 21 articleF. Alter, V. Caselles and A. Chambolle. Evolution of Convex Sets in the Plane by Minimizing the Total Variation FlowInterfaces and Free Boundaries3322005, 329--366
  back to text
• 22 articleF. R.F. Bach. Consistency of Trace Norm MinimizationJ. Mach. Learn. Res.9June 2008, 1019--1048URL: http://dl.acm.org/citation.cfm?id=1390681.1390716
  back to text
• 23 articleF. R.F. Bach. Consistency of the Group Lasso and Multiple Kernel LearningJ. Mach. Learn. Res.9June 2008, 1179--1225URL: http://dl.acm.org/citation.cfm?id=1390681.1390721
  back to text
• 24 articleH. H.H. Bauschke and P. L.P. Combettes. A Dykstra-like algorithm for two monotone operatorsPacific Journal of Optimization432008, 383--391
  back to text
• 25 articleM. F.M. Beg, M. I.M. Miller, A. Trouvé and L. Younes. Computing Large Deformation Metric Mappings via Geodesic Flows of DiffeomorphismsInternational Journal of Computer Vision612February 2005, 139--157URL: http://dx.doi.org/10.1023/B:VISI.0000043755.93987.aa
  back to textback to text
• 26 articleM. Beiglbock, P. Henry-Labordère and F. Penkner. Model-independent bounds for option prices mass transport approachFinance and Stochastics1732013, 477-501URL: http://dx.doi.org/10.1007/s00780-013-0205-8
  DOIback to textback to text
• 27 articleG. Bellettini, V. Caselles and M. Novaga. The Total Variation Flow in $R^N$'J. Differential Equations18422002, 475--525
  back to text
• 28 articleJean-DavidJ.-D. Benamou and Y. Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problemNumer. Math.8432000, 375--393URL: http://dx.doi.org/10.1007/s002110050002
  DOIback to textback to textback to textback to textback to text
• 29 articleJean-DavidJ.-D. Benamou and Y. Brenier. Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problemSIAM J. Appl. Math.5851998, 1450--1461
  back to text
• 30 article Jean-DavidJ.-D. Benamou and GuillaumeG. Carlier. Augmented Lagrangian algorithms for variational problems with divergence constraints JOTA 2015
  back to text back to text back to text back to text back to text back to text
• 31 techreportJean-DavidJ.-D. Benamou, GuillaumeG. Carlier and N. Bonne. An Augmented Lagrangian Numerical approach to solving Mean-Fields GamesINRIADecember 2013, 30
  HALback to text
• 32 article Jean-DavidJ.-D. Benamou, GuillaumeG. Carlier, MarcoM. Cuturi, LucaL. Nenna and GabrielG. Peyré. Iterative Bregman Projections for Regularized Transportation Problems SIAM J. Sci. Comp. to appear 2015
  back to text back to text back to text back to text back to text back to text back to text back to text
• 33 techreport Jean-DavidJ.-D. Benamou, GuillaumeG. Carlier, QuentinQ. Mérigot and ÉdouardÉ. Oudet. Discretization of functionals involving the Monge-Ampère operator HAL July 2014
  HAL back to text back to text
• 34 article Jean-DavidJ.-D. Benamou, F. Collino and Jean-MarieJ.-M. Mirebeau. Monotone and Consistent discretization of the Monge-Ampère operator arXiv preprint arXiv:1409.6694 to appear in Math of Comp 2014
  back to text
• 35 articleJean-DavidJ.-D. Benamou, Brittany DB. Froese and AdamA. Oberman. Numerical solution of the optimal transportation problem using the Monge--Ampere equationJournal of Computational Physics2602014, 107--126
  back to textback to textback to text
• 36 articleJean-DavidJ.-D. Benamou, B. D.B. Froese and AdamA. Oberman. Two numerical methods for the elliptic Monge-Ampère equationM2AN Math. Model. Numer. Anal.4442010, 737--758
  back to text
• 37 articleF. Benmansour, GuillaumeG. Carlier, GabrielG. Peyré and F. Santambrogio. Numerical approximation of continuous traffic congestion equilibriaNetw. Heterog. Media432009, 605--623
  back to textback to text
