Activity report
RNSR: 201321083P
Research center
In partnership with:
Université Paris-Dauphine, CNRS
Team name:
Advances in Numerical Calculus of Variations
In collaboration with:
Applied Mathematics, Computation and Simulation
Numerical schemes and simulations
Creation of the Project-Team: 2015 December 01


Computer Science and Digital Science

  • A5.3. Image processing and analysis
  • A5.9. Signal processing
  • A6.1.1. Continuous Modeling (PDE, ODE)
  • A6.2.1. Numerical analysis of PDE and ODE
  • A6.2.6. Optimization
  • A9. Artificial intelligence

Other Research Topics and Application Domains

  • B1.2. Neuroscience and cognitive science
  • B9.5.2. Mathematics
  • B9.5.3. Physics
  • B9.5.4. Chemistry
  • B9.6.3. Economy, Finance

1 Team members, visitors, external collaborators

Research Scientists

  • Jean-David Benamou [Team leader, INRIA, Senior Researcher, HDR]
  • Antonin Chambolle [CNRS & DAUPHINE PSL, Senior Researcher, from Sep 2022, HDR]
  • Vincent Duval [INRIA, Senior Researcher, HDR]
  • Thomas Gallouët [INRIA, Researcher]
  • Flavien Léger [INRIA, Researcher]
  • Irene Waldspurger [CNRS & DAUPHINE PSL, Researcher]

Faculty Members

  • Guillaume Carlier [DAUPHINE PSL, Professor, HDR]
  • Paul Pegon [DAUPHINE PSL, Associate Professor]
  • François-Xavier Vialard [UNIV GUSTAVE EIFFEL, Professor]

Post-Doctoral Fellow

  • Robert Tovey [INRIA, until Aug 2022]

PhD Students

  • Guillaume Chazareix [DAUPHINE PSL]
  • Katharina Eichinger [DAUPHINE PSL]
  • Hugo Malamut [DAUPHINE PSL, from Jul 2022]
  • Romain Petit [DAUPHINE PSL]
  • Joao Pinto Anastacio Machado [DAUPHINE PSL]
  • Erwan Stampfli [UNIV PARIS SACLAY]
  • Maxime Sylvestre [DAUPHINE PSL, from Sep 2022]
  • Gabriele Todeschi [FSMP, from Apr 2022]
  • Adrien Vacher [UNIV PARIS EST]

Administrative Assistant

  • Derya Gok [INRIA]

2 Overall objectives

2.1 Introduction

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

Optimal Transport, Old and New.

Optimal Mass Transportation is a mathematical research topic which started two centuries ago with Monge's work on the “Théorie des déblais et des remblais" (see  101). This engineering problem consists in minimizing the transport cost between two given mass densities. In the 40's, Kantorovich  110 introduced a powerful linear relaxation and introduced its dual formulation. 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. Since the seminal discoveries of Brenier in the 90's  64, 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 monographs  134, 135, 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. Optimal Transportation has also received a lot of attention from probabilists (see for instance the recent survey 114 for an overview of the Schrödinger problem which is a stochastic variant of the Benamou-Brenier dynamical formulation of optimal transport). The development of numerical methods for Optimal Transportation and Optimal Transportation related problems is a difficult topic and comparatively underdeveloped. This research field has experienced a surge of activity in the last five years, with important contributions of the Mokaplan group (see the list of important publications of the team). We describe below a few of recent and less recent Optimal Transportation concepts and methods which are connected to the future activities of Mokaplan  :

Brenier's theorem  67 characterizes the unique optimal map as the gradient of a convex potential. As such Optimal Transportation may be interpreted as an infinite dimensional optimisation problem under “convexity constraint": i.e. the solution of this infinite dimensional optimisation problem is a convex potential. This connects Optimal Transportation to “convexity constrained" non-linear variational problems such as, for instance, Newton's problem of the body of minimal resistance. The value function of the optimal transport problem is also known to define a distance between source and target densities called the Wasserstein distance which plays a key role in many applications such as image processing.

Monge-Ampère Methods.

A formal substitution of the optimal transport map as the gradient of a convex potential in the mass conservation constraint (a Jacobian equation) gives a non-linear Monge-Ampère equation. Caffarelli  71 used this result to extend the regularity theory for the Monge-Ampère equation. In the last ten years, it also motivated new research on numerical solvers for non-linear degenerate Elliptic equations  9311857  56 and the references therein. Geometric approaches based on Laguerre diagrams and discrete data 121 have also been developed. Monge-Ampère based Optimal Transportation solvers have recently given the first linear cost computations of Optimal Transportation (smooth) maps.

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  100. Optimal Transportation techniques have been applied for example to a Coulomb ground cost in Quantum chemistry in relation with Density Functional theory  87. 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  126 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 42. 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  125.

Figure 1

Example of color transfer between two images, computed using the method developed in  , see also  . The image framed in red and blue are the input images. Top and middle row: adjusted image where the color of the transported histogram has been imposed. Bottom row: geodesic (displacement) interpolation between the histogram of the chrominance of the image.

Figure 1:

Example of color transfer between two images, computed using the method developed in  55, see also  130. The image framed in red and blue are the input images. Top and middle row: adjusted image where the color of the transported histogram has been imposed. Bottom row: geodesic (displacement) interpolation between the histogram of the chrominance of the image.

Numerical Applications of Optimal Transportation.

Optimal transport has found many applications, starting from its relation with several physical models such as the semi-geostrophic equations in meteorology  105, 90, 89, 53, 117, mesh adaptation 116, the reconstruction of the early mass distribution of the Universe  98, 65 in Astrophysics, and the numerical optimisation of reflectors following the Optimal Transportation interpretation of Oliker  70 and Wang  136. Extensions of OT such as multi-marginal transport has potential applications in Density Functional Theory , Generalized solution of Euler equations 66 (DFT) and in statistics and finance  51, 99 .... Recently, there has been a spread of interest in applications of OT methods in imaging sciences  60, statistics  58 and machine learning  91. This is largely due to the emergence of fast numerical schemes to approximate the transportation distance and its generalizations, see for instance  55. Figure 1 shows an example of application of OT to color transfer. Figure 2 shows an example of application in computer graphics to interpolate between input shapes.

Figure 2

Example of barycenter between shapes computed using optimal transport barycenters of the uniform densities inside the 3 extremal shapes, computed as detailed in  . Note that the barycenters are not in general uniform distributions, and we display them as the surface defined by a suitable level-set of the density.

Figure 2:

Example of barycenter between shapes computed using optimal transport barycenters of the uniform densities inside the 3 extremal shapes, computed as detailed in  130. Note that the barycenters are not in general uniform distributions, and we display them as the surface defined by a suitable level-set of the density.

