2023Activity reportProject-TeamMOKAPLAN

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  103. Optimal Transportation techniques have been applied for example to a Coulomb ground cost in Quantum chemistry in relation with Density Functional theory  90. 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  129 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 44. 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  128.

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  108, 93, 92, 55, 120, mesh adaptation 119, the reconstruction of the early mass distribution of the Universe  101, 68 in Astrophysics, and the numerical optimisation of reflectors following the Optimal Transportation interpretation of Oliker  73 and Wang  139. Extensions of OT such as multi-marginal transport has potential applications in Density Functional Theory , Generalized solution of Euler equations 69 (DFT) and in statistics and finance  53, 102 .... Recently, there has been a spread of interest in applications of OT methods in imaging sciences  63, statistics  61 and machine learning  94. This is largely due to the emergence of fast numerical schemes to approximate the transportation distance and its generalizations, see for instance  57. 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.

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  54. 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  56. 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  115 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  64.

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  77, 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  76, 98, 97 have used this framework to provide theoretical performance guarantees by basically studying how the distance between neighboring spikes impacts noise stability.

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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  54, 131). 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  57) and machine learning (see  94).

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 88, 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 99. 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 43 in particular in relation with their use in ML. In particular for Wasserstein 2 metric but also the repulsive Density Functional theory cost 100.
• Back-and-forth
  The back-and-forth method  110, 109 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  114, 126.
• 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 49) the diffusion becomes an explicit constraint or the control itself 105. The entropic regularisation of these problems can then be understood as metric/ground cost learning 79 (see also 130) and offers a tractable numerical method.
• Wasserstein Hamiltonian systems
  We started to investigate the use of modern OT solvers for the SG equation 91, 55 Semi-Discrete and entropic regularization. This is a special instance Hamiltonian Systems in the sense of 45. 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 80.
• 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 86, and more recent developments 95, 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 142 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 58 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 51 and 50. 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 (66 and 81), 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 66 for inverse problems in imaging which involve Optimal Transport as a regularizer (see 135 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 65 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 62. 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 116. 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) 83, 127 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  46, cf  84 for the case of the total variation), or on finite differences and possibly a clever design of dual constraints as studied in  89, 85 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 72 falls into this category, as well as classical branched transport. The biologically-inspired network evolution model of 106 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.

7.1 Entropic Optimal Transport Solutions of the Semigeostrophic Equations

Participants: Jean-David Benamou, Colin Cotter, Hugo Malamut.

The Semigeostrophic equations are a frontogenesis model in atmospheric science. Existence of solutions both from the theoretical and numerical point of view is given under a change of variable involving the interpretation of the pressure gradient as an Optimal Transport map between the density of the fluid and its push forward. Thanks to recent advances in numerical Optimal Transportation, the computation of large scale discrete approximations can be envisioned. We study in 25 the use of Entropic Optimal Transport and its Sinkhorn algorithm companion.

7.2 Wasserstein gradient flow of the Fisher information from a non-smooth convex minimization viewpoint

Participants: Jean-David Benamou, Guillaume Carlier, Daniel Matthes.

Motivated by the Derrida-Lebowitz-Speer-Spohn (DLSS) quantum drift equation, which is the Wasserstein gradient flow of the Fisher information, we study in 27 in details solutions of the corresponding implicit Euler scheme. We also take advantage of the convex (but non-smooth) nature of the corresponding variational problem to propose a numerical method based on the Chambolle-Pock primal-dual algorithm.

7.3 Total variation regularization with Wasserstein penalization

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

In 20, a new derivation of the Euler-Lagrange equation of a total-variation regularization problem with a Wasserstein penalization is obtained, it is interesting as on easily deduces some regularity of the Lagrange multiplier for the non-negativity constraint. A numerical implementation is also described.

7.4 1D approximation of measures in Wasserstein spaces

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

We propose in 32 a variational approach to approximate measures with measures uniformly distributed over a 1 dimensional set. The problem consists in minimizing a Wasserstein distance as a data term with a regularization given by the length of the support. As it is challenging to prove existence of solutions to this problem, we propose a relaxed formulation, which always admits a solution. In the sequel we show that if the ambient space is $R^2$ , under technical assumptions, any solution to the relaxed problem is a solution to the original one. Finally we prove that any optimal solution to the relaxed problem, and hence also to the original, is Ahlfors regular.