• 38 articleM. Benning and M. Burger. Ground states and singular vectors of convex variational regularization methodsMeth. Appl. Analysis202013, 295--334
  back to text
• 39 article B. Berkels, A. Effland and M. Rumpf. Time discrete geodesic paths in the space of images Arxiv preprint 2014
  back to text
• 40 article J. Bigot and T. Klein. Consistent estimation of a population barycenter in the Wasserstein space Preprint arXiv:1212.2562 2012
  back to text
• 41 articleAdrienA. Blanchet and P. Laurençot. The parabolic-parabolic Keller-Segel system with critical diffusion as a gradient flow in $R^d,d3$'Comm. Partial Differential Equations3842013, 658--686URL: http://dx.doi.org/10.1080/03605302.2012.757705
  DOIback to text
• 42 articleJérémyJ. Bleyer, GuillaumeG. Carlier, VincentV. Duval, Jean-MarieJ.-M. Mirebeau and GabrielG. Peyré. A $$ Convergence Result for the Upper Bound Limit Analysis of Plates'ESAIM: Mathematical Modelling and Numerical Analysis501January 2016, 215--235URL: https://www.esaim-m2an.org/articles/m2an/abs/2016/01/m2an141087/m2an141087.html
  DOIback to textback to text
• 43 articleN. Bonneel, J. Rabin, GabrielG. Peyré and H. Pfister. Sliced and Radon Wasserstein Barycenters of MeasuresJournal of Mathematical Imaging and Vision5112015, 22--45URL: http://hal.archives-ouvertes.fr/hal-00881872/
  back to text
• 44 techreportU. Boscain, R. Chertovskih, J-P.J.-P. Gauthier, D. Prandi and A. Remizov. Highly corrupted image inpainting through hypoelliptic diffusionPreprint CMAP2014, URL: http://hal.archives-ouvertes.fr/hal-00842603/
  back to text
• 45 articleGuyG. Bouchitté and GiuseppeG. Buttazzo. Characterization of optimal shapes and masses through Monge-Kantorovich equationJ. Eur. Math. Soc. (JEMS)322001, 139--168URL: http://dx.doi.org/10.1007/s100970000027
  DOIback to text
• 46 articleL. Brasco, GuillaumeG. Carlier and F. Santambrogio. Congested traffic dynamics, weak flows and very degenerate elliptic equationsJ. Math. Pures Appl. (9)9362010, 652--671
  back to text
• 47 articleL. M.L. Bregman. The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programmingUSSR computational mathematics and mathematical physics731967, 200--217
  back to text
• 48 articleY. Brenier. Décomposition polaire et réarrangement monotone des champs de vecteursC. R. Acad. Sci. Paris Sér. I Math.305191987, 805--808
  back to text
• 49 articleY. Brenier, U. Frisch, M. Henon, G. Loeper, S. Matarrese, R. Mohayaee and A. Sobolevski. Reconstruction of the early universe as a convex optimization problem Mon. Not. Roy. Astron. Soc.3462003, 501--524URL: http://arxiv.org/pdf/astro-ph/0304214.pdf
  back to text
• 50 articleY. Brenier. Generalized solutions and hydrostatic approximation of the Euler equationsPhys. D23714-172008, 1982--1988URL: http://dx.doi.org/10.1016/j.physd.2008.02.026
  DOIback to textback to text
• 51 articleY. Brenier. Polar factorization and monotone rearrangement of vector-valued functionsComm. Pure Appl. Math.4441991, 375--417URL: http://dx.doi.org/10.1002/cpa.3160440402
  DOIback to text
• 52 articleM. Bruveris, L. Risser and F.-X. Vialard. Mixture of Kernels and Iterated Semidirect Product of Diffeomorphisms GroupsMultiscale Modeling & Simulation1042012, 1344-1368
  back to text
• 53 article M. Burger, M. DiFrancesco, P. Markowich and M. T.M. Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics DCDS B 19 2014
  back to text
• 54 articleM. Burger, M. Franek and C. Schonlieb. Regularized regression and density estimation based on optimal transportAppl. Math. Res. Expr.22012, 209--253
  back to text
• 55 article M. Burger and S. Osher. A guide to the TV zoo Level-Set and PDE-based Reconstruction Methods, Springer 2013
