Keywords
 A5.3. Image processing and analysis
 A5.9. Signal processing
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.2.1. Numerical analysis of PDE and ODE
 A6.2.6. Optimization
 A9. Artificial intelligence
 B1.2. Neuroscience and cognitive science
 B9.5.2. Mathematics
 B9.5.3. Physics
 B9.5.4. Chemistry
 B9.6.3. Economy, Finance
1 Team members, visitors, external collaborators
Research Scientists
 JeanDavid Benamou [Team leader, INRIA, Senior Researcher, HDR]
 Antonin Chambolle [CNRS & DAUPHINE PSL, Senior Researcher, from Sep 2022, HDR]
 Vincent Duval [INRIA, Senior Researcher, HDR]
 Thomas Gallouët [INRIA, Researcher]
 Flavien Léger [INRIA, Researcher]
 Irene Waldspurger [CNRS & DAUPHINE PSL, Researcher]
Faculty Members
 Guillaume Carlier [DAUPHINE PSL, Professor, HDR]
 Paul Pegon [DAUPHINE PSL, Associate Professor]
 FrançoisXavier Vialard [UNIV GUSTAVE EIFFEL, Professor]
PostDoctoral Fellow
 Robert Tovey [INRIA, until Aug 2022]
PhD Students
 Guillaume Chazareix [DAUPHINE PSL]
 Katharina Eichinger [DAUPHINE PSL]
 Hugo Malamut [DAUPHINE PSL, from Jul 2022]
 Romain Petit [DAUPHINE PSL]
 Joao Pinto Anastacio Machado [DAUPHINE PSL]
 Erwan Stampfli [UNIV PARIS SACLAY]
 Maxime Sylvestre [DAUPHINE PSL, from Sep 2022]
 Gabriele Todeschi [FSMP, from Apr 2022]
 Adrien Vacher [UNIV PARIS EST]
Administrative Assistant
 Derya Gok [INRIA]
2 Overall objectives
2.1 Introduction
The last decade has witnessed a remarkable convergence between several subdomains 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 sparsitybased 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 nonsmooth optimization and geometric discretization schemes. Optimal Transportation, diffeomorphisms and sparsitybased methods are powerful modeling tools, that impact a rapidly expanding list of scientific applications and call for efficient numerical strategies. Our research program shows the important part played by the team members in the development of these numerical methods and their application to challenging problems.
2.2 Static Optimal Transport and Generalizations
Optimal Transport, Old and New.
Optimal Mass Transportation is a mathematical research topic which started two centuries ago with Monge's work on the “Théorie des déblais et des remblais" (see 101). This engineering problem consists in minimizing the transport cost between two given mass densities. In the 40's, Kantorovich 110 introduced a powerful linear relaxation and introduced its dual formulation. The MongeKantorovich problem became a specialized research topic in optimization and Kantorovich obtained the 1975 Nobel prize in economics for his contributions to resource allocations problems. Since the seminal discoveries of Brenier in the 90's 64, Optimal Transportation has received renewed attention from mathematical analysts and the Fields Medal awarded in 2010 to C. Villani, who gave important contributions to Optimal Transportation and wrote the modern reference monographs 134, 135, arrived at a culminating moment for this theory. Optimal Mass Transportation is today a mature area of mathematical analysis with a constantly growing range of applications. Optimal Transportation has also received a lot of attention from probabilists (see for instance the recent survey 114 for an overview of the Schrödinger problem which is a stochastic variant of the BenamouBrenier dynamical formulation of optimal transport). The development of numerical methods for Optimal Transportation and Optimal Transportation related problems is a difficult topic and comparatively underdeveloped. This research field has experienced a surge of activity in the last five years, with important contributions of the Mokaplan group (see the list of important publications of the team). We describe below a few of recent and less recent Optimal Transportation concepts and methods which are connected to the future activities of Mokaplan :
Brenier's theorem 67 characterizes the unique optimal map as the gradient of a convex potential. As such Optimal Transportation may be interpreted as an infinite dimensional optimisation problem under “convexity constraint": i.e. the solution of this infinite dimensional optimisation problem is a convex potential. This connects Optimal Transportation to “convexity constrained" nonlinear variational problems such as, for instance, Newton's problem of the body of minimal resistance. The value function of the optimal transport problem is also known to define a distance between source and target densities called the Wasserstein distance which plays a key role in many applications such as image processing.
MongeAmpère Methods.
A formal substitution of the optimal transport map as the gradient of a convex potential in the mass conservation constraint (a Jacobian equation) gives a nonlinear MongeAmpère equation. Caffarelli 71 used this result to extend the regularity theory for the MongeAmpère equation. In the last ten years, it also motivated new research on numerical solvers for nonlinear degenerate Elliptic equations 9311857 56 and the references therein. Geometric approaches based on Laguerre diagrams and discrete data 121 have also been developed. MongeAmpère based Optimal Transportation solvers have recently given the first linear cost computations of Optimal Transportation (smooth) maps.
Generalizations of OT.
In recent years, the classical Optimal Transportation problem has been extended in several directions. First, different ground costs measuring the “physical" displacement have been considered. In particular, well posedness for a large class of convex and concave costs has been established by McCann and Gangbo 100. Optimal Transportation techniques have been applied for example to a Coulomb ground cost in Quantum chemistry in relation with Density Functional theory 87. Given the densities of electrons Optimal Transportation models the potential energy and their relative positions. For more than more than 2 electrons (and therefore more than 2 densities) the natural extension of Optimal Transportation is the so called Multimarginal Optimal Transport (see 126 and the references therein). Another instance of multimarginal Optimal Transportation arises in the socalled 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.
Numerical Applications of Optimal Transportation.
Optimal transport has found many applications, starting from its relation with several physical models such as the semigeostrophic 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 multimarginal 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.
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 nonlinear 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 recast as a convex but non smooth optimization problem. This convex dynamical formulation finds many nontrivial 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 HamiltonJacobi with a FokkerPlanck equation. In contrast, the Monge case where the ground cost is the euclidan distance leads to a static system of PDEs 61.
Gradient Flows for the Wasserstein Distance.
Another extension is, instead of considering geodesic for transportation metric (i.e. minimizing the Wasserstein distance to a target measure), to make the density evolve in order to minimize some functional. Computing the steepest descent direction with respect to the Wasserstein distance defines a socalled Wasserstein gradient flow, also known as JKO gradient flows after its authors 109. This is a popular tool to study a large class of nonlinear diffusion equations. Two interesting examples are the KellerSegel system for chemotaxis 108, 79 and a model of congested crowd motion proposed by Maury, Santambrogio and RoudneffChupin 120. From the numerical point of view, these schemes are understood to be the natural analogue of implicit scheme for linear parabolic equations. The resolution is however costly as it involves taking the derivative in the Wasserstein sense of the relevant energy, which in turn requires the resolution of a large scale convex but nonsmooth minimization.
Geodesic on infinite dimensional Riemannian spaces.
To tackle more complicated warping problems, such as those encountered in medical image analysis, one unfortunately has to drop the convexity of the functional involved in defining the gradient flow. This gradient flow can either be understood as defining a geodesic on the (infinite dimensional) group of diffeomorphisms 50, or on a (infinite dimensional) space of curves or surfaces 137. The defacto standard to define, analyze and compute these geodesics is the “Large Deformation Diffeomorphic Metric Mapping” (LDDMM) framework of Trouvé, Younes, Holm and coauthors 50, 104. While in the CFD formulation of optimal transport, the metric on infinitesimal deformations is just the ${L}^{2}$ norm (measure according to the density being transported), in LDDMM, one needs to use a stronger regularizing metric, such as Sobolevlike norms or reproducing kernel Hilbert spaces (RKHS). This enables a control over the smoothness of the deformation which is crucial for many applications. The price to pay is the need to solve a nonconvex optimization problem through geodesic shooting method 122, which requires to integrate backward and forward the geodesic ODE. The resulting strong Riemannian geodesic structure on spaces of diffeomorphisms or shapes is also pivotal to allow us to perform statistical analysis on the tangent space, to define mean shapes and perform dimensionality reduction when analyzing large collection of input shapes (e.g. to study evolution of a diseases in time or the variation across patients) 72.
2.4 Sparsity in Imaging
Sparse ${\ell}^{1}$ regularization.
Beside image warping and registration in medical image analysis, a key problem in nearly all imaging applications is the reconstruction of high quality data from low resolution observations. This field, commonly referred to as “inverse problems”, is very often concerned with the precise location of features such as point sources (modeled as Dirac masses) or sharp contours of objects (modeled as gradients being Dirac masses along curves). The underlying intuition behind these ideas is the socalled sparsity model (either of the data itself, its gradient, or other more complicated representations such as wavelets, curvelets, bandlets 119 and learned representation 138).
The huge interest in these ideas started mostly from the introduction of convex methods to serve as proxy for these sparse regularizations. The most well known is the ${\ell}^{1}$ norm introduced independently in imaging by Donoho and coworkers under the name “Basis Pursuit” 84 and in statistics by Tibshirani 131 under the name “Lasso”. A more recent resurgence of this interest dates back to 10 years ago with the introduction of the socalled “compressed sensing” acquisition techniques 75, which make use of randomized forward operators and ${\ell}^{1}$type reconstruction.
Regularization over measure spaces.
However, the theoretical analysis of sparse reconstructions involving reallife acquisition operators (such as those found in seismic imaging, neuroimaging, astrophysical imaging, etc.) is still mostly an open problem. A recent research direction, triggered by a paper of Candès and FernandezGranda 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 2D 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 
Two example of application of the total variation regularization of functions. Left: image segmentation into homogeneous color regions. Right: image zooming (increasing the number of pixels while keeping the edges sharp).
Low complexity regularization and partial smoothness.
In image processing, one of the most popular methods is the total variation regularization 129, 68. It favors lowcomplexity images that are piecewise constant, see Figure 4 for some examples on how to solve some image processing problems. Beside applications in image processing, sparsityrelated ideas also had a deep impact in statistics 131 and machine learning 46. As a typical example, for applications to recommendation systems, it makes sense to consider sparsity of the singular values of matrices, which can be relaxed using the socalled nuclear norm (a.k.a. trace norm) 45. The underlying methodology is to make use of lowcomplexity regularization models, which turns out to be equivalent to the use of partlysmooth regularization functionals 115, 133 enforcing the solution to belong to a lowdimensional manifold.
2.5 Mokaplan unified point of view
The dynamical formulation of optimal transport creates a link between optimal transport and geodesics on diffeomorphisms groups. This formal link has at least two strong implications that Mokaplan will elaborate on: (i) the development of novel models that bridge the gap between these two fields ; (ii) the introduction of novel fast numerical solvers based on ideas from both nonsmooth 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 nonsmooth and large scale optimization problems. Ideas from proximal optimization has proved crucial to address problems in both fields (see for instance 52, 128). Transportation metrics are also the correct way to compare and regularize variational problems that arise in image processing (see for instance the Radon inversion method proposed in 55) and machine learning (see 91).
3 Research program
Since its creation, the Mokaplan team has made important contributions in Optimal Transport both on the theoretical and the numerical side, together with applications such as fluid mechanics, the simulation biological systems, machine learning. We have also contributed to to the field of inverse problems in signal and image processing (superresolution, nonconvex low rank matrix recovery). In 2022, the team was renewed with the following research program which broadens our spectrum and addresses exciting new problems.
3.1 OT and related variational problems solvers encore et toujours
Participants: Flavien Léger, JeanDavid Benamou, Guillaume Carlier, Thomas Gallouët, FrançoisXavier 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