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  52. 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  54. 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  112 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  61.

Figure 3

Examples of displacement interpolation (geodesic for optimal transport) according to a non-Euclidean Riemannian metric (the mass is constrained to move inside a maze) between to input Gaussian distributions. Note that the maze is dynamic: its topology change over time, the mass being “trapped” at time t=1/3t=1/3.

Figure 3:

Examples of displacement interpolation (geodesic for optimal transport) according to a non-Euclidean Riemannian metric (the mass is constrained to move inside a maze) between to input Gaussian distributions. Note that the maze is dynamic: its topology change over time, the mass being “trapped” at time t=1/3.

Gradient Flows for the Wasserstein Distance.

Another extension is, instead of considering geodesic for transportation metric (i.e. minimizing the Wasserstein distance to a target measure), to make the density evolve in order to minimize some functional. Computing the steepest descent direction with respect to the Wasserstein distance defines a so-called Wasserstein gradient flow, also known as JKO gradient flows after its authors  109. This is a popular tool to study a large class of non-linear diffusion equations. Two interesting examples are the Keller-Segel system for chemotaxis  108, 79 and a model of congested crowd motion proposed by Maury, Santambrogio and Roudneff-Chupin  120. From the numerical point of view, these schemes are understood to be the natural analogue of implicit scheme for linear parabolic equations. The resolution is however costly as it involves taking the derivative in the Wasserstein sense of the relevant energy, which in turn requires the resolution of a large scale convex but non-smooth minimization.

Geodesic on infinite dimensional Riemannian spaces.

To tackle more complicated warping problems, such as those encountered in medical image analysis, one unfortunately has to drop the convexity of the functional involved in defining the gradient flow. This gradient flow can either be understood as defining a geodesic on the (infinite dimensional) group of diffeomorphisms  50, or on a (infinite dimensional) space of curves or surfaces  137. The de-facto standard to define, analyze and compute these geodesics is the “Large Deformation Diffeomorphic Metric Mapping” (LDDMM) framework of Trouvé, Younes, Holm and co-authors  50, 104. While in the CFD formulation of optimal transport, the metric on infinitesimal deformations is just the L2 norm (measure according to the density being transported), in LDDMM, one needs to use a stronger regularizing metric, such as Sobolev-like norms or reproducing kernel Hilbert spaces (RKHS). This enables a control over the smoothness of the deformation which is crucial for many applications. The price to pay is the need to solve a non-convex optimization problem through geodesic shooting method  122, which requires to integrate backward and forward the geodesic ODE. The resulting strong Riemannian geodesic structure on spaces of diffeomorphisms or shapes is also pivotal to allow us to perform statistical analysis on the tangent space, to define mean shapes and perform dimensionality reduction when analyzing large collection of input shapes (e.g. to study evolution of a diseases in time or the variation across patients)  72.

2.4 Sparsity in Imaging

Sparse 1 regularization.

Beside image warping and registration in medical image analysis, a key problem in nearly all imaging applications is the reconstruction of high quality data from low resolution observations. This field, commonly referred to as “inverse problems”, is very often concerned with the precise location of features such as point sources (modeled as Dirac masses) or sharp contours of objects (modeled as gradients being Dirac masses along curves). The underlying intuition behind these ideas is the so-called sparsity model (either of the data itself, its gradient, or other more complicated representations such as wavelets, curvelets, bandlets  119 and learned representation  138).

The huge interest in these ideas started mostly from the introduction of convex methods to serve as proxy for these sparse regularizations. The most well known is the 1 norm introduced independently in imaging by Donoho and co-workers under the name “Basis Pursuit”  84 and in statistics by Tibshirani  131 under the name “Lasso”. A more recent resurgence of this interest dates back to 10 years ago with the introduction of the so-called “compressed sensing” acquisition techniques  75, which make use of randomized forward operators and 1-type reconstruction.

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  74, 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  73, 95, 94 have used this framework to provide theoretical performance guarantees by basically studying how the distance between neighboring spikes impacts noise stability.

Figure 4:

Two example of application of the total variation regularization of functions. Left: image segmentation into homogeneous color regions. Right: image zooming (increasing the number of pixels while keeping the edges sharp).

Low complexity regularization and partial smoothness.

In image processing, one of the most popular methods is the total variation regularization  129, 68. It favors low-complexity images that are piecewise constant, see Figure 4 for some examples on how to solve some image processing problems. Beside applications in image processing, sparsity-related ideas also had a deep impact in statistics  131 and machine learning  46. As a typical example, for applications to recommendation systems, it makes sense to consider sparsity of the singular values of matrices, which can be relaxed using the so-called nuclear norm (a.k.a. trace norm)  45. The underlying methodology is to make use of low-complexity regularization models, which turns out to be equivalent to the use of partly-smooth regularization functionals  115, 133 enforcing the solution to belong to a low-dimensional manifold.

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.

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  52, 128). 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  55) and machine learning (see  91).

3 Research program

Since its creation, the Mokaplan team has made important contributions in Optimal Transport both on the theoretical and the numerical side, together with applications such as fluid mechanics, the simulation biological systems, machine learning. We have also contributed to to the field of inverse problems in signal and image processing (super-resolution, nonconvex low rank matrix recovery). In 2022, the team was renewed with the following research program which broadens our spectrum and addresses exciting new problems.

3.1 OT and related variational problems solvers encore et toujours

Participants: Flavien Léger, Jean-David Benamou, Guillaume Carlier, Thomas Gallouët, François-Xavier Vialard, Guillaume Chazareix , Adrien Vacher , Paul Pegon.