  back to text
• 56 articleG. Buttazzo, C. Jimenez and ÉdouardÉ. Oudet. An optimization problem for mass transportation with congested dynamicsSIAM J. Control Optim.4832009, 1961--1976
  back to text
• 57 articleH. Byrne and D. Drasdo. Individual-based and continuum models of growing cell populations: a comparisonJournal of Mathematical Biology584-52009, 657-687
  back to textback to textback to text
• 58 incollectionL. A.L. Caffarelli, S. A.S. Kochengin and V.I.V. Oliker. On the numerical solution of the problem of reflector design with given far-field scattering dataMonge Ampère equation: applications to geometry and optimization (Deerfield Beach, FL, 1997)226Contemp. Math.Providence, RIAmer. Math. Soc.1999, 13--32URL: http://dx.doi.org/10.1090/conm/226/03233
  DOIback to text
• 59 articleL. A.L. Caffarelli. The regularity of mappings with a convex potentialJ. Amer. Math. Soc.511992, 99--104URL: http://dx.doi.org/10.2307/2152752
  DOIback to text
• 60 article C. CanCeritoglu. Computational Analysis of LDDMM for Brain Mapping Frontiers in Neuroscience 7 2013
  back to text
• 61 articleE. J.E. Candès and C. Fernandez-Granda. Super-Resolution from Noisy DataJournal of Fourier Analysis and Applications1962013, 1229--1254
  back to text
• 62 articleE. J.E. Candès and C. Fernandez-Granda. Towards a Mathematical Theory of Super-ResolutionCommunications on Pure and Applied Mathematics6762014, 906--956
  back to textback to textback to text
• 63 articleE. Candes and M. Wakin. An Introduction to Compressive SensingIEEE Signal Processing Magazine2522008, 21--30
  back to text
• 64 articleP. Cardaliaguet, GuillaumeG. Carlier and B. Nazaret. Geodesics for a class of distances in the space of probability measuresCalc. Var. Partial Differential Equations483-42013, 395--420
  back to text
• 65 articleGuillaumeG. Carlier. A general existence result for the principal-agent problem with adverse selectionJ. Math. Econom.3512001, 129--150
  back to textback to text
• 66 techreport GuillaumeG. Carlier, V. Chernozhukov and A. Galichon. Vector Quantile Regression Arxiv 1406.4643 2014
  back to text
• 67 articleGuillaumeG. Carlier, M. Comte, I. Ionescu and GabrielG. Peyré. A Projection Approach to the Numerical Analysis of Limit Load ProblemsMathematical Models and Methods in Applied Sciences2162011, 1291--1316URL: http://hal.archives-ouvertes.fr/hal-00450000/
  DOIback to text
• 68 techreport GuillaumeG. Carlier and XavierX. Dupuis. An iterated projection approach to variational problems under generalized convexity constraints and applications In preparation 2015
  back to text back to text
• 69 articleGuillaumeG. Carlier, C. Jimenez and F. Santambrogio. Optimal Transportation with Traffic Congestion and Wardrop EquilibriaSIAM Journal on Control and Optimization4732008, 1330-1350
  back to text
• 70 articleGuillaumeG. Carlier, T. Lachand-Robert and B. Maury. A numerical approach to variational problems subject to convexity constraintNumer. Math.8822001, 299--318URL: http://dx.doi.org/10.1007/PL00005446
  DOIback to textback to text
• 71 article GuillaumeG. Carlier, AdamA. Oberman and ÉdouardÉ. Oudet. Numerical methods for matching for teams and Wasserstein barycenters M2AN to appear 2015
  back to text back to text back to text
• 72 articleJ. A.J. Carrillo, S. Lisini and E. Mainini. Uniqueness for Keller-Segel-type chemotaxis modelsDiscrete Contin. Dyn. Syst.3442014, 1319--1338URL: http://dx.doi.org/10.3934/dcds.2014.34.1319
  DOIback to text
• 73 articleV. Caselles, A. Chambolle and M. Novaga. The discontinuity set of solutions of the TV denoising problem and some extensionsMultiscale Modeling and Simulation632007, 879--894
