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

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.

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.

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.

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.

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.

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.

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  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.

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.

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).

3.1 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.
• $L^{\infty }$ 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 $\infty$-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 $\infty$-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 $M{K}_{\omega }$ with a metric $\omega ={\omega }_{\Sigma }$ which is modified as a fraction of the euclidean metric on the network $\Sigma$. We would like to consider general problems involving a construction cost to generate a conductance field $\sigma$ (having in mind 1-dimensional integral of some function of $\sigma$), 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 $L^2$ 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.

4.1 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.

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 $min{\int }_{{ℝ}^N}f(u,\nabla u)$ under a mass constraint ${\int }_{{ℝ}^N}u=m\in ℝ$. We prove that the minimal energy function $H\left(m\right)$ is always concave on $(-\infty ,0)$ and $(0,+\infty )$, and that relevant rescalings of the energy, depending on a small parameter $\epsilon$, $\Gamma$-converge in the weak topology of measures towards the $H$-mass, defined for atomic measures ${\sum }_i{m}_i{\delta }_{x_i}$ as ${\sum }_{i}H\left(m_i\right)$. We also consider space dependent Lagrangians $f(x,u,\nabla u)$, which cover the case of space dependent $H$-masses ${\sum }_{i}H(x_i,m_i)$, and also the case of a family of Lagrangians ${\left(f_{\epsilon }\right)}_{\epsilon }$ converging as $\epsilon \to 0$. The $\Gamma$-convergence result holds under mild assumptions on $f$, and covers several situations including homogeneous $H$-masses in any dimension $N\ge 2$ 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.

6.11 Convergence rate of general entropic optimal transport costs

Participants: Guillaume Carlier, Paul Pegon, Luca Tamanini.

7.1 International research visitors

7.2 European initiatives

7.3 National initiatives

• ANR MAGA
  (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 .
• ANR CIPRESSI
  (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.

8.1 Promoting scientific activities

8.2 Teaching - Supervision - Juries

8.3 Popularization

9.1 Major publications

• 1 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
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• 2 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 Journal on Scientific Computing2372015, A1111-A1138
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• 3 articleJ.-D.Jean-David Benamou, T.Thomas Gallouët and F.-X.François-Xavier Vialard. Second order models for optimal transport and cubic splines on the Wasserstein space.Foundations of Computational MathematicsOctober 2019
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• 4 articleC.Claire Boyer, A.Antonin Chambolle, Y.Yohann de Castro, V.Vincent Duval, F.Frédéric de Gournay and P.Pierre Weiss. On Representer Theorems and Convex Regularization.SIAM Journal on Optimization292May 2019, 1260–1281
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• 5 articleC.Clément Cancès, T.Thomas Gallouët and G.Gabriele Todeschi. A variational finite volume scheme for Wasserstein gradient flows.Numerische Mathematik14632020, pp 437 - 480
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• 6 articleG.Guillaume Carlier, V.Vincent Duval, G.Gabriel Peyré and B.Bernhard Schmitzer. Convergence of Entropic Schemes for Optimal Transport and Gradient Flows.SIAM Journal on Mathematical Analysis492April 2017
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• 7 miscG.Guillaume Carlier, P.Paul Pegon and L.Luca Tamanini. Convergence rate of general entropic optimal transport costs.June 2022
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• 8 miscF.Flavien Léger and F.-X.François-Xavier Vialard. A geometric Laplace method.December 2022
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• 9 articleI.Irène Waldspurger. Phase retrieval with random Gaussian sensing vectors by alternating projections.IEEE Transactions on Information Theory6452018, 3301-3312
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• 10 articleI.Irène Waldspurger and A.Alden Waters. Rank optimality for the Burer-Monteiro factorization.SIAM Journal on Optimization3032020, 2577-2602
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9.2 Publications of the year

9.3 Cited publications

• 42 articleM.M. Agueh and G.Guillaume Carlier. Barycenters in the Wasserstein space.SIAM J. Math. Anal.4322011, 904--924
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• 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
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• 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
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• 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
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• 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
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• 47 miscJ. D.Julio Daniel Backhoff-Veraguas and G.Gudmund Pammer. Applications of weak transport theory.2020
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• 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
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• 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
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• 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
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• 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
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• 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
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• 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
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• 54 articleJ.-D.Jean-David Benamou and G.Guillaume Carlier. Augmented Lagrangian algorithms for variational problems with divergence constraints.JOTA2015
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• 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
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• 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
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• 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
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• 58 articleJ.J. Bigot and T.T. Klein. Consistent estimation of a population barycenter in the Wasserstein space.Preprint arXiv:1212.25622012
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• 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
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• 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/
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• 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
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• 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
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• 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
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