  • Asymptotic analysis of entropic OT
    for a small entropic parameter is well understood for regular data on compact manifolds and standard quadratic ground cost 85, the team will extend this study to more general settings and also establish rigorous asymptotic estimates for the transports maps. This is important to provide a sound theoretical background to efficient and useful debiasing approaches like Sinkhorn Divergences 96. Guillaume Carlier , Paul Pegon and Luca Tamanini are investigating speed of convergence and quantitative stability results under general conditions on the cost (so that optimal maps may not be continuous or even fail to exist). Some sharp bounds have already been obtained, the next challenging goal is to extend the Laplace method to a nonsmooth setting and understand what entropic OT really selects when there are several optimal OT plans.
  • High dimensional - Curse of dimensionality
    We will continue to investigate the computation or approximation of high-dimensional OT losses and the associated transports 132 in particular in relation with their use in ML. In particular for Wasserstein 2 metric but also the repulsive Density Functional theory cost 97.
  • Back-and-forth
    The back-and-forth method  107, 106 is a state-of-the-art solver to compute optimal transport with convex costs and 2-Wasserstein gradient flows on grids. Based on simple but new ideas it has great potential to be useful for related problems. We plan to investigate: OT on point clouds in low dimension, the principal-agent problem in economics and more generally optimization under convex constraints  111, 123.
  • Transport and diffusion
    The diffusion induced by the entropic regularization is fixed and now well understood. For recent variations of the OT problem (Martingale OT, Weak OT see 47) the diffusion becomes an explicit constraint or the control itself 102. The entropic regularisation of these problems can then be understood as metric/ground cost learning 77 (see also 127) and offers a tractable numerical method.
  • Wasserstein Hamiltonian systems
    We started to investigate the use of modern OT solvers for the SG equation 88, 53 Semi-Discrete and entropic regularization. This is a special instance Hamiltonian Systems in the sense of 43. with an OT component in the Energy.
  • Nonlinear fourth-order diffusion equations
    such as thin-films or (the more involved) DLSS quantum drift equations are WGF. Such WGF are challenging both in terms of mathematical analysis (lack of maximum principle...) and of numerics. They are currently investigated by Jean-David Benamou , Guillaume Carlier in collaboration with Daniel Matthes. Note also that Mokaplan already contributed to a related topic through the TV-JKO scheme 78.
  • Lagrangian approaches for fluid mechanics
    More generally we want to extend the design and implementation of Lagrangian numerical scheme for a large class of problem coming from fluids mechanics (WHS or WGF) using semi-discrete OT or entropic regularization. We will also take a special attention to link this approaches with problems in machine/statistical learning. To achieve this part of the project we will join forces with colleagues in Orsay University: Y. Brenier, H. Leclerc, Q. Mérigot, L. Nenna.
  • L optimal transport
    is a variant of OT where we want to minimize the maximal displacement of the transport plan, instead of the average distance. Following the seminal work of 83, and more recent developments 92, Guillaume Carlier , Paul Pegon and Luigi De Pascale are working on the description of restrictable solutions (which are cyclically -monotone) through some potential maps, in the spirit of Mange-Kantorovich potentials provided by a duality theory. Some progress has been made to partially describe cyclically quasi-motonone maps (related in some sense to cyclically -monotone maps), through quasi-convex potentials.


3.2 Application of OT numerics to non-variational and non convex problems

Participants: Flavien Léger, Guillaume Carlier, Jean-David Benamou, François-Xavier Vialard.

  • Market design
    Z-mappings form a theory of non-variational problems initiated in the '70s but that has been for the most part overlooked by mathematicians. We are developing a new theory of the algorithms associated with convergent regular splitting of Z-mappings. Various well-established algorithms for matching models can be grouped under this point of view (Sinkhorn, Gale–Shapley, Bertsekas' auction) and this new perspective has the potential to unlock new convergence results, rates and accelerated methods.
  • Non Convex inverse problems
    The PhD 139 provided a first exploration of Unbalanced Sinkhorn Divergence in this context. Given enough resources, a branch of PySit, a public domain software to test misfit functions in the context of Seismic imaging will be created and will allow to test other signal processing strategies in Full Waveform Inversion. Likewise the numerical method tested for 1D reflectors in 11 coule be develloped further (in particular in 2D).
  • Equilibrium and transport
    Equilibrium in labor markets can often be expressed in terms of the Kantorovich duality. In the context of urban modelling or spatial pricing, this observation can be fruitfully used to compute equilibrium prices or densities as fixed points of operators involving OT, this was used in 49 and 48. Quentin Petit, Guillaume Carlier and Yves Achdou are currently developing a (non-variational) new semi-discrete model for the structure of cities with applications to tele-working.
  • Non-convex Principal-Agent problems
    Guillaume Carlier , Xavier Dupuis, Jean-Charles Rochet and John Thanassoulis are developing a new saddle-point approach to non-convex multidimensional screening problems arising in regulation (Barron-Myerson) and taxation (Mirrlees).


3.3 Inverse problems with structured priors

Participants: Irène Waldspurger, Antonin Chambolle, Vincent Duval, Robert Tovey , Romain Petit .

  • Off-the-grid reconstruction of complex objects
    Whereas, very recently, some methods were proposed for the reconstruction of curves and piecewise constant images on a continuous domain (63 and 14), those are mostly proofs of concept, and there is still some work to make them competitive in real applications. As they are much more complex than point source reconstruction methods, there is room for improvements (parametrization, introduction of several atoms...). In particular, we are currently working on an improvement of the algorithm 63 for inverse problems in imaging which involve Optimal Transport as a regularizer (see 40 for preliminary results). Moreover, we need to better understand their convergence and the robustness of such methods, using sensitivity analysis.
  • Correctness guarantees for Burer-Monteiro methods
    Burer-Monteiro methods work well in practice and are therefore widely used, but existing correctness guarantees 62 hold under unrealistic assumptions only. In the long term, we aim at proposing new guarantees, which would be slightly weaker but would hold in settings more relevant to practice. A first step is to understand the “average” behavior of Burer-Monteiro methods, when applied to random problems, and could be the subject of a PhD thesis.


3.4 Geometric variational problems, and their interactions with transport

Participants: Vincent Duval, Paul Pegon, Antonin Chambolle, Joao-Miguel Machado .

  • Approximation of measures with geometric constraints
    Optimal Transport is a powerful tool to compare and approximate densities, but its interaction with geometric constraints is still not well understood. In applications such as optimal design of structures, one aims at approximating an optimal pattern while taking into account fabrication constraints 59. In Magnetic Resonance Imaging (MRI), one tries to sample the Fourier transform of the unknown image according to an optimal density but the acquisition device can only proceed along curves with bounded speed and bounded curvature 113. Our goal is to understand how OT interacts with energy terms which involve, e.g. the length, the perimeter or the curvature of the support... We want to understand the regularity of the solutions and to quantify the approximation error. Moreover, we want to design numerical methods for the resolution of such problems, with guaranteed performance.
  • Discretization of singular measures
    Beyond the (B)Lasso and the total variation (possibly off-the-grid), numerically solving branched transportation problems requires the ability to faithfully discretize and represent 1-dimensional structures in the space. The research program of A. Chambolle consists in part in developing the numerical analysis of variational problems involving singular measures, such as lower-dimensional currents or free surfaces. We will explore both phase-field methods (with P. Pegon, V. Duval) 80, 124 which easily represent non-convex problems, but lack precision, and (with V. Duval) precise discretizations of convex problems, based either on finite elements (and relying to the FEM discrete exterior calculus  44, cf  81 for the case of the total variation), or on finite differences and possibly a clever design of dual constraints as studied in  86, 82 again for the total variation.
  • Transport problems with metric optimization
    In urban planning models, one looks at building a network (of roads, metro or train lines, etc.) so as to minimize a transport cost between two distributions, penalized by the cost for building the network, usually its length. A typical transport cost is Monge cost MKω with a metric ω=ωΣ which is modified as a fraction of the euclidean metric on the network Σ. We would like to consider general problems involving a construction cost to generate a conductance field σ (having in mind 1-dimensional integral of some function of σ), and a transport cost depending on this conductance field. The afore-mentioned case studied in 69 falls into this category, as well as classical branched transport. The biologically-inspired network evolution model of 103 seems to provide such an energy in the vanishing diffusivity limit, with a cost for building a 1-dimensional permeability tensor and an L2 congested transport cost with associated resistivity metric ; such a cost seems particularly relevant to model urban planning. Finally, we would like to design numerical methods to solve such problems, taking advantage of the separable structure of the whole cost.