  back to text
• 74 articleF. A. C. C.F. Chalub, P. A.P. Markowich, B. Perthame and C. Schmeiser. Kinetic models for chemotaxis and their drift-diffusion limitsMonatsh. Math.1421-22004, 123--141URL: http://dx.doi.org/10.1007/s00605-004-0234-7
  DOIback to text
• 75 article A. Chambolle and T. Pock. On the ergodic convergence rates of a first-order primal-dual algorithm Preprint OO/2014/09/4532 2014
  back to text
• 76 techreportG. Charpiat, G. Nardi, GabrielG. Peyré and F.-X. Vialard. Finsler Steepest Descent with Applications to Piecewise-regular Curve EvolutionPreprint hal-008498852013, URL: http://hal.archives-ouvertes.fr/hal-00849885/
  back to textback to text
• 77 articleS. S.S. Chen, D. L.D. Donoho and M. A.M. Saunders. Atomic decomposition by basis pursuitSIAM journal on scientific computing2011999, 33--61
  back to text
• 78 articleP. Choné and H. V. J.H. Le Meur. Non-convergence result for conformal approximation of variational problems subject to a convexity constraintNumer. Funct. Anal. Optim.225-62001, 529--547URL: http://dx.doi.org/10.1081/NFA-100105306
  DOIback to text
• 79 articleC. Cotar, G. Friesecke and C. Kluppelberg. Density Functional Theory and Optimal Transportation with Coulomb CostCommunications on Pure and Applied Mathematics6642013, 548--599URL: http://dx.doi.org/10.1002/cpa.21437
  DOIback to textback to text
• 80 articleM. J. P.M. Cullen, W. Gangbo and G. Pisante. The semigeostrophic equations discretized in reference and dual variablesArch. Ration. Mech. Anal.18522007, 341--363URL: http://dx.doi.org/10.1007/s00205-006-0040-6
  DOIback to text
• 81 articleM. J. P.M. Cullen, J. Norbury and R. J.R. Purser. Generalised Lagrangian solutions for atmospheric and oceanic flowsSIAM J. Appl. Math.5111991, 20--31
  back to text
• 82 inproceedingsMarcoM. Cuturi. Sinkhorn Distances: Lightspeed Computation of Optimal TransportProc. NIPS2013, 2292--2300
  back to textback to textback to text
• 83 articleE. J.E. Dean and R. Glowinski. Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère typeComput. Methods Appl. Mech. Engrg.19513-162006, 1344--1386
  back to text
• 84 articleVincentV. Duval and GabrielG. Peyré. Exact Support Recovery for Sparse Spikes DeconvolutionFoundations of Computational Mathematics2014, 1-41URL: http://dx.doi.org/10.1007/s10208-014-9228-6
  DOIback to textback to textback to textback to textback to textback to textback to textback to textback to textback to text
• 85 articleVincentV. Duval and GabrielG. Peyré. Sparse regularization on thin grids I: the L assoInverse Problems3352017, 055008URL: http://stacks.iop.org/0266-5611/33/i=5/a=055008
  DOIback to textback to textback to text
• 86 articleJ. Fehrenbach and Jean-MarieJ.-M. Mirebeau. Sparse Non-negative Stencils for Anisotropic DiffusionJournal of Mathematical Imaging and Vision4912014, 123-147URL: http://dx.doi.org/10.1007/s10851-013-0446-3
  back to textback to text
• 87 articleC. Fernandez-Granda. Support detection in super-resolutionProc. Proceedings of the 10th International Conference on Sampling Theory and Applications2013, 145--148
  back to text
• 88 article A. Figalli, R.J.R. McCann and Y.H.Y. Kim. When is multi-dimensional screening a convex program? Journal of Economic Theory 2011
  back to text back to text
• 89 articleJ-B.J.-B. Fiot, H. Raguet, L. Risser, L. D.L. Cohen, J. Fripp and F.-X. Vialard. Longitudinal deformation models, spatial regularizations and learning strategies to quantify Alzheimer's disease progressionNeuroImage: Clinical402014, 718 - 729URL: http://www.sciencedirect.com/science/article/pii/S2213158214000205
  DOIback to textback to textback to text