7.5 Exact recovery of the support of piecewise constant images via total variation regularization

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

This work 23, 30 is concerned with the recovery of piecewise constant images from noisy linear measurements. We study the noise robustness of a variational reconstruction method, which is based on total (gradient) variation regularization. We show that, if the unknown image is the superposition of a few simple shapes, and if a non-degenerate source condition holds, then, in the low noise regime, the reconstructed images have the same structure: they are the superposition of the same number of shapes, each a smooth deformation of one of the unknown shapes. Moreover, the reconstructed shapes and the associated intensities converge to the unknown ones as the noise goes to zero.

7.6 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 9, 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.

7.7 Convergence rate of general entropic optimal transport costs

Participants: Guillaume Carlier, Paul Pegon, Luca Tamanini, Luca Nenna.

We investigate in 15 the convergence rate of the optimal entropic cost $O{T}_{\epsilon }$ to the optimal transport cost as the noise parameter $\epsilon \to 0$. For a large class of cost functions (for which optimal plans are not necessarily unique or induced by a transport map), we establish lower and upper bounds on the difference with the unregularized cost that depends on the dimensions of the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under a non-degeneracy condition on the cost function (invertibility of the cross-derivative) we get the lower bound by establishing a quadratic detachment of the duality gap in $d$ dimensions thanks to Minty's trick. These results were improved and extended to the multi-marginal setting in 40. In particular, we establish lower bounds for $C^2$ costs defined on the product of $M$ submanifolds satisfying some signature condition on the mixed second derivatives that may include degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic, including the case of Wasserstein barycenters.

7.8 A geometric approach to apriori estimates for optimal transport maps

Participants: Simon Brendle, Flavien Leger, Robert J. McCann, Cale Rankin.

A key inequality which underpins the regularity theory of optimal transport for costs satisfying the Ma–Trudinger–Wang condition is the Pogorelov second derivative bound. This translates to an apriori interior C1 estimate for smooth optimal maps. Here we give a new derivation of this estimate which relies in part on Kim, McCann and Warren's observation that the graph of an optimal map becomes a volume maximizing spacelike submanifold when the product of the source and target domains is endowed with a suitable pseudo-Riemannian geometry that combines both the marginal densities and the cost.

7.9 Gradient descent with a general cost

Participants: Flavien Leger, Pierre-Cyril Aubin.

We present a new class of gradient-type optimization methods that extends vanilla gradient descent, mirror descent, Riemannian gradient descent, and natural gradient descent. Our approach involves constructing a surrogate for the objective function in a systematic manner, based on a chosen cost function. This surrogate is then minimized using an alternating minimization scheme. Using optimal transport theory we establish convergence rates based on generalized notions of smoothness and convexity. We provide local versions of these two notions when the cost satisfies a condition known as nonnegative cross-curvature. In particular our framework provides the first global rates for natural gradient descent and the standard Newton's method.

7.10 Second-order methods for Burer-Monteiro factorization

Participants: Florentin Goyens, Clément Royer, Irène Waldspurger.

The Burer-Monteiro factorization is a classical reformulation of optimization problems where the unknown is a matrix, when this matrix is known to have low rank. Rigorous analysis has been provided for this reformulation, when solved with first-order algorithms, but second-order algorithms oftentimes perform better in practice. We have established convergence rates for a representative second-order algorithm in a simplified setting. An article is in preparation.

7.11 Optimization for imaging and machine learning, analysis of inverse problems

Participants: Antonin Chambolle.

In 31, is analysed a stochastic primal-dual hybrid gradient for large-scale inverse problems (with application mostly to medical imaging), which was proposed some years ago by A. Chambolle and collaborators. The new result describes how the parameters can be modified/updated at each iteration in a way which still ensures (almost sure) convergence, and proposes some heuristic rules which fit into the framework and effectively improve the rate of convergence in practical experiments. The proceeding 21, in collaboration with U. Bordeaux, shows some convergence guarantees for a particular implementation of a “plug-and-play” image restoration method, where the regularizer for inverse problems is based on a denoising neural network. A more developed journal version has been submitted 39.