4 Application domains

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 Highlights of the year

Guillaume Carlier has published a new reference book in optimization 76.

6 New results

6.1 Stability of optimal traffic plans in the irrigation problem

Participants: Maria Colombo, Antonio De Rosa, Andrea Marchese, Paul Pegon , Antoine Prouff.

We prove in 17 the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [Colombo, De Rosa, Marchese ; 2021], extending it to the Lagrangian framework.

6.2 Convergence of a Lagrangian discretization for barotropic fluids and porous media flow

Participants: Thomas Gallouët, Quentin Merigot, Andrea Natale.

When expressed in Lagrangian variables, the equations of motion for compressible (barotropic) fluids have the structure of a classical Hamiltonian system in which the potential energy is given by the internal energy of the fluid. The dissipative counterpart of such a system coincides with the porous medium equation, which can be cast in the form of a gradient flow for the same internal energy. Motivated by these related variational structures, we propose in 19 a particle method for both problems in which the internal energy is replaced by its Moreau-Yosida regularization in the L2 sense, which can be efficiently computed as a semi-discrete optimal transport problem. Using a modulated energy argument which exploits the convexity of the problem in Eulerian variables, we prove quantitative convergence estimates towards smooth solutions. We verify such estimates by means of several numerical tests.

6.3 Point Source Regularization of the Finite Source Reflector Problem

Participants: Jean-David Benamou, Guillaume Chazareix, Wilbert L Ijzerman, Giorgi Rukhaia.

We address in 11 the “freeform optics” inverse problem of designing a reflector surface mapping a prescribed source distribution of light to a prescribed far field distribution, for a finite light source. When the finite source reduces to a point source, the light source distribution has support only on the optics ray directions. In this setting the inverse problem is well posed for arbitrary source and target probability distributions. It can be recast as an Optimal Transportation problem and has been studied both mathematically and nu-merically. We are not aware of any similar mathematical formulation in the finite source case: i.e. the source has an “´etendue” with support both in space and directions. We propose to leverage the well-posed variational formulation of the point source problem to build a smooth parameterization of the reflec-tor and the reflection map. Under this parameterization we can construct a smooth loss/misfit function to optimize for the best solution in this class of reflectors. Both steps, the parameterization and the loss, are related to Optimal Transportation distances. We also take advantage of recent progress in the numerical approximation and resolution of these mathematical objects to perform a numerical study.

6.4 On the linear convergence of the multi-marginal Sinkhorn algorithm

Participants: Guillaume Carlier.

The aim of 13 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.

6.5 "FISTA" in Banach spaces with adaptive discretisations

Participants: Antonin Chambolle, Robert Tovey.

FISTA is a popular convex optimisation algorithm which is known to converge at an optimal rate whenever the optimisation domain is contained in a suitable Hilbert space. We propose in 16 a modified algorithm where each iteration is performed in a subspace, and that subspace is allowed to change at every iteration. Analytically, this allows us to guarantee convergence in a Banach space setting, although at a reduced rate depending on the conditioning of the specific problem. Numerically we show that a greedy adaptive choice of discretisation can greatly increase the time and memory efficiency in infinite dimensional Lasso optimisation problems.

6.6 Accelerated Bregman primal-dual methods applied to optimal transport and Wasserstein Barycenter problems

Participants: Antonin Chambolle, Juan Pablo Contreras.

We discuss in 15 the efficiency of Hybrid Primal-Dual (HPD) type algorithms to approximate solve discrete Optimal Transport (OT) and Wasserstein Barycenter (WB) problems, with and without entropic regularization. Our first contribution is an analysis showing that these methods yield state-of-the-art convergence rates, both theoretically and practically. Next, we extend the HPD algorithm with linesearch proposed by Malitsky and Pock in 2018 to the setting where the dual space has a Bregman divergence, and the dual function is relatively strongly convex to the Bregman's kernel. This extension yields a new method for OT and WB problems based on smoothing of the objective that also achieves state-of-the-art convergence rates. Finally, we introduce a new Bregman divergence based on a scaled entropy function that makes the algorithm numerically stable and reduces the smoothing, leading to sparse solutions of OT and WB problems. We complement our findings with numerical experiments and comparisons.

6.7 Towards Off-the-grid Algorithms for Total Variation Regularized Inverse Problems

Participants: Yohann De Castro, Vincent Duval, Romain Petit.

We introduce in 14 an algorithm to solve linear inverse problems regularized with the total (gradient) variation in a gridless manner. Contrary to most existing methods, that produce an approximate solution which is piecewise constant on a fixed mesh, our approach exploits the structure of the solutions and consists in iteratively constructing a linear combination of indicator functions of simple polygons.

6.8 Mass concentration in rescaled first order integral functionals

Participants: Antonin Monteil, Paul Pegon.

We consider in 39 first order local minimization problems minNf(u,u) under a mass constraint Nu=m. We prove that the minimal energy function H(m) is always concave on (-,0) and (0,+), and that relevant rescalings of the energy, depending on a small parameter ε, Γ-converge in the weak topology of measures towards the H-mass, defined for atomic measures imiδxi as iH(mi). We also consider space dependent Lagrangians f(x,u,u), which cover the case of space dependent H-masses iH(xi,mi), and also the case of a family of Lagrangians (fε)ε converging as ε0. The Γ-convergence result holds under mild assumptions on f, and covers several situations including homogeneous H-masses in any dimension N2 for exponents above a critical threshold, and all concave H-masses in dimension N=1. Our result yields in particular the concentration of Cahn-Hilliard fluids into droplets, and is related to the approximation of branched transport by elliptic energies.

6.9 Dynamical Programming for off-the-grid dynamic Inverse Problems

Participants: Robert Tovey, Vincent Duval.