Backandforth
The backandforth method 107, 106 is a stateoftheart solver to compute optimal transport with convex costs and 2Wasserstein 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 principalagent 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

Nonlinear fourthorder diffusion equations
such as thinfilms 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 JeanDavid Benamou , Guillaume Carlier in collaboration with Daniel Matthes. Note also that Mokaplan already contributed to a related topic through the TVJKO 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 semidiscrete 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 MangeKantorovich potentials provided by a duality theory. Some progress has been made to partially describe cyclically quasimotonone maps (related in some sense to cyclically $\infty $monotone maps), through quasiconvex potentials.
3.2 Application of OT numerics to nonvariational and non convex problems
Participants: Flavien Léger, Guillaume Carlier, JeanDavid Benamou, FrançoisXavier Vialard.

Market design
Zmappings form a theory of nonvariational 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 Zmappings. Various wellestablished 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 (nonvariational) new semidiscrete model for the structure of cities with applications to teleworking.

Nonconvex PrincipalAgent problems
Guillaume Carlier , Xavier Dupuis, JeanCharles Rochet and John Thanassoulis are developing a new saddlepoint approach to nonconvex multidimensional screening problems arising in regulation (BarronMyerson) and taxation (Mirrlees).
3.3 Inverse problems with structured priors
Participants: Irène Waldspurger, Antonin Chambolle, Vincent Duval, Robert Tovey , Romain Petit .

Offthegrid 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 BurerMonteiro methods
BurerMonteiro 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 BurerMonteiro 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, JoaoMiguel 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 offthegrid), numerically solving branched transportation problems requires the ability to faithfully discretize and represent 1dimensional 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 lowerdimensional currents or free surfaces. We will explore both phasefield methods (with P. Pegon, V. Duval) 80, 124 which easily represent nonconvex 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 1dimensional integral of some function of $\sigma $), and a transport cost depending on this conductance field. The aforementioned case studied in 69 falls into this category, as well as classical branched transport. The biologicallyinspired network evolution model of 103 seems to provide such an energy in the vanishing diffusivity limit, with a cost for building a 1dimensional 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 Application domains
4.1 Natural Sciences
FreeForm Optics, Fluid Mechanics (Incompressible Euler, SemiGeostrophic equations), Quantum Chemistry (Density Functional Theory), Statistical Physics (Schroedinger problem), Porous Media.
4.2 Signal Processing and inverse problems
Full Waveform Inversion (Geophysics), Superresolution microscopy (Biology), Satellite imaging (Meteorology)
4.3 Social Sciences
Meanfield games, spatial economics, principalagent models, taxation, nonlinear pricing.
5 Highlights of the year
Guillaume Carlier has published a new reference book in optimization 76.
6 New results
6.1 Stability of optimal traffic plans in the irrigation problem
Participants: Maria Colombo, Antonio De Rosa, Andrea Marchese, Paul Pegon , Antoine Prouff.
We prove in 17 the stability of optimal traffic plans in branched transport. In particular, we show that any limit of optimal traffic plans is optimal as well. This result goes beyond the Eulerian stability proved in [Colombo, De Rosa, Marchese ; 2021], extending it to the Lagrangian framework.
6.2 Convergence of a Lagrangian discretization for barotropic fluids and porous media flow
Participants: Thomas Gallouët, Quentin Merigot, Andrea Natale.
When expressed in Lagrangian variables, the equations of motion for compressible (barotropic) fluids have the structure of a classical Hamiltonian system in which the potential energy is given by the internal energy of the fluid. The dissipative counterpart of such a system coincides with the porous medium equation, which can be cast in the form of a gradient flow for the same internal energy. Motivated by these related variational structures, we propose in 19 a particle method for both problems in which the internal energy is replaced by its MoreauYosida regularization in the L2 sense, which can be efficiently computed as a semidiscrete 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: JeanDavid 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 numerically. 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 wellposed variational formulation of the point source problem to build a smooth parameterization of the reﬂector 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 multimarginal 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 multimarginal 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 (GaussSeidel) 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 primaldual methods applied to optimal transport and Wasserstein Barycenter problems
Participants: Antonin Chambolle, Juan Pablo Contreras.
We discuss in 15 the efficiency of Hybrid PrimalDual (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 stateoftheart 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 stateoftheart 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 Offthegrid 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 $min{\int}_{{\mathbb{R}}^{N}}f(u,\nabla u)$ under a mass constraint ${\int}_{{\mathbb{R}}^{N}}u=m\in \mathbb{R}$. 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 CahnHilliard fluids into droplets, and is related to the approximation of branched transport by elliptic energies.
6.9 Dynamical Programming for offthegrid dynamic Inverse Problems
Participants: Robert Tovey, Vincent Duval.
In 40, we consider algorithms for reconstructing timevarying data into a finite sum of discrete trajectories, alternatively, an offthegrid sparsespikes 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, FrançoisXavier Vialard.
A classical tool for approximating integrals is the Laplace method. The firstorder, as well as the higherorder Laplace formula is most often written in coordinates without any geometrical interpretation. In 8, motivated by a situation arising, among others, in optimal transport, we give a geometric formulation of the firstorder 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 firstorder term with standard geometric objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two different metrics arising naturally in the Kim–McCann framework. Passing by, we give an explicitly quantified version of the Laplace formula, as well as examples of applications.6.11 Convergence rate of general entropic optimal transport costs
Participants: Guillaume Carlier, Paul Pegon, Luca Tamanini.
We investigate in 32 the convergence rate of the optimal entropic cost vε to the optimal transport cost as the noise parameter $\epsilon \to 0$. We show that for a large class of cost functions $c$ on ${\mathbb{R}}^{d}\times {\mathbb{R}}^{d}$ (for which optimal plans are not necessarily unique or induced by a transport map) and compactly supported and ${L}^{\infty}$ marginals, one has ${v}_{\epsilon}{v}_{0}=d/2\epsilon log(1/\epsilon )+O\left(\epsilon \right)$. Upper bounds are obtained by a block approximation strategy and an integral variant of Alexandrov's theorem. Under an infinitesimal twist condition on c, i.e. invertibility of ${\nabla}_{xy}^{2}c$, we get the lower bound by establishing a quadratic detachment of the duality gap in d dimensions thanks to Minty's trick.7 Partnerships and cooperations
7.1 International research visitors
7.1.1 Visits of international scientists
Other international visits to the team
Luigi De Pascale

Status
Associate Professor

Institution of origin:
Università di Firenze

Country:
Italy

Dates:
October 5thoctober 20th

Context of the visit:
research with Guillaume Carlier, Luca Nenna and Paul Pegon, mentoring of the PhD of Camilla Brizzi

Mobility program/type of mobility:
research stay and lecture
Giuseppe Buttazzo

Status
Full Professor

Institution of origin:
Università di Pisa

Country:
Italy

Dates:
march 13thmarch 17th and december 11th14th

Context of the visit:
research with Guillaume Carlier and Katharina Eichinger, PhD defense of Katharina Eichinger

Mobility program/type of mobility:
research stay, lecture and Ph.D defense.
7.1.2 Visits to international teams
Research stays abroad
Flavien Léger

Visited institution:
Fields Institute, Toronto, ON

Country:
Canada

Dates:
November 7, 2022 to November 24, 2022

Context of the visit:
longterm visit to the Thematic Program on Nonsmooth Riemannian and Lorentzian Geometry

Mobility program/type of mobility:
research stay
JeanDavid Benamou

Visited institution:
Imperial College

Country:
RoyaumeUni

Dates:
April 23 to April 28 and September 26 to December 19.