The proceeding 22, in collaboration with the computer imaging group at TU Graz (Austria), implements as a toy model a stochastic diffusion equation for sampling image priors based on Gaussian Mixture models, with exact formulas.

In a different direction, the proceeding 19, also with TU Graz, considers the issue of parameters learning for a better discretization of variational regularizers allowing for singularities (the “total-generalized-variation” of Bredies, Kunisch and Pock). A theoretical analysis of this model and of more standard total-variation regularization models is found in the new preprint 37, which introduces a novel approach (and much simpler than the previous ones) for studying the stability of the discontinuity sets in elementary denoising models.

7.12 Free discontinuity problems, fractures and shape optimization

Participants: Antonin Chambolle.

The publications 14, 17, 16, 33 are related to “free discontinuity problems” in materials science, with application to fracture growth or shape optimisation. In 17, 16 we discuss compactness for functionals which appear in the variational approach to fracture, in particular 16 is a new, very general, and in some sense more natural proof of compactness with respect to the previous results. The preprint 33 was submitted to the proceedings of the ICIAM conference, and it contains in an appendix a relatively simple presentation (in a simpler case) of the proof of Poincaré / Poincaré-Korn inequalities with small jump set developped in the past 10 years by A. Chambolle.

An application to shape optimization (of an object dragged in a Stokes flow) is presented in 14, while 34, 35 address other type of shape optimization problems in a more classical framework.

7.13 Interface evolution problems

Participants: Antonin Chambolle.

The new article 18 of Chambolle, DeGennaro, Morini generalizes to non-homogeneous flows an implicit approach for mean curvature flow of surfaces introduced in the 1990's by Luckhaus and Sturzenhecker, and Almgren, Taylor and Wang. Current developments in the fully discrete case are under study, with striking results which should appear in 2024, in the meantime, a short description of the possible anisotropies (or surface tension) which arise on discrete lattices was published in 13.

A different dynamics of interfaces, based on $L^1$ gradient flow instead of $L^2$-type, is studied in 36.

7.14 Optimal quantization via branched optimal transport distance

Participants: Paul Pegon, Mircea Petrache.

In 41 we consider the problem of optimal approximation of a target measure by an atomic measure with $N$ atoms, in branched optimal transport distance. This is a new branched transport version of optimal quantization problems. New difficulties arise, as in previously known Wasserstein semi-discrete transport results the interfaces between cells associated with neighboring atoms had Voronoi structure and satisfied an explicit description. This description is missing for our problem, in which the cell interfaces are thought to have fractal boundary. We study the asymptotic behaviour of optimal quantizers for absolutely continuous measures as the number $N$ of atoms grows to infinity. We compute the limit distribution of the corresponding point clouds and show in particular a branched transport version of Zador's theorem. Moreover, we establish uniformity bounds of optimal quantizers in terms of separation distance and covering radius of the atoms, when the measure is $d$-Ahlfors regular. A crucial technical tool is the uniform in $N$ Hölder regularity of the landscape function, a branched transport analog to Kantorovich potentials in classical optimal transport.

7.15 From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows

Participants: Thomas Gallouët, Andrea Natale, Gabriele Todeschi.

We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differentiation Formula of order two. The main building block of the scheme is the notion of geodesic extrapolation in the Wasserstein space, which in general is not uniquely defined. We propose several possible definitions for such an operation, and we prove convergence of the resulting scheme to the limit PDE, in the case of the Fokker-Planck equation. For a specific choice of extrapolation we also prove a more general result, that is convergence towards EVI flows. Finally, we propose a variational finite volume discretization of the scheme which numerically achieves second order accuracy in both space and time.

7.16 Regularity theory and geometry of unbalanced optimal transport

Participants: Thomas Gallouët, Roberta Ghezzi, Francois-Xavier Vialard.