• 90 incollectionJ-B.J.-B. Fiot, L. Risser, L. D.L. Cohen, J. Fripp and F.-X. Vialard. Local vs Global Descriptors of Hippocampus Shape Evolution for Alzheimer's Longitudinal Population AnalysisSpatio-temporal Image Analysis for Longitudinal and Time-Series Image Data7570Lecture Notes in Computer ScienceSpringer Berlin Heidelberg2012, 13-24URL: http://dx.doi.org/10.1007/978-3-642-33555-6_2
  DOIback to text
• 91 article U. Frisch, S. Matarrese, R. Mohayaee and A. Sobolevski. Monge-Ampère-Kantorovitch (MAK) reconstruction of the eary universe Nature 417 260 2002
  back to text
• 92 articleB. D.B. Froese and AdamA. Oberman. Convergent filtered schemes for the Monge-Ampère partial differential equationSIAM J. Numer. Anal.5112013, 423--444
  back to text
• 93 article A. Galichon, P. Henry-Labordère and N. Touzi. A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Loopback options submitted to Annals of Applied Probability 2011
  back to text back to text
• 94 articleW. Gangbo and R.J.R. McCann. The geometry of optimal transportationActa Math.17721996, 113--161URL: http://dx.doi.org/10.1007/BF02392620
  DOIback to text
• 95 articleE. Ghys. Gaspard Monge, Le mémoire sur les déblais et les remblaisImage des mathématiques, CNRS2012, URL: http://images.math.cnrs.fr/Gaspard-Monge,1094.html
  back to text
• 96 incollectionO. Guéant, J-M.J.-M. Lasry and P-L.P.-L. Lions. Mean field games and applicationsParis-Princeton Lectures on Mathematical Finance 20102003Lecture Notes in Math.BerlinSpringer2011, 205--266URL: http://dx.doi.org/10.1007/978-3-642-14660-2_3
  DOIback to text
• 97 book G. Herman. Image reconstruction from projections: the fundamentals of computerized tomography Academic Press 1980
  back to text
• 98 articleD. D.D. Holm, J. T.J. Ratnanather, A. Trouvé and L. Younes. Soliton dynamics in computational anatomyNeuroImage232004, S170--S178
  back to text
• 99 incollectionB. J.B. Hoskins. The mathematical theory of frontogenesisAnnual review of fluid mechanics, Vol. 14Palo Alto, CAAnnual Reviews1982, 131--151
  back to text
• 100 articleW. Jäger and S. Luckhaus. On explosions of solutions to a system of partial differential equations modelling chemotaxisTrans. Amer. Math. Soc.32921992, 819--824URL: http://dx.doi.org/10.2307/2153966
  DOIback to text
• 101 articleR. Jordan, D. Kinderlehrer and F. Otto. The variational formulation of the Fokker-Planck equationSIAM J. Math. Anal.2911998, 1--17
  back to textback to textback to textback to text
• 102 articleL. Kantorovitch. On the translocation of massesC. R. (Doklady) Acad. Sci. URSS (N.S.)371942, 199--201
  back to text
• 103 articleE. Klann. A Mumford-Shah-Like Method for Limited Data Tomography with an Application to Electron TomographySIAM J. Imaging Sciences442011, 1029--1048
  back to text
• 104 articleJ-M.J.-M. Lasry and P-L.P.-L. Lions. Mean field gamesJpn. J. Math.212007, 229--260URL: http://dx.doi.org/10.1007/s11537-007-0657-8
  DOIback to textback to text
• 105 articleJ. Lasserre. Global Optimization with Polynomials and the Problem of MomentsSIAM Journal on Optimization1132001, 796-817
  back to text
• 106 articleJanJ. Lellmann, Dirk A.D. Lorenz, Carola-BibianeC.-B. Schönlieb and TuomoT. Valkonen. Imaging with Kantorovich-Rubinstein DiscrepancySIAM J. Imaging Sciences742014, 2833--2859
  back to text
• 107 articleChristianC. Léonard. A survey of the Schrödinger problem and some of its connections with optimal transportDiscrete Contin. Dyn. Syst.3442014, 1533--1574URL: http://dx.doi.org/10.3934/dcds.2014.34.1533
  DOIback to text
• 108 articleA. S.A. Lewis. Active sets, nonsmoothness, and sensitivitySIAM Journal on Optimization1332003, 702--725