Context of the visit:
Invitation and Nelder Fellowship

Mobility program/type of mobility:
research stay and lecture.
Guillaume Carlier

Visited institution:
TUM (Munich)

Country:
Allemagne

Dates:
may 22ndmay 26th and november 6thnovember 11th

Context of the visit:
research with Daniel Matthes and with Gero Friesecke, workshop calculus of variations

Mobility program/type of mobility:
research stay, lecture
7.2 European initiatives
7.2.1 H2020 projects
ROMSOC
Participants: JeanDavid Benamou, Giorgi Rukhaia.
ROMSOC project on cordis.europa.eu

Title:
Reduced Order Modelling, Simulation and Optimization of Coupled systems

Duration:
From September 1, 2017 to August 31, 2022

Partners:
 INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET AUTOMATIQUE (INRIA), France
 DANIELI OFFICINE MECCANICHE SpA, Italy
 ABB SCHWEIZ AG (ABB SUISSE SA ABB SWITZERLAND LTD), Switzerland
 FRIEDRICHALEXANDERUNIVERSITAET ERLANGENNUERNBERG (FAU), Germany
 MATHCONSULT GMBH (MATHCONSULT GMBH), Austria
 Math.Tec GmbH (Math.Tec), Austria
 MICROGATE SRL (MICROGATE S.R.L.), Italy
 FORSCHUNGSVERBUND BERLIN EV, Germany
 STICHTING EUROPEAN SERVICE NETWORK OF MATHEMATICS FOR INDUSTRY AND INNOVATION (EUROPEAN SERVICE NETWORK OF MATHEMATICS FOR INDUSTRY AND INNOVATION), Netherlands
 SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI DI TRIESTE (SISSA), Italy
 SAGIV TECH LTD (SAGIVTECH), Israel
 UNIVERSITAT LINZ (JOHANNES KEPLER UNIVERSITAT LINZ UNIVERSITY OF LINZ JOHANNES KEPLER UNIVERSITY OF LINZ JKU), Austria
 TECHNISCHE UNIVERSITAT BERLIN (TUB), Germany
 CONSORCIO CENTRO DE INVESTIGACIÓN E TECNOLOXÍA MATEMÁTICA DE GALICIA (CITMAGA), Spain
 UNIVERSITAET BREMEN (UBREMEN), Germany
 BERGISCHE UNIVERSITAET WUPPERTAL (BUW), Germany
 POLITECNICO DI MILANO (POLIMI), Italy
 ARCELORMITTAL INNOVACION INVESTIGACION E INVERSION SL (AMIII), Spain
 STMICROELECTRONICS SRL, Italy
 MICROFLOWN TECHNOLOGIES BV (MICROFLOWN), Netherlands
 DB Schenker Rail Polska S.A. (DB Schenker Rail Polska), Poland
 SIGNIFY NETHERLANDS BV (Signify Netherlands BV), Netherlands
 CorWave (CorWave), France

Inria contact:
JeanDavid Benamou
 Coordinator:

Summary:
The development of high quality products and processes is essential for the future competitiveness of the European economy. In most key technology areas product development is increasingly based on simulation and optimization via mathematical models that allow to optimize design and functionality using free design parameters. Best performance of modelling, simulation and optimization (MSO) techniques is obtained by using a model hierarchy ranging from very fine to very coarse models obtained by model order reduction (MOR) techniques and to adapt the model and the methods to the userdefined requirements in accuracy and computational speed.
ROMSOC will work towards this goal for high dimensional and coupled systems that describe different physical phenomena on different scales; it will derive a common framework for different industrial applications and train the next generation of researchers in this highly interdisciplinary field. It will focus on the three major methodologies: coupling methods, model reduction methods, and optimization methods, for industrial applications in well selected areas, such as optical and electronic systems, economic processes, and materials. ROMSOC will develop novel MSO techniques and associated software with adaptability to userdefined accuracy and efficiency needs in different scientific disciplines. It will transfer synergies between different industrial sectors, in particular for SMEs.
To lift this common framework to a new qualitative level, a joint training programme will be developed which builds on the strengths of the academic and industrial partners and their strong history of academic/industrial cooperation. By delivering earlycareer training embedded in a cuttingedge research programme, ROMSOC will educate highly skilled interdisciplinary researchers in mathematical MSO that will become facilitators in the transfer of innovative concepts to industry. It will thus enhance the capacity of European research and development.
7.3 National initiatives