Using the dual formulation only, we show that regularity of unbalanced optimal transport also called entropy-transport inherits from regularity of standard optimal transport. We then provide detailed examples of Riemannian manifolds and costs for which unbalanced optimal transport is regular.Among all entropy-transport formulations, Wasserstein-Fisher-Rao metric, also called Hellinger-Kantorovich, stands out since it admits a dynamic formulation, which extends the Benamou-Brenier formulation of optimal transport. After demonstrating the equivalence between dynamic and static formulations on a closed Riemannian manifold, we prove a polar factorization theorem, similar to the one due to Brenier and Mc-Cann. As a byproduct, we formulate the Monge-Ampère equation associated with Wasserstein-Fisher-Rao metric, which also holds for more general costs.

7.17 Entropic approximation of $\infty$ optimal transport problems

Participants: Camilla Brizzi, Guillaume Carlier, Luigi De-Pascale.

We propose an entropic approximation approach for optimal transportation problems with a supremal cost. We establish $\Gamma$-convergence for suitably chosen parameters for the entropic penalization and that this procedure selects $\infty$ cyclically monotone plans at the limit. We also present some numerical illustrations performed with Sinkhorn's algorithm.

7.18 Quantitative Stability of the Pushforward Operation by an Optimal Transport Map

Participants: Guillaume Carlier, Alex Delalande, Quentin Mérigot.

We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain.

7.19 Quantitative Stability of Barycenters in the Wasserstein Space

Participants: Guillaume Carlier, Alex Delalande, Quentin Mérigot.

Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability measures of interest are often not accessible in their entirety and the practitioner may have to deal with statistical or computational approximations instead. In this article, we quantify the effect of such approximations on the corresponding barycenters. We show that Wasserstein barycenters depend in a Hölder-continuous way on their marginals under relatively mild assumptions. Our proof relies on recent estimates that quantify the strong convexity of the dual quadratic optimal transport problem and a new result that allows to control the modulus of continuity of the push-forward operation under a (not necessarily smooth) optimal transport map.

7.20 Wasserstein medians: robustness, PDE characterization and numerics

Participants: Guillaume Carlier, Enis Chenchene, Katharina Eichinger.

We investigate the notion of Wasserstein median as an alternative to the Wasserstein barycenter, which has become popular but may be sensitive to outliers. In terms of robustness to corrupted data, we indeed show that Wasserstein medians have a breakdown point of approximately 1 2. We give explicit constructions of Wasserstein medians in dimension one which enable us to obtain $L^p$ estimates (which do not hold in higher dimensions). We also address dual and multimarginal reformulations. In convex subsets of ${ℝ}^d$ , we connect Wasserstein medians to a minimal (multi) flow problem à la Beckmann and a system of PDEs of Monge-Kantorovich-type, for which we propose a p-Laplacian approximation. Our analysis eventually leads to a new numerical method to compute Wasserstein medians, which is based on a Douglas-Rachford scheme applied to the minimal flow formulation of the problem.

9.1 International research visitors

9.2 National initiatives

• PRAIRIE chair
  : Irène Waldspurger .
• ANR CIPRESSI
  (2019-2024) 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.
• PDE AI
  (2023-2027) Antonin Chambolle is the main coordinator of the PDE-AI project, funded by the PEPR IA (France 2030, ANR) and gathering 10 groups throughout France working on PDEs and nonlinear analysis for artificial intelligence.

10.1 Promoting scientific activities

10.2 Teaching - Supervision - Juries

11.1 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 States2022HAL
• 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-A1138HALDOI
• 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 2019HALDOI
• 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, 1260–1281HALDOI
• 5 articleC.Clément Cancès, T.Thomas Gallouët and G.Gabriele Todeschi. A variational finite volume scheme for Wasserstein gradient flows.Numerische Mathematik14632020, pp 437 - 480HALDOI
• 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 2017HALDOI
• 7 miscG.Guillaume Carlier, P.Paul Pegon and L.Luca Tamanini. Convergence rate of general entropic optimal transport costs.June 2022HAL
• 8 miscF.Flavien Léger and P.-C.Pierre-Cyril Aubin-Frankowski. Gradient descent with a general cost.December 2023HAL
• 9 miscF.Flavien Léger and F.-X.François-Xavier Vialard. A geometric Laplace method.December 2022HALback to text
• 10 articleI.Irène Waldspurger. Phase retrieval with random Gaussian sensing vectors by alternating projections.IEEE Transactions on Information Theory6452018, 3301-3312HAL
• 11 articleI.Irène Waldspurger and A.Alden Waters. Rank optimality for the Burer-Monteiro factorization.SIAM Journal on Optimization3032020, 2577-2602HALDOI