  back to text
• 109 articleB. Li, F. Habbal and M. Ortiz. Optimal transportation meshfree approximation schemes for Fluid and plastic FlowsInt. J. Numer. Meth. Engng 83:1541--579832010, 1541--1579
  back to text
• 110 articleG. Loeper. A fully nonlinear version of the incompressible Euler equations: the semigeostrophic systemSIAM J. Math. Anal.3832006, 795--823 (electronic)
  back to text
• 111 articleG. Loeper and F. Rapetti. Numerical solution of the Monge-Ampére equation by a Newton's algorithmC. R. Math. Acad. Sci. Paris34042005, 319--324
  back to text
• 112 article D. Lombardi and E. Maitre. Eulerian models and algorithms for unbalanced optimal transport Preprint hal-00976501 2013
  back to text
• 113 article J. Maas, M. Rumpf, C. Schonlieb and S. Simon. A generalized model for optimal transport of images including dissipation and density modulation Arxiv preprint 2014
  back to text
• 114 book S. G.S. Mallat. A wavelet tour of signal processing Elsevier/Academic Press, Amsterdam 2009
  back to text
• 115 articleB. Maury, A. Roudneff-Chupin and F. Santambrogio. A macroscopic crowd motion model of gradient flow typeMath. Models Methods Appl. Sci.20102010, 1787--1821URL: http://dx.doi.org/10.1142/S0218202510004799
  DOIback to textback to text
• 116 articleQuentinQ. Mérigot. A multiscale approach to optimal transportComputer Graphics Forum3052011, 1583--1592
  back to text
• 117 articleQuentinQ. Mérigot and ÉdouardÉ. Oudet. Handling Convexity-Like Constraints in Variational ProblemsSIAM J. Numer. Anal.5252014, 2466--2487
  back to textback to text
• 118 articleM. I.M. Miller, A. Trouvé and L. Younes. Geodesic Shooting for Computational AnatomyJournal of Mathematical Imaging and Vision242March 2006, 209--228URL: http://dx.doi.org/10.1007/s10851-005-3624-0
  back to text
• 119 article Jean-MarieJ.-M. Mirebeau. Adaptive, Anisotropic and Hierarchical cones of Discrete Convex functions Preprint 2014
  back to text back to text back to text
• 120 articleJean-MarieJ.-M. Mirebeau. Anisotropic Fast-Marching on Cartesian Grids Using Lattice Basis ReductionSIAM Journal on Numerical Analysis5242014, 1573-1599
  back to text
• 121 articleN. Papadakis, GabrielG. Peyré and ÉdouardÉ. Oudet. Optimal Transport with Proximal SplittingSIAM Journal on Imaging Sciences712014, 212--238URL: http://hal.archives-ouvertes.fr/hal-00816211/
  DOIback to textback to textback to text
• 122 articleB. Pass and N. Ghoussoub. Optimal transport: From moving soil to same-sex marriageCMS Notes452013, 14--15
  back to text
• 123 articleB. Pass. Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation ProblemSIAM Journal on Mathematical Analysis4362011, 2758-2775
  back to text
• 124 articleB. Perthame, F. Quiros and J. L.J. Vazquez. The Hele-Shaw Asymptotics for Mechanical Models of Tumor GrowthArchive for Rational Mechanics and Analysis21212014, 93-127URL: http://dx.doi.org/10.1007/s00205-013-0704-y
  DOIback to textback to text
• 125 articleJ. Petitot. The neurogeometry of pinwheels as a sub-riemannian contact structureJournal of Physiology-Paris97232003, 265--309
  back to text
• 126 articleGabrielG. Peyré. Texture Synthesis with GroupletsPattern Analysis and Machine Intelligence, IEEE Transactions on324April 2010, 733--746
  back to textback to text
• 127 articleB. Piccoli and F. Rossi. Generalized Wasserstein distance and its application to transport equations with sourceArchive for Rational Mechanics and Analysis21112014, 335--358
  back to text
• 128 article ClariceC. Poon. Structure dependent sampling in compressed sensing: theoretical guarantees for tight frames Applied and Computational Harmonic Analysis 2015