ANR MAGA
(20162022) To study and to implement discretizations of optimal transport and MongeAmpère 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 offthegrid methods for inverse problems involving the reconstruction of complex objects.
8 Dissemination
8.1 Promoting scientific activities
8.1.1 Scientific events: organisation
Member of the organizing committees
 Antonin Chambolle was a coorganizer of the Oberwolfach 2234 seminar: Mathematical Imaging and Surface Processing (August, 2127 2022)
 Antonin Chambolle is a coorganizer of the Séminaire Parisien d'Optimisation (SPO)
 Vincent Duval is a coorganizer of the Imaging in Paris seminar.
 Paul Pegon is a coorganizer of the workgoup on Calculus of Variation GT CalVa
 Thomas Gallouët is coorganizer of the journées MAGA at Autrans (February, 24 2022)
8.1.2 Journal
Member of the editorial boards
 Irène Waldspurger is associate editor for the journal IEEE Transactions on Signal Processing.
Reviewer  reviewing activities
 Vincent Duval has reviewed submissions for the journals Math in Action and SIAM Journal on Imaging Sciences (SIIMS).
 Irène Waldspurger has reviewed submissions for the journals Applied and Computational Harmonic Analysis, IEEE Transactions on Information Theory, IEEE Transactions on Signal Processing and the SIAM journal on Optimization, as well as for the NeurIPS conference.
 Flavien Léger has reviewed submission for the journals IMA Journal of Applied Mathematics, ESAIM: Control, Optimisation and Calculus of Variations and Communications in Partial Differential Equations.
 Paul Pegon has reviewed submissions for the SIAM Journal on Mathematical Analysis (SIMA) and the Journal of Mathematical Analysis and Applications (JMAA).
 Thomas Gallouët has reviewed submissions for Foundations of Computational Mathematics, Journal of Convex Analysis.
8.1.3 Invited talks
 Antonin Chambolle was invited to give talks at the Inverse Problem on Large Scales workshop at RICAM (Linz, Austria), at the IEEE ICIP 2022 Int. Conference on Image Processing, and at the Summerschool on Analysis and Applied Mathematics at Münster, Germany.
 Vincent Duval was invited to give a talk at the Inverse Problems: Modelling and Simulation (IPMS) conference, and the "Probabilités et Statistiques" seminar at IECL (Nancy).
 Irène Waldspurger was invited to give a talk at the Learning and Optimization in Luminy workshop, and at the CIS/MINDS seminar at Johns Hopkins University.
 Flavien Léger was invited to give a talk at the Institut de mathématique d’Orsay, the Laboratoire Jean Kuntzmann Université Grenoble Alpes, the Laboratoire Paul Painlevé, Université de Lille and the Fields Institute Applied Mathematics Colloquium.
 Paul Pegon was invited to give a talk at the Séminaire parisien d'optimisation (IHP) and at the Gdt Transport Optimal (Laboratoire de Mathématiques d'Orsay, Université ParisSaclay).
 Thomas Gallouët was invited to give a talk at the Séminaire ANEDP at Lille University, at the the workgoup on Calculus of VariationGT CalVa at Dauphine University and at the conference on Optimal transport, geometric and stochastic Hydrodynamics at Lisbon, June 2124, 2022.
8.1.4 Research administration
 Irène Waldspurger was a member of a selection committee at Université Aix Marseille.
 Vincent Duval was a member of a selection committee at Sorbonne Université (LJLL)
 Thomas Gallouët is a member of the Commission d'Évaluation Scientifique (CES) of Inria Paris.
8.2 Teaching  Supervision  Juries
8.2.1 Teaching
 Master: Antonin Chambolle Optimisation Continue, ??h, niveau M2, Université Paris DauphinePSL, FR
 Master : Vincent Duval , Problèmes Inverses, 22,5 h équivalent TD, niveau M1, Université PSL/Mines ParisTech, FR
 Master : Vincent Duval , Optimization for Machine Learning, 6h, niveau M2, Université PSL/ENS, FR
 Licence : Irène Waldspurger , Prérentrée raisonnement, 31,2 h équivalent TD, niveau L1, Université ParisDauphine, FR
 Licence : Irène Waldspurger , Analyse 4, 58,5 h équivalent TD, niveau L1, Université ParisDauphine, FR
 Master : Irène Waldspurger , Optimization for Machine Learning, 6h, niveau M2, Université PSL/ENS, FR
 Licence : Guillaume Carlier , algebre 1, L1 78h, Dauphine, FR
 Master : Guillaume Carlier Variational and transport methods in economics, M2 Masef, 27h, Dauphine, FR
 Licence : Paul Pegon , Analyse 2, 51 H. équivalent TD, TD niveau L1, Université ParisDauphine, FR
 Licence : Paul Pegon , Intégrale de Lebesgue et probabilités, 44 H. équivalent TD, TD niveau L3, Université ParisDauphine, FR
 Licence : Paul Pegon , Analyse fonctionnelle et hilbertienne, 44 H. équivalent TD, TD niveau L3, Université ParisDauphine, FR
 Licence : Paul Pegon , Méthodes numériques pour l'optimisation, 33 H. équivalent TD, TD/TP niveau L3, Université ParisDauphine, FR
 Agregation : Thomas Gallouët , Optimisation, Analyse numérique, 48h équivalent TD, niveau M2, Université d'Orsay), FR
 Guillaume Carlier : Licence Algèbre 1, Dauphine 70h, M2 Masef: Variatioanl and transport problems in economics, 18h
 Flavien Léger : Graduate course, two lectures in ‘math+econ+code’ masterclass on equilibrium transport and matching models in economics, NYU Paris. 5h.
8.2.2 Supervision
 PhD completed: Quentin Petit, Mean field games and optimal transport in urban modelling, defended on 18/02/2022, Supervised by Guillaume Carlier , Y. Achdou, D. Tonon.
 PhD completed: Romain Petit, Reconstruction of piecewise constant images via total variation regularization, defended on 12/12/2022, Supervised by Vincent Duval and Yohann De Castro.
 PhD in completed: Katharina Eichinger, Problèmes variationnels pour l'interpolation dans l’espace de Wasserstein 1/09/2019. Supervised by Guillaume Carlier .
 PhD in progress : JoaoMiguel Machado, Transport optimal et structures géométriques, 01/10/2021, Cosupervised by Vincent Duval and Antonin Chambolle .
 PhD in progress : Adrien Vacher 1/10/2020. Cosupervised by FrançoisXavier Vialard and JeanDavid Benamou
 Summer internship: Mitchell Gaudet, an undergraduate student at the University of Toronto, Canada. The Kim–McCann geometry in applied optimal transport. Supervised by Flavien Léger and Robert McCann.