In 40, we consider algorithms for reconstructing time-varying data into a finite sum of discrete trajectories, alternatively, an off-the-grid sparse-spikes decomposition which is continuous in time. Recent work showed that this decomposition was possible by minimising a convex variational model which combined a quadratic data fidelity with dynamical Optimal Transport. We generalise this framework and propose new numerical methods which leverage efficient classical algorithms for computing shortest paths on directed acyclic graphs. Our theoretical analysis confirms that these methods converge to globally optimal reconstructions which represent a finite number of discrete trajectories. Numerically, we show new examples for unbalanced Optimal Transport penalties, and for balanced examples we are 100 times faster in comparison to the previously known method.

6.10 A geometric Laplace method

Participants: Flavien Léger, François-Xavier Vialard.

A classical tool for approximating integrals is the Laplace method. The first-order, as well as the higher-order Laplace formula is most often written in coordinates without any geometrical interpretation. In 8, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation of the first-order term of the Laplace method. The central tool is the Kim–McCann Riemannian metric which was introduced in the field of optimal transportation. Our main result expresses the first-order term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two different metrics arising naturally in the Kim–McCann framework. Passing by, we give an explicitly quantified version of the Laplace formula, as well as examples of applications.

6.11 Convergence rate of general entropic optimal transport costs

Participants: Guillaume Carlier, Paul Pegon, Luca Tamanini.

We investigate in 32 the convergence rate of the optimal entropic cost vε to the optimal transport cost as the noise parameter ε0. We show that for a large class of cost functions c on d×d (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and L marginals, one has vε-v0=d/2εlog(1/ε)+O(ε). Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on c, i.e. invertibility of xy2c, we get the lower bound by establishing a quadratic detachment of the duality gap in d dimensions thanks to Minty's trick.

7 Partnerships and cooperations

7.1 International research visitors

7.1.1 Visits of international scientists

Other international visits to the team
Luigi De Pascale
  • Status
    Associate Professor
  • Institution of origin:
    Università di Firenze
  • Country:
  • Dates:
    October 5th-october 20th
  • Context of the visit:
    research with Guillaume Carlier, Luca Nenna and Paul Pegon, mentoring of the PhD of Camilla Brizzi
  • Mobility program/type of mobility:
    research stay and lecture
Giuseppe Buttazzo
  • Status
    Full Professor
  • Institution of origin:
    Università di Pisa
  • Country:
  • Dates:
    march 13th-march 17th and december 11th-14th
  • Context of the visit:
    research with Guillaume Carlier and Katharina Eichinger, PhD defense of Katharina Eichinger
  • Mobility program/type of mobility:
    research stay, lecture and Ph.D defense.

7.1.2 Visits to international teams

Research stays abroad
Flavien Léger
  • Visited institution:
    Fields Institute, Toronto, ON
  • Country:
  • Dates:
    November 7, 2022 to November 24, 2022
  • Context of the visit:
    long-term visit to the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry
  • Mobility program/type of mobility:
    research stay
Jean-David Benamou
  • Visited institution:
    Imperial College
  • Country:
  • Dates:
    April 23 to April 28 and September 26 to December 19.
  • Context of the visit:
    Invitation and Nelder Fellowship
  • Mobility program/type of mobility:
    research stay and lecture.
Guillaume Carlier
  • Visited institution:
    TUM (Munich)
  • Country:
  • Dates:
    may 22nd-may 26th and november 6th-november 11th
  • Context of the visit:
    research with Daniel Matthes and with Gero Friesecke, workshop calculus of variations
  • Mobility program/type of mobility:
    research stay, lecture

7.2 European initiatives

7.2.1 H2020 projects


Participants: Jean-David Benamou, Giorgi Rukhaia.

ROMSOC project on cordis.europa.eu

  • Title:
    Reduced Order Modelling, Simulation and Optimization of Coupled systems
  • Duration:
    From September 1, 2017 to August 31, 2022
  • Partners:
    • Math.Tec GmbH (Math.Tec), Austria
    • DB Schenker Rail Polska S.A. (DB Schenker Rail Polska), Poland
    • SIGNIFY NETHERLANDS BV (Signify Netherlands BV), Netherlands
    • CorWave (CorWave), France
  • Inria contact:
    Jean-David Benamou
  • Coordinator:
  • Summary:

    The development of high quality products and processes is essential for the future competitiveness of the European economy. In most key technology areas product development is increasingly based on simulation and optimization via mathematical models that allow to optimize design and functionality using free design parameters. Best performance of modelling, simulation and optimization (MSO) techniques is obtained by using a model hierarchy ranging from very fine to very coarse models obtained by model order reduction (MOR) techniques and to adapt the model and the methods to the user-defined requirements in accuracy and computational speed.

    ROMSOC will work towards this goal for high dimensional and coupled systems that describe different physical phenomena on different scales; it will derive a common framework for different industrial applications and train the next generation of researchers in this highly interdisciplinary field. It will focus on the three major methodologies: coupling methods, model reduction methods, and optimization methods, for industrial applications in well selected areas, such as optical and electronic systems, economic processes, and materials. ROMSOC will develop novel MSO techniques and associated software with adaptability to user-defined accuracy and efficiency needs in different scientific disciplines. It will transfer synergies between different industrial sectors, in particular for SMEs.

    To lift this common framework to a new qualitative level, a joint training programme will be developed which builds on the strengths of the academic and industrial partners and their strong history of academic/industrial cooperation. By delivering early-career training embedded in a cutting-edge research programme, ROMSOC will educate highly skilled interdisciplinary researchers in mathematical MSO that will become facilitators in the transfer of innovative concepts to industry. It will thus enhance the capacity of European research and development.

7.3 National initiatives

    (2016-2022) To study and to implement discretizations of optimal transport and Monge-Ampère equations which rely on tools from computational geometry (Laguerre diagrams). to apply these solvers to concrete problems from various fields involving optimal transport. .
  • PRAIRIE chair
    : Irène Waldspurger .
    (2019-) is a JCJC grant (149k€) carried by Vincent Duval . Its aim is to develop off-the-grid methods for inverse problems involving the reconstruction of complex objects.

8 Dissemination

8.1 Promoting scientific activities

8.1.1 Scientific events: organisation

Member of the organizing committees

8.1.2 Journal

Member of the editorial boards
  • Irène Waldspurger is associate editor for the journal IEEE Transactions on Signal Processing.
Reviewer - reviewing activities
  • Vincent Duval has reviewed submissions for the journals Math in Action and SIAM Journal on Imaging Sciences (SIIMS).
  • Irène Waldspurger has reviewed submissions for the journals Applied and Computational Harmonic Analysis, IEEE Transactions on Information Theory, IEEE Transactions on Signal Processing and the SIAM journal on Optimization, as well as for the NeurIPS conference.
  • Flavien Léger has reviewed submission for the journals IMA Journal of Applied Mathematics, ESAIM: Control, Optimisation and Calculus of Variations and Communications in Partial Differential Equations.
  • Paul Pegon has reviewed submissions for the SIAM Journal on Mathematical Analysis (SIMA) and the Journal of Mathematical Analysis and Applications (JMAA).
  • Thomas Gallouët has reviewed submissions for Foundations of Computational Mathematics, Journal of Convex Analysis.