  back to text
• 129 articleH. Raguet, J. Fadili and GabrielG. Peyré. A Generalized Forward-Backward SplittingSIAM Journal on Imaging Sciences632013, 1199--1226URL: http://hal.archives-ouvertes.fr/hal-00613637/
  DOIback to textback to textback to text
• 130 article J.-C. Rochet and P. Choné. Ironing, Sweeping and multi-dimensional screening Econometrica 1998
  back to text
• 131 articleL.I.L. Rudin, S. Osher and E. Fatemi. Nonlinear total variation based noise removal algorithmsPhysica D: Nonlinear Phenomena6011992, 259--268URL: http://dx.doi.org/10.1016/0167-2789(92)90242-F
  back to textback to text
• 132 book O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen. Variational Methods in Imaging Springer 2008
  back to text
• 133 article TanyaT. Schmah, LaurentL. Risser and François-XavierF.-X. Vialard. Diffeomorphic image matching with left-invariant metrics Fields Institute Communications series, special volume in memory of Jerrold E. Marsden January 2014
  back to text back to text
• 134 inproceedingsT. Schmah, L. Risser and F.-X. Vialard. Left-Invariant Metrics for Diffeomorphic Image Registration with Spatially-Varying RegularisationMICCAI (1)2013, 203-210
  back to text
• 135 article JustinJ. Solomon, F. de~Goes, GabrielG. Peyré, MarcoM. Cuturi, A. Butscher, A. Nguyen, T. Du and L. Guibas. Convolutional Wasserstein Distances: Efficient Optimal Transportation on Geometric Domains ACM Transaction on Graphics, Proc. SIGGRAPH'15 to appear 2015
  back to text back to text back to text
• 136 articleR. Tibshirani. Regression shrinkage and selection via the LassoJournal of the Royal Statistical Society. Series B. Methodological5811996, 267--288
  back to textback to text
• 137 article A. Trouvé and F.-X. Vialard. Shape splines and stochastic shape evolutions: A second order point of view Quarterly of Applied Mathematics 2012
  back to text
• 138 articleS. Vaiter, M. Golbabaee, J. Fadili and GabrielG. Peyré. Model Selection with Piecewise Regular GaugesInformation and Inferenceto appear2015, URL: http://hal.archives-ouvertes.fr/hal-00842603/
  back to textback to textback to text
• 139 articleF.-X. Vialard, L. Risser, D. Rueckert and C.J.C. Cotter. Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint CalculationInternational Journal of Computer Vision9722012, 229-241URL: http://dx.doi.org/10.1007/s11263-011-0481-8
  DOIback to text
• 140 incollectionF.-X. Vialard and L. Risser. Spatially-Varying Metric Learning for Diffeomorphic Image Registration: A Variational FrameworkMedical Image Computing and Computer-Assisted Intervention MICCAI 20148673Lecture Notes in Computer ScienceSpringer International Publishing2014, 227-234
  back to textback to text
• 141 bookC. Villani. Optimal transport338Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]Old and newBerlinSpringer-Verlag2009, xxii+973URL: http://dx.doi.org/10.1007/978-3-540-71050-9
  DOIback to text
• 142 bookC. Villani. Topics in optimal transportation58Graduate Studies in MathematicsAmerican Mathematical Society, Providence, RI2003, xvi+370
  back to text
• 143 articleX-J.X.-J. Wang. On the design of a reflector antenna. IICalc. Var. Partial Differential Equations2032004, 329--341URL: http://dx.doi.org/10.1007/s00526-003-0239-4
  DOIback to text
• 144 articleB. Wirth, L. Bar, M. Rumpf and G. Sapiro. A continuum mechanical approach to geodesics in shape spaceInternational Journal of Computer Vision9332011, 293--318
  back to text
• 145 articleJ. Wright, Y. Ma, J. Mairal, G. Sapiro, T. ST. Huang and S. Yan. Sparse representation for computer vision and pattern recognitionProceedings of the IEEE9862010, 1031--1044
  back to text