11.2 Publications of the year

11.3 Cited publications

• 44 articleM.M. Agueh and G.Guillaume Carlier. Barycenters in the Wasserstein space.SIAM J. Math. Anal.4322011, 904--924back to text
• 45 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--53back to text
• 46 articleD. N.Douglas N. Arnold, R. S.Richard S. Falk and R.Ragnar Winther. Finite element exterior calculus, homological techniques, and applications.Acta Numerica152006, 1–155DOIback to text
• 47 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=1390681.1390716back to text
• 48 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=1390681.1390721back to text
• 49 miscJ. D.Julio Daniel Backhoff-Veraguas and G.Gudmund Pammer. Applications of weak transport theory.2020back to text
• 50 unpublishedX.Xavier Bacon, G. G.Guillaume Guillaume Carlier and B.Bruno Nazaret. A spatial Pareto exchange economy problem.December 2021, working paper or preprintHALback to text
• 51 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 Games2021HALback to text
• 52 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.aaback to textback to text
• 53 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/s00780-013-0205-8DOIback to text
• 54 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/s002110050002DOIback to textback to text
• 55 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--1461back to textback to text
• 56 articleJ.-D.Jean-David Benamou and G.Guillaume Carlier. Augmented Lagrangian algorithms for variational problems with divergence constraints.JOTA2015back to text
• 57 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 appear2015back to textback to textback to text
• 58 articleJ.-D.Jean-David Benamou, G.Guillaume Chazareix, G.Giorgi Rukhaia and W. L.Wilbert L Ijzerman. Point Source Regularization of the Finite Source Reflector Problem.Journal of Computational PhysicsMay 2022HALback to text
• 59 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--126back to text
• 60 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--758back to text
• 61 articleJ.J. Bigot and T.T. Klein. Consistent estimation of a population barycenter in the Wasserstein space.Preprint arXiv:1212.25622012back to text
• 62 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/s00158-020-02614-3DOIback to text
• 63 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/back to text
• 64 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/s100970000027DOIback to text
• 65 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/1804.020082018back to text
• 66 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:2012.117062020back to textback to text
• 67 articleY.Y. Brenier. Décomposition polaire et réarrangement monotone des champs de vecteurs.C. R. Acad. Sci. Paris Sér. I Math.305191987, 805--808back to text
• 68 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.pdfback to text
• 69 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.2008.02.026DOIback to text
• 70 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.3160440402DOIback to text
• 71 articleM.M. Burger and S.S. Osher. A guide to the TV zoo.Level-Set and PDE-based Reconstruction Methods, Springer2013back to text
• 72 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/10.1007/978-3-540-85799-0DOIback to text
• 73 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/226/03233DOIback to text
• 74 articleL. A.L. A. Caffarelli. The regularity of mappings with a convex potential.J. Amer. Math. Soc.511992, 99--104URL: http://dx.doi.org/10.2307/2152752DOIback to text
• 75 articleC.C. CanCeritoglu. Computational Analysis of LDDMM for Brain Mapping.Frontiers in Neuroscience72013back to text
• 76 articleE. J.E. J. Candès and C.C. Fernandez-Granda. Super-Resolution from Noisy Data.Journal of Fourier Analysis and Applications1962013, 1229--1254back to text