 PhD in progress : Erwan Stämplfi 1/10/2021, Around multiphasic flows. Cosupervised by Yann Brenier and Thomas Gallouët .
 PhD in progress : Siwan Boufadene 1/10/2022, Gradient flows for energy distance. Cosupervised by FrançoisXavier Vialard and Thomas Gallouët .
 PhD in progress : Chazareix Guillaume 1/08/2021, Non Linear Parabolic equations and Volatility Calibration. Cosupervised by JeanDavid Benamou and G. Loeper.
 PhD in progress : Malamut Hugo 1/09/2022, Régularisation Entropique et Transport Optimal Généralisé. Supervised by JeanDavid Benamou .
8.2.3 Juries
 Antonin Chambolle was a member of the jury for the PhD theses of Ulysse MarteauFerey (ENS), Romain Petit (Dauphine ParisPSL), and for the habilitation thesis of Andrea Simonetto. He was the reviewer of the PhD theses of Jordan Michelet (Univ. La Rochelle) and Garry Terii (Univ. Lyon 1).
 Vincent Duval was a member of the jury for the PhD Theses of Thomas Debarre (EPFL) and Zoé Lambert (INSA Rouen).
 Irène Waldspurger was a member of the jury of the PhD thesis of PierreHugo Vial (ENSEEIHT, Toulouse).
 JeanDavid Benamou was a reviewer of the PhD thesis JeanFrançois Quilbert (U. Joseph Fourier, Grenoble) and Charlie Egan (Heriot Watt, Edimbourg).
8.3 Popularization
8.3.1 Internal or external Inria responsibilities
 Vincent Duval is a member of the CSD (comité de suivi doctoral) of the Inria Paris Centre.
 Thomas Gallouët is a member of the Commission d'Évaluation Scientifique (CES) of Inria Paris.
8.3.2 Education
 Vincent Duval has given a 10h lecture on Inverse Problems at the CIMPA school "Mathématiques en analyse et traitement du signal, des images et des données" at Thiès (Sénégal).
8.3.3 Interventions
 Irène Waldspurger has given a talk at the Journée des doctorant·e·s en mathématiques des HautsdeFrance.
9 Scientific production
9.1 Major publications
 1 inproceedingsMirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM.NeurIPS 2022  Thirtysixth Conference on Neural Information Processing SystemsNew Orleans, United States2022
 2 articleIterative Bregman Projections for Regularized Transportation Problems.SIAM Journal on Scientific Computing2372015, A1111A1138
 3 articleSecond order models for optimal transport and cubic splines on the Wasserstein space.Foundations of Computational MathematicsOctober 2019
 4 articleOn Representer Theorems and Convex Regularization.SIAM Journal on Optimization292May 2019, 1260–1281
 5 articleA variational finite volume scheme for Wasserstein gradient flows.Numerische Mathematik14632020, pp 437  480
 6 articleConvergence of Entropic Schemes for Optimal Transport and Gradient Flows.SIAM Journal on Mathematical Analysis492April 2017
 7 miscConvergence rate of general entropic optimal transport costs.June 2022
 8 miscA geometric Laplace method.December 2022
 9 articlePhase retrieval with random Gaussian sensing vectors by alternating projections.IEEE Transactions on Information Theory6452018, 33013312
 10 articleRank optimality for the BurerMonteiro factorization.SIAM Journal on Optimization3032020, 25772602
9.2 Publications of the year
International journals
 11 articlePoint Source Regularization of the Finite Source Reflector Problem.Journal of Computational PhysicsMay 2022
 12 articleCorrection to: Vector Quantile Regression and Optimal Transport, from Theory to Numerics.Empirical Economics621January 2022, 6363
 13 articleOn the linear convergence of the multimarginal Sinkhorn algorithm.SIAM Journal on Optimization3222022, 786794
 14 articleTowards Offthegrid Algorithms for Total Variation Regularized Inverse Problems.Journal of Mathematical Imaging and VisionJuly 2022
 15 articleAccelerated Bregman primaldual methods applied to optimal transport and Wasserstein Barycenter problems.SIAM Journal on Mathematics of Data Science2022
 16 article"FISTA" in Banach spaces with adaptive discretisations.Computational Optimization and Applications2022
 17 articleStability of optimal traffic plans in the irrigation problem.Discrete and Continuous Dynamical Systems  Series A2022
 18 articleRobust LassoZero for sparse corruption and model selection with missing covariates.Scandinavian Journal of Statistics2022
 19 articleConvergence of a Lagrangian discretization for barotropic fluids and porous media flow.SIAM Journal on Mathematical Analysis5432022
 20 articleA mixed finite element discretization of dynamical optimal transport.Journal of Scientific Computing2022
International peerreviewed conferences
 21 inproceedingsMirror Descent with Relative Smoothness in Measure Spaces, with application to Sinkhorn and EM.NeurIPS 2022  Thirtysixth Conference on Neural Information Processing SystemsNew Orleans, United States2022
Edition (books, proceedings, special issue of a journal)
 22 proceedingsA.Alexis AymeC.Claire BoyerA.Aymeric DieuleveutE.Erwan ScornetNearoptimal rate of consistency for linear models with missing values.2022
Doctoral dissertations and habilitation theses
 23 thesisFaces and extreme points of convex sets for the resolution of inverse problems.Ecole doctorale SDOSEJune 2022
Reports & preprints
 24 miscA mean field model for the interactions between firms on the markets of their inputs.July 2022
 25 miscA simple city equilibrium model with an application to teleworking.July 2022
 26 misc Is interpolation benign for random forests? April 2022
 27 miscA FREE DISCONTINUITY APPROACH TO OPTIMAL PROFILES IN STOKES FLOWS.January 2023
 28 miscWasserstein interpolation with constraints and application to a parking problem.November 2022
 29 miscLipschitz Continuity of the Schrödinger Map in Entropic Optimal Transport.October 2022
 30 miscQuantitative Stability of Barycenters in the Wasserstein Space.September 2022
 31 miscFenchelYoung inequality with a remainder and applications to convex duality and optimal transport.April 2022
 32 miscConvergence rate of general entropic optimal transport costs.June 2022
 33 miscA general compactness theorem in <i>G(S)BD</i>.October 2022
 34 miscStochastic Primal Dual Hybrid Gradient Algorithm with Adaptive StepSizes.January 2023
 35 miscL1Gradient Flow of Convex Functionals.October 2022