8.1.3 Invited talks

  • Antonin Chambolle was invited to give talks at the Inverse Problem on Large Scales workshop at RICAM (Linz, Austria), at the IEEE ICIP 2022 Int. Conference on Image Processing, and at the Summerschool on Analysis and Applied Mathematics at Münster, Germany.
  • Vincent Duval was invited to give a talk at the Inverse Problems: Modelling and Simulation (IPMS) conference, and the "Probabilités et Statistiques" seminar at IECL (Nancy).
  • Irène Waldspurger was invited to give a talk at the Learning and Optimization in Luminy workshop, and at the CIS/MINDS seminar at Johns Hopkins University.
  • Flavien Léger was invited to give a talk at the Institut de mathématique d’Orsay, the Laboratoire Jean Kuntzmann Université Grenoble Alpes, the Laboratoire Paul Painlevé, Université de Lille and the Fields Institute Applied Mathematics Colloquium.
  • Paul Pegon was invited to give a talk at the Séminaire parisien d'optimisation (IHP) and at the Gdt Transport Optimal (Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay).
  • Thomas Gallouët was invited to give a talk at the Séminaire ANEDP at Lille University, at the the workgoup on Calculus of VariationGT CalVa at Dauphine University and at the conference on Optimal transport, geometric and stochastic Hydrodynamics at Lisbon, June 21-24, 2022.

8.1.4 Research administration

  • Irène Waldspurger was a member of a selection committee at Université Aix Marseille.
  • Vincent Duval was a member of a selection committee at Sorbonne Université (LJLL)
  • Thomas Gallouët is a member of the Commission d'Évaluation Scientifique (CES) of Inria Paris.

8.2 Teaching - Supervision - Juries

8.2.1 Teaching

  • Master: Antonin Chambolle Optimisation Continue, ??h, niveau M2, Université Paris Dauphine-PSL, FR
  • Master : Vincent Duval , Problèmes Inverses, 22,5 h équivalent TD, niveau M1, Université PSL/Mines ParisTech, FR
  • Master : Vincent Duval , Optimization for Machine Learning, 6h, niveau M2, Université PSL/ENS, FR
  • Licence : Irène Waldspurger , Pré-rentrée raisonnement, 31,2 h équivalent TD, niveau L1, Université Paris-Dauphine, FR
  • Licence : Irène Waldspurger , Analyse 4, 58,5 h équivalent TD, niveau L1, Université Paris-Dauphine, FR
  • Master : Irène Waldspurger , Optimization for Machine Learning, 6h, niveau M2, Université PSL/ENS, FR
  • Licence : Guillaume Carlier , algebre 1, L1 78h, Dauphine, FR
  • Master : Guillaume Carlier Variational and transport methods in economics, M2 Masef, 27h, Dauphine, FR
  • Licence : Paul Pegon , Analyse 2, 51 H. équivalent TD, TD niveau L1, Université Paris-Dauphine, FR
  • Licence : Paul Pegon , Intégrale de Lebesgue et probabilités, 44 H. équivalent TD, TD niveau L3, Université Paris-Dauphine, FR
  • Licence : Paul Pegon , Analyse fonctionnelle et hilbertienne, 44 H. équivalent TD, TD niveau L3, Université Paris-Dauphine, FR
  • Licence : Paul Pegon , Méthodes numériques pour l'optimisation, 33 H. équivalent TD, TD/TP niveau L3, Université Paris-Dauphine, FR
  • Agregation : Thomas Gallouët , Optimisation, Analyse numérique, 48h équivalent TD, niveau M2, Université d'Orsay), FR
  • Guillaume Carlier : Licence Algèbre 1, Dauphine 70h, M2 Masef: Variatioanl and transport problems in economics, 18h
  • Flavien Léger : Graduate course, two lectures in ‘math+econ+code’ masterclass on equilibrium transport and matching models in economics, NYU Paris. 5h.

8.2.2 Supervision

  • PhD completed: Quentin Petit, Mean field games and optimal transport in urban modelling, defended on 18/02/2022, Supervised by Guillaume Carlier , Y. Achdou, D. Tonon.
  • PhD completed: Romain Petit, Reconstruction of piecewise constant images via total variation regularization, defended on 12/12/2022, Supervised by Vincent Duval and Yohann De Castro.
  • PhD in completed: Katharina Eichinger, Problèmes variationnels pour l'interpolation dans l’espace de Wasserstein 1/09/2019. Supervised by Guillaume Carlier .
  • PhD in progress : Joao-Miguel Machado, Transport optimal et structures géométriques, 01/10/2021, Co-supervised by Vincent Duval and Antonin Chambolle .
  • PhD in progress : Adrien Vacher 1/10/2020. Co-supervised by François-Xavier Vialard and Jean-David Benamou
  • Summer internship: Mitchell Gaudet, an undergraduate student at the University of Toronto, Canada. The Kim–McCann geometry in applied optimal transport. Supervised by Flavien Léger and Robert McCann.
  • PhD in progress : Erwan Stämplfi 1/10/2021, Around multiphasic flows. Co-supervised by Yann Brenier and Thomas Gallouët .
  • PhD in progress : Siwan Boufadene 1/10/2022, Gradient flows for energy distance. Co-supervised by François-Xavier Vialard and Thomas Gallouët .
  • PhD in progress : Chazareix Guillaume 1/08/2021, Non Linear Parabolic equations and Volatility Calibration. Co-supervised by Jean-David Benamou and G. Loeper.
  • PhD in progress : Malamut Hugo 1/09/2022, Régularisation Entropique et Transport Optimal Généralisé. Supervised by Jean-David Benamou .

8.2.3 Juries

  • Antonin Chambolle was a member of the jury for the PhD theses of Ulysse Marteau-Ferey (ENS), Romain Petit (Dauphine Paris-PSL), and for the habilitation thesis of Andrea Simonetto. He was the reviewer of the PhD theses of Jordan Michelet (Univ. La Rochelle) and Garry Terii (Univ. Lyon 1).
  • Vincent Duval was a member of the jury for the PhD Theses of Thomas Debarre (EPFL) and Zoé Lambert (INSA Rouen).
  • Irène Waldspurger was a member of the jury of the PhD thesis of Pierre-Hugo Vial (ENSEEIHT, Toulouse).
  • Jean-David Benamou was a reviewer of the PhD thesis Jean-François Quilbert (U. Joseph Fourier, Grenoble) and Charlie Egan (Heriot Watt, Edimbourg).