• 77 articleE. J.E. J. Candès and C.C. Fernandez-Granda. Towards a Mathematical Theory of Super-Resolution.Communications on Pure and Applied Mathematics6762014, 906--956back to text
• 78 articleE.E. Candes and M.M. Wakin. An Introduction to Compressive Sensing.IEEE Signal Processing Magazine2522008, 21--30back to text
• 79 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 preprintHALback to text
• 80 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 Variations2019HALback to text
• 81 articleY.Yohann de Castro, V.Vincent Duval and R.Romain Petit. Towards Off-the-grid Algorithms for Total Variation Regularized Inverse Problems.Journal of Mathematical Imaging and VisionJuly 2022HALDOIback to text
• 82 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/s00605-004-0234-7DOIback to text
• 83 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/2018027DOIback to text
• 84 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/s10851-019-00939-3DOIback to text
• 85 articleA.Antonin Chambolle and T.Thomas Pock. Learning consistent discretizations of the total variation.SIAM J. Imaging Sci.1422021, 778--813URL: https://doi.org/10.1137/20M1377199DOIback to text
• 86 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/10.1137/07069938XDOIback to text
• 87 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--61back to text
• 88 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 2020HALback to text
• 89 articleL.L. Condat. Discrete total variation: new definition and minimization.SIAM J. Imaging Sci.1032017, 1258--1290URL: https://doi.org/10.1137/16M1075247back to text
• 90 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.21437DOIback to text
• 91 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=JxBqDQAAQBAJback to text
• 92 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/s00205-006-0040-6DOIback to text
• 93 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--31back to text
• 94 inproceedingsM.Marco Cuturi. Sinkhorn Distances: Lightspeed Computation of Optimal Transport.Proc. NIPS2013, 2292--2300back to textback to text
• 95 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/S0022123619300643DOIback to text
• 96 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--1386back to text
• 97 articleV.Vincent Duval and G.Gabriel Peyré. Exact Support Recovery for Sparse Spikes Deconvolution.Foundations of Computational Mathematics2014, 1--41URL: http://dx.doi.org/10.1007/s10208-014-9228-6DOIback to text
• 98 articleC.C. Fernandez-Granda. Support detection in super-resolution.Proc. Proceedings of the 10th International Conference on Sampling Theory and Applications2013, 145--148back to text
• 99 unpublishedJ.Jean Feydy, T.Thibault Séjourné, F.-X.François-Xavier Vialard, S.-I.Shun-Ichi Amari, A.Alain Trouvé and G.Gabriel Peyré. Interpolating between Optimal Transport and MMD using Sinkhorn Divergences.October 2018, working paper or preprintHALback to text
• 100 articleG.Gero Friesecke and D.Daniela Vögler. Breaking the Curse of Dimension in Multi-Marginal Kantorovich Optimal Transport on Finite State Spaces.SIAM Journal on Mathematical Analysis5042018, 3996--4019URL: https://doi.org/10.1137/17M1150025DOIback to text
• 101 articleU.U. Frisch, S.S. Matarrese, R.R. Mohayaee and A.A. Sobolevski. Monge-Ampère-Kantorovitch (MAK) reconstruction of the eary universe.Nature4172602002back to text
• 102 articleA.A. Galichon, P.P. Henry-Labordère and N.N. Touzi. A stochastic control approach to No-Arbitrage bounds given marginals, with an application to Loopback options.submitted to Annals of Applied Probability2011back to text
• 103 articleW.W. Gangbo and R.R.J. McCann. The geometry of optimal transportation.Acta Math.17721996, 113--161URL: http://dx.doi.org/10.1007/BF02392620DOIback to text