 36 miscEfficient preconditioners for solving dynamical optimal transport via interior point methods.2022
 37 miscFrom geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows.September 2022
 38 miscA geometric Laplace method.December 2022
 39 miscMass concentration in rescaled first order integral functionals.March 2022
 40 miscDynamical Programming for offthegrid dynamic Inverse Problems.December 2022
 41 miscStability of SemiDual Unbalanced Optimal Transport: fast statistical rates and convergent algorithm..June 2022
9.3 Cited publications
 42 articleBarycenters in the Wasserstein space.SIAM J. Math. Anal.4322011, 904924
 43 articleHamiltonian ODEs in the Wasserstein space of probability measures.Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences6112008, 1853
 44 articleFinite element exterior calculus, homological techniques, and applications.Acta Numerica152006, 1–155
 45 articleConsistency of Trace Norm Minimization.J. Mach. Learn. Res.9June 2008, 10191048URL: http://dl.acm.org/citation.cfm?id=1390681.1390716
 46 articleConsistency of the Group Lasso and Multiple Kernel Learning.J. Mach. Learn. Res.9June 2008, 11791225URL: http://dl.acm.org/citation.cfm?id=1390681.1390721
 47 miscApplications of weak transport theory.2020
 48 unpublishedA spatial Pareto exchange economy problem.December 2021, working paper or preprint
 49 articleA mean field game model for the evolution of cities.Journal of Dynamics and Games2021
 50 articleComputing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms.International Journal of Computer Vision612February 2005, 139157URL: http://dx.doi.org/10.1023/B:VISI.0000043755.93987.aa
 51 articleModelindependent bounds for option prices mass transport approach.Finance and Stochastics1732013, 477501URL: http://dx.doi.org/10.1007/s0078001302058
 52 articleA computational fluid mechanics solution to the MongeKantorovich mass transfer problem.Numer. Math.8432000, 375393URL: http://dx.doi.org/10.1007/s002110050002
 53 articleWeak existence for the semigeostrophic equations formulated as a coupled MongeAmpère/transport problem.SIAM J. Appl. Math.5851998, 14501461
 54 articleAugmented Lagrangian algorithms for variational problems with divergence constraints.JOTA2015
 55 articleIterative Bregman Projections for Regularized Transportation Problems.SIAM J. Sci. Comp.to appear2015
 56 articleNumerical solution of the optimal transportation problem using the MongeAmpere equation.Journal of Computational Physics2602014, 107126
 57 articleTwo numerical methods for the elliptic MongeAmpère equation.M2AN Math. Model. Numer. Anal.4442010, 737758
 58 articleConsistent estimation of a population barycenter in the Wasserstein space.Preprint arXiv:1212.25622012
 59 articleAdditive manufacturing scanning paths optimization using shape optimization tools.Struct. Multidiscip. Optim.6162020, 24372466URL: https://doi.org/10.1007/s00158020026143
 60 articleSliced and Radon Wasserstein Barycenters of Measures.Journal of Mathematical Imaging and Vision5112015, 2245URL: http://hal.archivesouvertes.fr/hal00881872/
 61 articleCharacterization of optimal shapes and masses through MongeKantorovich equation.J. Eur. Math. Soc. (JEMS)322001, 139168URL: http://dx.doi.org/10.1007/s100970000027
 62 articleDeterministic guarantees for BurerMonteiro factorizations of smooth semidefinite programs.preprinthttps://arxiv.org/abs/1804.020082018
 63 articleA generalized conditional gradient method for dynamic inverse problems with optimal transport regularization.arXiv preprint arXiv:2012.117062020
 64 articleDécomposition polaire et réarrangement monotone des champs de vecteurs.C. R. Acad. Sci. Paris Sér. I Math.305191987, 805808
 65 articleReconstruction of the early universe as a convex optimization problem.Mon. Not. Roy. Astron. Soc.3462003, 501524URL: http://arxiv.org/pdf/astroph/0304214.pdf
 66 articleGeneralized solutions and hydrostatic approximation of the Euler equations.Phys. D23714172008, 19821988URL: http://dx.doi.org/10.1016/j.physd.2008.02.026
 67 articlePolar factorization and monotone rearrangement of vectorvalued functions.Comm. Pure Appl. Math.4441991, 375417URL: http://dx.doi.org/10.1002/cpa.3160440402
 68 articleA guide to the TV zoo.LevelSet and PDEbased Reconstruction Methods, Springer2013
 69 bookOptimal Urban Networks via Mass Transportation.1961Lecture Notes in MathematicsBerlin, HeidelbergSpringer Berlin Heidelberg2009, URL: http://link.springer.com/10.1007/9783540857990
 70 incollectionOn the numerical solution of the problem of reflector design with given farfield scattering data.Monge Ampère equation: applications to geometry and optimization (Deerfield Beach, FL, 1997)226Contemp. Math.Providence, RIAmer. Math. Soc.1999, 1332URL: http://dx.doi.org/10.1090/conm/226/03233
 71 articleThe regularity of mappings with a convex potential.J. Amer. Math. Soc.511992, 99104URL: http://dx.doi.org/10.2307/2152752
 72 articleComputational Analysis of LDDMM for Brain Mapping.Frontiers in Neuroscience72013
 73 articleSuperResolution from Noisy Data.Journal of Fourier Analysis and Applications1962013, 12291254
 74 articleTowards a Mathematical Theory of SuperResolution.Communications on Pure and Applied Mathematics6762014, 906956
 75 articleAn Introduction to Compressive Sensing.IEEE Signal Processing Magazine2522008, 2130
 76 bookClassical and modern optimization.Advanced textbooks in mathematicsTopological and functional analytic preliminaries  Differential calculus  Convexity  Optimality conditions for differentiable optimization  Problems depending on a parameter  Convex duality and applications  Iterative methods for convex minimization  When optimization and data meet  An invitation to the calculus of variationsHackensack, New JerseyWorld Scientific2022
 77 unpublishedSISTA: Learning Optimal Transport Costs under Sparsity Constraints.October 2020, working paper or preprint
 78 articleOn the total variation Wasserstein gradient flow and the TVJKO scheme.ESAIM: Control, Optimisation and Calculus of Variations2019
 79 articleKinetic models for chemotaxis and their driftdiffusion limits.Monatsh. Math.142122004, 123141URL: http://dx.doi.org/10.1007/s0060500402347