8.3 Popularization

8.3.1 Internal or external Inria responsibilities

  • Vincent Duval is a member of the CSD (comité de suivi doctoral) of the Inria Paris Centre.
  • Thomas Gallouët is a member of the Commission d'Évaluation Scientifique (CES) of Inria Paris.

8.3.2 Education

  • Vincent Duval has given a 10h lecture on Inverse Problems at the CIMPA school "Mathématiques en analyse et traitement du signal, des images et des données" at Thiès (Sénégal).

8.3.3 Interventions

  • Irène Waldspurger has given a talk at the Journée des doctorant·e·s en mathématiques des Hauts-de-France.

9 Scientific production

9.1 Major publications

9.2 Publications of the year

International journals

International peer-reviewed conferences

  • 21 inproceedingsP.-C.Pierre-Cyril Aubin-Frankowski, A.Anna Korba and F.Flavien Léger. Mirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM.NeurIPS 2022 - Thirty-sixth Conference on Neural Information Processing SystemsNew Orleans, United States2022

Edition (books, proceedings, special issue of a journal)

  • 22 proceedingsA.Alexis AymeC.Claire BoyerA.Aymeric DieuleveutE.Erwan ScornetNear-optimal rate of consistency for linear models with missing values.2022

Doctoral dissertations and habilitation theses

  • 23 thesisV.Vincent Duval. Faces and extreme points of convex sets for the resolution of inverse problems.Ecole doctorale SDOSEJune 2022