• 104 articleE.E. Ghys. Gaspard Monge, Le mémoire sur les déblais et les remblais.Image des mathématiques, CNRS2012, URL: http://images.math.cnrs.fr/Gaspard-Monge,1094.htmlback to text
• 105 articleI.Ivan Guo and G.Grégoire Loeper. Path Dependent Optimal Transport and Model Calibration on Exotic Derivatives.SSRN Electron.~J.Available at doi:10.2139/ssrn.330238401 2018DOIback to text
• 106 articleJ.Jan Haskovec, P.Peter Markowich, B.Benoît Perthame and M.Matthias Schlottbom. Notes on a PDE System for Biological Network Formation.138June 2016, 127--155URL: https://www.sciencedirect.com/science/article/pii/S0362546X15004344DOIback to text
• 107 articleD. D.D. D. Holm, J. T.J. T. Ratnanather, A.A. Trouvé and L.L. Younes. Soliton dynamics in computational anatomy.NeuroImage232004, S170--S178back to text
• 108 incollectionB. J.B. J. Hoskins. The mathematical theory of frontogenesis.Annual review of fluid mechanics, Vol. 14Palo Alto, CAAnnual Reviews1982, 131--151back to text
• 109 articleM.Matt Jacobs, W.Wonjun Lee and F.Flavien Léger. The back-and-forth method for Wasserstein gradient flows.ESAIM: Control, Optimisation and Calculus of Variations272021, 28back to text
• 110 articleM.Matt Jacobs and F.Flavien Léger. A fast approach to optimal transport: The back-and-forth method.Numer. Math.1462020, 513--544URL: https://doi.org/10.1007/s00211-020-01154-8DOIback to text
• 111 articleW.W. Jäger and S.S. Luckhaus. On explosions of solutions to a system of partial differential equations modelling chemotaxis.Trans. Amer. Math. Soc.32921992, 819--824URL: http://dx.doi.org/10.2307/2153966DOIback to text
• 112 articleR.R. Jordan, D.D. Kinderlehrer and F.F. Otto. The variational formulation of the Fokker-Planck equation.SIAM J. Math. Anal.2911998, 1--17back to text
• 113 articleL.L. Kantorovitch. On the translocation of masses.C. R. (Doklady) Acad. Sci. URSS (N.S.)371942, 199--201back to text
• 114 articleT.Thomas Lachand-Robert and E.Edouard Oudet. Minimizing within convex bodies using a convex hull method.SIAM Journal on Optimization162January 2005, 368--379HALback to text
• 115 articleJ.-M.J-M. Lasry and P.-L.P-L. Lions. Mean field games.Jpn. J. Math.212007, 229--260URL: http://dx.doi.org/10.1007/s11537-007-0657-8DOIback to text
• 116 articleL.Léo Lebrat, F.Frédéric de Gournay, J.Jonas Kahn and P.Pierre Weiss. Optimal Transport Approximation of 2-Dimensional Measures.SIAM Journal on Imaging Sciences122January 2019, 762--787URL: https://epubs.siam.org/doi/10.1137/18M1193736DOIback to text
• 117 articleC.Christian Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport.Discrete Contin. Dyn. Syst.3442014, 1533--1574URL: http://dx.doi.org/10.3934/dcds.2014.34.1533DOIback to text
• 118 articleA. S.A. S. Lewis. Active sets, nonsmoothness, and sensitivity.SIAM Journal on Optimization1332003, 702--725back to text
• 119 articleB.B. Li, F.F. Habbal and M.M. Ortiz. Optimal transportation meshfree approximation schemes for Fluid and plastic Flows.Int. J. Numer. Meth. Engng 83:1541--579832010, 1541--1579back to text
• 120 articleG.G. Loeper. A fully nonlinear version of the incompressible Euler equations: the semigeostrophic system.SIAM J. Math. Anal.3832006, 795--823 (electronic)back to text
• 121 articleG.G. Loeper and F.F. Rapetti. Numerical solution of the Monge-Ampére equation by a Newton's algorithm.C. R. Math. Acad. Sci. Paris34042005, 319--324back to text
• 122 bookS. G.S. G. Mallat. A wavelet tour of signal processing.Elsevier/Academic Press, Amsterdam2009back to text
• 123 articleB.B. Maury, A.A. Roudneff-Chupin and F.F. Santambrogio. A macroscopic crowd motion model of gradient flow type.Math. Models Methods Appl. Sci.20102010, 1787--1821URL: http://dx.doi.org/10.1142/S0218202510004799DOIback to text