80
articleVariational approximation of sizemass energies for
$k$ dimensional currents.ESAIM Control Optim. Calc. Var.252019, Paper No. 43, 39URL: https://doi.org/10.1051/cocv/2018027  81 articleCrouzeixRaviart approximation of the total variation on simplicial meshes.J. Math. Imaging Vision62672020, 872899URL: https://doi.org/10.1007/s10851019009393
 82 articleLearning consistent discretizations of the total variation.SIAM J. Imaging Sci.1422021, 778813URL: https://doi.org/10.1137/20M1377199

83
articleThe
$$ Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps.401January 2008, 120URL: https://epubs.siam.org/doi/10.1137/07069938X  84 articleAtomic decomposition by basis pursuit.SIAM journal on scientific computing2011999, 3361
 85 inproceedingsFaster Wasserstein Distance Estimation with the Sinkhorn Divergence.Neural Information Processing SystemsAdvances in Neural Information Processing SystemsVancouver, CanadaDecember 2020
 86 articleDiscrete total variation: new definition and minimization.SIAM J. Imaging Sci.1032017, 12581290URL: https://doi.org/10.1137/16M1075247
 87 articleDensity Functional Theory and Optimal Transportation with Coulomb Cost.Communications on Pure and Applied Mathematics6642013, 548599URL: http://dx.doi.org/10.1002/cpa.21437
 88 bookA Mathematical Theory of LargeScale Atmosphere/Ocean Flow.Imperial College Press2006, URL: https://books.google.fr/books?id=JxBqDQAAQBAJ
 89 articleThe semigeostrophic equations discretized in reference and dual variables.Arch. Ration. Mech. Anal.18522007, 341363URL: http://dx.doi.org/10.1007/s0020500600406
 90 articleGeneralised Lagrangian solutions for atmospheric and oceanic flows.SIAM J. Appl. Math.5111991, 2031
 91 inproceedingsSinkhorn Distances: Lightspeed Computation of Optimal Transport.Proc. NIPS2013, 22922300