Reports & preprints

9.3 Cited publications

  • 42 articleM.M. Agueh and G.Guillaume Carlier. Barycenters in the Wasserstein space.SIAM J. Math. Anal.4322011, 904--924
  • 43 articleL.Luigi Ambrosio and W.Wilfrid Gangbo. Hamiltonian ODEs in the Wasserstein space of probability measures.Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences6112008, 18--53
  • 44 articleD. N.Douglas N. Arnold, R. S.Richard S. Falk and R.Ragnar Winther. Finite element exterior calculus, homological techniques, and applications.Acta Numerica152006, 1–155
  • 45 articleF. R.F. R. Bach. Consistency of Trace Norm Minimization.J. Mach. Learn. Res.9June 2008, 1019--1048URL: http://dl.acm.org/citation.cfm?id=1390681.1390716
  • 46 articleF. R.F. R. Bach. Consistency of the Group Lasso and Multiple Kernel Learning.J. Mach. Learn. Res.9June 2008, 1179--1225URL: http://dl.acm.org/citation.cfm?id=1390681.1390721
  • 47 miscJ. D.Julio Daniel Backhoff-Veraguas and G.Gudmund Pammer. Applications of weak transport theory.2020
  • 48 unpublishedX.Xavier Bacon, G. G.Guillaume Guillaume Carlier and B.Bruno Nazaret. A spatial Pareto exchange economy problem.December 2021, working paper or preprint
  • 49 articleC.César Barilla, G.Guillaume Carlier and J.-M.Jean-Michel Lasry. A mean field game model for the evolution of cities.Journal of Dynamics and Games2021
  • 50 articleM. F.M. F. Beg, M. I.M. I. Miller, A.A. Trouvé and L.L. Younes. Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms.International Journal of Computer Vision612February 2005, 139--157URL: http://dx.doi.org/10.1023/B:VISI.0000043755.93987.aa
  • 51 articleM.M. Beiglbock, P.P. Henry-Labordère and F.F. Penkner. Model-independent bounds for option prices mass transport approach.Finance and Stochastics1732013, 477--501URL: http://dx.doi.org/10.1007/s00780-013-0205-8
  • 52 articleJ.-D.Jean-David Benamou and Y.Y. Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem.Numer. Math.8432000, 375--393URL: http://dx.doi.org/10.1007/s002110050002
  • 53 articleJ.-D.Jean-David Benamou and Y.Y. Brenier. Weak existence for the semigeostrophic equations formulated as a coupled Monge-Ampère/transport problem.SIAM J. Appl. Math.5851998, 1450--1461
  • 54 articleJ.-D.Jean-David Benamou and G.Guillaume Carlier. Augmented Lagrangian algorithms for variational problems with divergence constraints.JOTA2015
  • 55 articleJ.-D.Jean-David Benamou, G.Guillaume Carlier, M.Marco Cuturi, L.Luca Nenna and G.Gabriel Peyré. Iterative Bregman Projections for Regularized Transportation Problems.SIAM J. Sci. Comp.to appear2015
  • 56 articleJ.-D.Jean-David Benamou, B. D.Brittany D Froese and A.Adam Oberman. Numerical solution of the optimal transportation problem using the Monge--Ampere equation.Journal of Computational Physics2602014, 107--126
  • 57 articleJ.-D.Jean-David Benamou, B. D.B. D. Froese and A.Adam Oberman. Two numerical methods for the elliptic Monge-Ampère equation.M2AN Math. Model. Numer. Anal.4442010, 737--758
  • 58 articleJ.J. Bigot and T.T. Klein. Consistent estimation of a population barycenter in the Wasserstein space.Preprint arXiv:1212.25622012
  • 59 articleM.M. Boissier, G.G. Allaire and C.C. Tournier. Additive manufacturing scanning paths optimization using shape optimization tools.Struct. Multidiscip. Optim.6162020, 2437--2466URL: https://doi.org/10.1007/s00158-020-02614-3
  • 60 articleN.N. Bonneel, J.J. Rabin, G.Gabriel Peyré and H.H. Pfister. Sliced and Radon Wasserstein Barycenters of Measures.Journal of Mathematical Imaging and Vision5112015, 22--45URL: http://hal.archives-ouvertes.fr/hal-00881872/
  • 61 articleG.Guy Bouchitté and G.Giuseppe Buttazzo. Characterization of optimal shapes and masses through Monge-Kantorovich equation.J. Eur. Math. Soc. (JEMS)322001, 139--168URL: http://dx.doi.org/10.1007/s100970000027
  • 62 articleN.N. Boumal, V.V. Voroninski and A. S.A. S. Bandeira. Deterministic guarantees for Burer-Monteiro factorizations of smooth semidefinite programs.preprinthttps://arxiv.org/abs/1804.020082018
  • 63 articleK.Kristian Bredies, M.Marcello Carioni, S.Silvio Fanzon and F.Francisco Romero. A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization.arXiv preprint arXiv:2012.117062020
  • 64 articleY.Y. Brenier. Décomposition polaire et réarrangement monotone des champs de vecteurs.C. R. Acad. Sci. Paris Sér. I Math.305191987, 805--808
  • 65 articleY.Y. Brenier, U.U. Frisch, M.M. Henon, G.G. Loeper, S.S. Matarrese, R.R. Mohayaee and A.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
  • 66 articleY.Y. Brenier. Generalized solutions and hydrostatic approximation of the Euler equations.Phys. D23714-172008, 1982--1988URL: http://dx.doi.org/10.1016/j.physd.2008.02.026
  • 67 articleY.Y. Brenier. Polar factorization and monotone rearrangement of vector-valued functions.Comm. Pure Appl. Math.4441991, 375--417URL: http://dx.doi.org/10.1002/cpa.3160440402
  • 68 articleM.M. Burger and S.S. Osher. A guide to the TV zoo.Level-Set and PDE-based Reconstruction Methods, Springer2013
  • 69 bookG.Giuseppe Buttazzo, A.Aldo Pratelli, E.Eugene Stepanov and S.Sergio Solimini. Optimal Urban Networks via Mass Transportation.1961Lecture Notes in MathematicsBerlin, HeidelbergSpringer Berlin Heidelberg2009, URL: http://link.springer.com/10.1007/978-3-540-85799-0
  • 70 incollectionL. A.L. A. Caffarelli, S. A.S. A. Kochengin and V.V.I. Oliker. On the numerical solution of the problem of reflector design with given far-field scattering data.Monge 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
  • 71 articleL. A.L. A. Caffarelli. The regularity of mappings with a convex potential.J. Amer. Math. Soc.511992, 99--104URL: http://dx.doi.org/10.2307/2152752
  • 72 articleC.C. CanCeritoglu. Computational Analysis of LDDMM for Brain Mapping.Frontiers in Neuroscience72013
  • 73 articleE. J.E. J. Candès and C.C. Fernandez-Granda. Super-Resolution from Noisy Data.Journal of Fourier Analysis and Applications1962013, 1229--1254
  • 74 articleE. J.E. J. Candès and C.C. Fernandez-Granda. Towards a Mathematical Theory of Super-Resolution.Communications on Pure and Applied Mathematics6762014, 906--956
  • 75 articleE.E. Candes and M.M. Wakin. An Introduction to Compressive Sensing.IEEE Signal Processing Magazine2522008, 21--30
  • 76 bookG.Guillaume Carlier. Classical and modern optimization.Advanced textbooks in mathematicsTopological and functional analytic preliminaries -- Differential calculus -- Convexity -- Optimality conditions for differentiable optimization -- Problems depending on a parameter -- Convex duality and applications -- Iterative methods for convex minimization -- When optimization and data meet -- An invitation to the calculus of variationsHackensack, New JerseyWorld Scientific2022
  • 77 unpublishedG.Guillaume Carlier, A.Arnaud Dupuy, A.Alfred Galichon and Y.Yifei Sun. SISTA: Learning Optimal Transport Costs under Sparsity Constraints.October 2020, working paper or preprint
  • 78 articleG.Guillaume Carlier and C.Clarice Poon. On the total variation Wasserstein gradient flow and the TV-JKO scheme.ESAIM: Control, Optimisation and Calculus of Variations2019
  • 79 articleF. A.F. A. C. C. Chalub, P. A.P. A. Markowich, B.B. Perthame and C.C. Schmeiser. Kinetic models for chemotaxis and their drift-diffusion limits.Monatsh. Math.1421-22004, 123--141URL: http://dx.doi.org/10.1007/s00605-004-0234-7
  • 80 articleA.Antonin Chambolle, L. A.Luca A. D. Ferrari and B.Benoit Merlet. Variational approximation of size-mass energies for k-dimensional currents.ESAIM Control Optim. Calc. Var.252019, Paper No. 43, 39URL: https://doi.org/10.1051/cocv/2018027
  • 81 articleA.Antonin Chambolle and T.Thomas Pock. Crouzeix-Raviart approximation of the total variation on simplicial meshes.J. Math. Imaging Vision626-72020, 872--899URL: https://doi.org/10.1007/s10851-019-00939-3
  • 82 articleA.Antonin Chambolle and T.Thomas Pock. Learning consistent discretizations of the total variation.SIAM J. Imaging Sci.1422021, 778--813URL: https://doi.org/10.1137/20M1377199
  • 83 articleT.Thierry Champion, L.Luigi De Pascale and P.Petri Juutinen. The -Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps.401January 2008, 1--20URL: https://epubs.siam.org/doi/10.1137/07069938X
  • 84 articleS. S.S. S. Chen, D. L.D. L. Donoho and M. A.M. A. Saunders. Atomic decomposition by basis pursuit.SIAM journal on scientific computing2011999, 33--61
  • 85 inproceedingsL.Lenaic Chizat, P.Pierre Roussillon, F.Flavien Léger, F.-X.François-Xavier Vialard and G.Gabriel Peyré. Faster Wasserstein Distance Estimation with the Sinkhorn Divergence.Neural Information Processing SystemsAdvances in Neural Information Processing SystemsVancouver, CanadaDecember 2020
  • 86 articleL.L. Condat. Discrete total variation: new definition and minimization.SIAM J. Imaging Sci.1032017, 1258--1290URL: https://doi.org/10.1137/16M1075247
  • 87 articleC.C. Cotar, G.G. Friesecke and C.C. Kluppelberg. Density Functional Theory and Optimal Transportation with Coulomb Cost.Communications on Pure and Applied Mathematics6642013, 548--599URL: http://dx.doi.org/10.1002/cpa.21437
  • 88 bookM. J.M. J. P. Cullen. A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow.Imperial College Press2006, URL: https://books.google.fr/books?id=JxBqDQAAQBAJ
  • 89 articleM. J.M. J. P. Cullen, W.W. Gangbo and G.G. Pisante. The semigeostrophic equations discretized in reference and dual variables.Arch. Ration. Mech. Anal.18522007, 341--363URL: http://dx.doi.org/10.1007/s00205-006-0040-6
  • 90 articleM. J.M. J. P. Cullen, J.J. Norbury and R. J.R. J. Purser. Generalised Lagrangian solutions for atmospheric and oceanic flows.SIAM J. Appl. Math.5111991, 20--31
  • 91 inproceedingsM.Marco Cuturi. Sinkhorn Distances: Lightspeed Computation of Optimal Transport.Proc. NIPS2013, 2292--2300
  • 92 articleL.Luigi De Pascale and J.Jean Louet. A Study of the Dual Problem of the One-Dimensional L -Optimal Transport Problem with Applications.27611June 2019, 3304--3324URL: https://www.sciencedirect.com/science/article/pii/S0022123619300643
  • 93 articleE. J.E. J. Dean and R.R. Glowinski. Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type.Comput. Methods Appl. Mech. Engrg.19513-162006, 1344--1386
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