• 124 articleQ.Quentin Mérigot. A multiscale approach to optimal transport.Computer Graphics Forum3052011, 1583--1592back to text
• 125 articleM. I.M. I. Miller, A.A. Trouvé and L.L. Younes. Geodesic Shooting for Computational Anatomy.Journal of Mathematical Imaging and Vision242March 2006, 209--228URL: http://dx.doi.org/10.1007/s10851-005-3624-0back to text
• 126 articleJ.-M.Jean-Marie Mirebeau. Adaptive, Anisotropic and Hierarchical cones of Discrete Convex functions.Numerische Mathematik132435 pages, 11 figures. (Second version fixes a small bug in Lemma 3.2. Modifications are anecdotic.)2016, 807--853HALback to text
• 127 articleE.Edouard Oudet and F.Filippo Santambrogio. A Modica-Mortola Approximation for Branched Transport and Applications.Archive for Rational Mechanics and Analysis2011July 2011, 115--142URL: http://link.springer.com/10.1007/s00205-011-0402-6DOIback to text
• 128 articleB.B. Pass and N.N. Ghoussoub. Optimal transport: From moving soil to same-sex marriage.CMS Notes452013, 14--15back to text
• 129 articleB.B. Pass. Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem.SIAM Journal on Mathematical Analysis4362011, 2758--2775back to text
• 130 miscF.-P.François-Pierre Paty and M.Marco Cuturi. Regularized Optimal Transport is Ground Cost Adversarial.2020back to text
• 131 articleH.H. Raguet, J.J. Fadili and G.Gabriel Peyré. A Generalized Forward-Backward Splitting.SIAM Journal on Imaging Sciences632013, 1199--1226URL: http://hal.archives-ouvertes.fr/hal-00613637/DOIback to text
• 132 articleL.L.I. Rudin, S.S. Osher and E.E. Fatemi. Nonlinear total variation based noise removal algorithms.Physica D: Nonlinear Phenomena6011992, 259--268URL: http://dx.doi.org/10.1016/0167-2789(92)90242-Fback to text
• 133 articleJ.Justin Solomon, F.F. d eGoes, G.Gabriel Peyré, M.Marco Cuturi, A.A. Butscher, A.A. Nguyen, T.T. Du and L.L. Guibas. Convolutional Wasserstein Distances: Efficient Optimal Transportation on Geometric Domains.ACM Transaction on Graphics, Proc. SIGGRAPH'15to appear2015back to textback to text
• 134 articleR.R. Tibshirani. Regression shrinkage and selection via the Lasso.Journal of the Royal Statistical Society. Series B. Methodological5811996, 267--288back to textback to text
• 135 unpublishedR.Robert Tovey and V.Vincent Duval. Dynamical Programming for off-the-grid dynamic Inverse Problems.December 2022, working paper or preprintHALback to text
• 136 articleS.S. Vaiter, M.M. Golbabaee, J.J. Fadili and G.Gabriel Peyré. Model Selection with Piecewise Regular Gauges.Information and Inferenceto appear2015, URL: http://hal.archives-ouvertes.fr/hal-00842603/back to text
• 137 bookC.C. Villani. Optimal transport.338Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]Old and newBerlinSpringer-Verlag2009, xxii+973URL: http://dx.doi.org/10.1007/978-3-540-71050-9DOIback to text
• 138 bookC.C. Villani. Topics in optimal transportation.58Graduate Studies in MathematicsAmerican Mathematical Society, Providence, RI2003, xvi+370back to text
• 139 articleX.-J.X-J. Wang. On the design of a reflector antenna. II.Calc. Var. Partial Differential Equations2032004, 329--341URL: http://dx.doi.org/10.1007/s00526-003-0239-4DOIback to text
• 140 articleB.B. Wirth, L.L. Bar, M.M. Rumpf and G.G. Sapiro. A continuum mechanical approach to geodesics in shape space.International Journal of Computer Vision9332011, 293--318back to text
• 141 articleJ.J. Wright, Y.Y. Ma, J.J. Mairal, G.G. Sapiro, T. S.T. S Huang and S.S. Yan. Sparse representation for computer vision and pattern recognition.Proceedings of the IEEE9862010, 1031--1044back to text
• 142 phdthesisM.Miao Yu. Entropic Unbalanced Optimal Transport: Application to Full-Waveform Inversion and Numerical Illustration.Université de ParisDecember 2021HALback to text