92
articleA Study of the Dual Problem of the OneDimensional
${L}^{}$ Optimal Transport Problem with Applications.27611June 2019, 33043324URL: https://www.sciencedirect.com/science/article/pii/S0022123619300643  93 articleNumerical methods for fully nonlinear elliptic equations of the MongeAmpère type.Comput. Methods Appl. Mech. Engrg.19513162006, 13441386
 94 articleExact Support Recovery for Sparse Spikes Deconvolution.Foundations of Computational Mathematics2014, 141URL: http://dx.doi.org/10.1007/s1020801492286
 95 articleSupport detection in superresolution.Proc. Proceedings of the 10th International Conference on Sampling Theory and Applications2013, 145148
 96 unpublishedInterpolating between Optimal Transport and MMD using Sinkhorn Divergences.October 2018, working paper or preprint
 97 articleBreaking the Curse of Dimension in MultiMarginal Kantorovich Optimal Transport on Finite State Spaces.SIAM Journal on Mathematical Analysis5042018, 39964019URL: https://doi.org/10.1137/17M1150025
 98 articleMongeAmpèreKantorovitch (MAK) reconstruction of the eary universe.Nature4172602002
 99 articleA stochastic control approach to NoArbitrage bounds given marginals, with an application to Loopback options.submitted to Annals of Applied Probability2011
 100 articleThe geometry of optimal transportation.Acta Math.17721996, 113161URL: http://dx.doi.org/10.1007/BF02392620
 101 articleGaspard Monge, Le mémoire sur les déblais et les remblais.Image des mathématiques, CNRS2012, URL: http://images.math.cnrs.fr/GaspardMonge,1094.html

102
articlePath Dependent Optimal Transport and Model Calibration on Exotic Derivatives.SSRN Electron.~J.Available at
doi:10.2139/ssrn.3302384 01 2018  103 articleNotes on a PDE System for Biological Network Formation.138June 2016, 127155URL: https://www.sciencedirect.com/science/article/pii/S0362546X15004344
 104 articleSoliton dynamics in computational anatomy.NeuroImage232004, S170S178
 105 incollectionThe mathematical theory of frontogenesis.Annual review of fluid mechanics, Vol. 14Palo Alto, CAAnnual Reviews1982, 131151
 106 articleThe backandforth method for Wasserstein gradient flows.ESAIM: Control, Optimisation and Calculus of Variations272021, 28
 107 articleA fast approach to optimal transport: The backandforth method.Numer. Math.1462020, 513544URL: https://doi.org/10.1007/s00211020011548
 108 articleOn explosions of solutions to a system of partial differential equations modelling chemotaxis.Trans. Amer. Math. Soc.32921992, 819824URL: http://dx.doi.org/10.2307/2153966
 109 articleThe variational formulation of the FokkerPlanck equation.SIAM J. Math. Anal.2911998, 117
 110 articleOn the translocation of masses.C. R. (Doklady) Acad. Sci. URSS (N.S.)371942, 199201
 111 articleMinimizing within convex bodies using a convex hull method.SIAM Journal on Optimization162January 2005, 368379
 112 articleMean field games.Jpn. J. Math.212007, 229260URL: http://dx.doi.org/10.1007/s1153700706578
 113 articleOptimal Transport Approximation of 2Dimensional Measures.SIAM Journal on Imaging Sciences122January 2019, 762787URL: https://epubs.siam.org/doi/10.1137/18M1193736
 114 articleA survey of the Schrödinger problem and some of its connections with optimal transport.Discrete Contin. Dyn. Syst.3442014, 15331574URL: http://dx.doi.org/10.3934/dcds.2014.34.1533
 115 articleActive sets, nonsmoothness, and sensitivity.SIAM Journal on Optimization1332003, 702725
 116 articleOptimal transportation meshfree approximation schemes for Fluid and plastic Flows.Int. J. Numer. Meth. Engng 83:1541579832010, 15411579
 117 articleA fully nonlinear version of the incompressible Euler equations: the semigeostrophic system.SIAM J. Math. Anal.3832006, 795823 (electronic)
 118 articleNumerical solution of the MongeAmpére equation by a Newton's algorithm.C. R. Math. Acad. Sci. Paris34042005, 319324
 119 bookA wavelet tour of signal processing.Elsevier/Academic Press, Amsterdam2009
 120 articleA macroscopic crowd motion model of gradient flow type.Math. Models Methods Appl. Sci.20102010, 17871821URL: http://dx.doi.org/10.1142/S0218202510004799
 121 articleA multiscale approach to optimal transport.Computer Graphics Forum3052011, 15831592
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