2023Activity reportProjectTeamMOKAPLAN
RNSR: 201321083P Research center Inria Paris Centre
 In partnership with:Université ParisDauphine, CNRS
 Team name: Advances in Numerical Calculus of Variations
 In collaboration with:CEREMADE
 Domain:Applied Mathematics, Computation and Simulation
 Theme:Numerical schemes and simulations
Keywords
Computer Science and Digital Science
 A5.3. Image processing and analysis
 A5.9. Signal processing
 A6.1.1. Continuous Modeling (PDE, ODE)
 A6.2.1. Numerical analysis of PDE and ODE
 A6.2.6. Optimization
 A6.3.1. Inverse problems
 A8.2.3. Calculus of variations
 A8.12. Optimal transport
 A9. Artificial intelligence
Other Research Topics and Application Domains
 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
 Vincent Duval [Team leader, INRIA, Senior Researcher, HDR]
 JeanDavid Benamou [INRIA, Senior Researcher, HDR]
 Antonin Chambolle [CNRS, Senior Researcher, HDR]
 Thomas Gallouët [INRIA, Researcher]
 Flavien Leger [INRIA, Researcher]
 Irène Waldspurger [CNRS, Researcher]
Faculty Members
 Guillaume Carlier [DAUPHINE PSL, Professor, HDR]
 Paul Pegon [DAUPHINE PSL, Associate Professor, from Sep 2023]
 Paul Pegon [DAUPHINE PSL, Associate Professor Delegation, until Aug 2023]
 FrançoisXavier Vialard [UNIV GUSTAVE EIFFEL, Professor Delegation, from Feb 2023 until Jul 2023]
PostDoctoral Fellow
 Adrien Vacher [INRIA, PostDoctoral Fellow, from Dec 2023]
PhD Students
 Guillaume Chazareix [BNP, CIFRE]
 Hugo Malamut [DAUPHINE PSL]
 Joao Pinto Anastacio Machado [DAUPHINE PSL]
 Faniriana Rakoto Endor [CNRS, from Oct 2023]
 Erwan Stampfli [UNIV PARIS SACLAY]
 Maxime Sylvestre [DAUPHINE PSL]
Administrative Assistant
 Derya Gok [INRIA]
External Collaborators
 Yann Brenier [CNRS]
 FrançoisXavier Vialard [UNIV GUSTAVE EIFFEL, from Sep 2023]
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 104). This engineering problem consists in minimizing the transport cost between two given mass densities. In the 40's, Kantorovich 113 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 67, 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 137, 138, 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 117 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 70 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 74 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 9612160 59 and the references therein. Geometric approaches based on Laguerre diagrams and discrete data 124 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 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 Multimarginal Optimal Transport (see 129 and the references therein). Another instance of multimarginal Optimal Transportation arises in the socalled 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 semigeostrophic 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 multimarginal 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 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 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 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 64.
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 112. This is a popular tool to study a large class of nonlinear diffusion equations. Two interesting examples are the KellerSegel system for chemotaxis 111, 82 and a model of congested crowd motion proposed by Maury, Santambrogio and RoudneffChupin 123. 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 52, or on a (infinite dimensional) space of curves or surfaces 140. 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 52, 107. 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 125, 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) 75.
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 122 and learned representation 141).
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” 87 and in statistics by Tibshirani 134 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 78, 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 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 2D 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.




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 132, 71. 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 134 and machine learning 48. 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) 47. The underlying methodology is to make use of lowcomplexity regularization models, which turns out to be equivalent to the use of partlysmooth regularization functionals 118, 136 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 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 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 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

Backandforth
The backandforth method 110, 109 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 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

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 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 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 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 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 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 (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 (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 BurerMonteiro methods
BurerMonteiro 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 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 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 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) 83, 127 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 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 1dimensional integral of some function of $\sigma $), and a transport cost depending on this conductance field. The aforementioned case studied in 72 falls into this category, as well as classical branched transport. The biologicallyinspired network evolution model of 106 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
Antonin Chambolle gave a plenary talk at the International Congress on Industrial and Applied Mathematics (ICIAM) 2023.
6 New software, platforms, open data
 Antonin Chambolle has implemented a new version of the fast and exact proximal operator of the Graph Total variation, built upon Boykov and Kolmogorov's maxflowv3.04 algorithm (available on his web page and soon on plmlab), this is used to implement efficient methods for computing crystalline mean curvature flows.
7 New results
7.1 Entropic Optimal Transport Solutions of the Semigeostrophic Equations
Participants: JeanDavid 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 nonsmooth convex minimization viewpoint
Participants: JeanDavid Benamou, Guillaume Carlier, Daniel Matthes.
Motivated by the DerridaLebowitzSpeerSpohn (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 nonsmooth) nature of the corresponding variational problem to propose a numerical method based on the ChambollePock primaldual algorithm.
7.3 Total variation regularization with Wasserstein penalization
Participants: Antonin Chambolle, Vincent Duval, JoaoMiguel Machado.
In 20, a new derivation of the EulerLagrange equation of a totalvariation regularization problem with a Wasserstein penalization is obtained, it is interesting as on easily deduces some regularity of the Lagrange multiplier for the nonnegativity constraint. A numerical implementation is also described.
7.4 1D approximation of measures in Wasserstein spaces
Participants: Antonin Chambolle, Vincent Duval, JoaoMiguel 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 nondegenerate 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, 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 9, 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.
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 nondegeneracy condition on the cost function (invertibility of the crossderivative) 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 multimarginal 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 pseudoRiemannian geometry that combines both the marginal densities and the cost.
7.9 Gradient descent with a general cost
Participants: Flavien Leger, PierreCyril Aubin.
We present a new class of gradienttype 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 crosscurvature. In particular our framework provides the first global rates for natural gradient descent and the standard Newton's method.
7.10 Secondorder methods for BurerMonteiro factorization
Participants: Florentin Goyens, Clément Royer, Irène Waldspurger.
The BurerMonteiro 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 firstorder algorithms, but secondorder algorithms oftentimes perform better in practice. We have established convergence rates for a representative secondorder 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 primaldual hybrid gradient for largescale 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 “plugandplay” 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 “totalgeneralizedvariation” of Bredies, Kunisch and Pock). A theoretical analysis of this model and of more standard totalvariation 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 / PoincaréKorn 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 nonhomogeneous 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 semidiscrete 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 FokkerPlanck 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, FrancoisXavier Vialard.
Using the dual formulation only, we show that regularity of unbalanced optimal transport also called entropytransport 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 entropytransport formulations, WassersteinFisherRao metric, also called HellingerKantorovich, stands out since it admits a dynamic formulation, which extends the BenamouBrenier 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 McCann. As a byproduct, we formulate the MongeAmpère equation associated with WassersteinFisherRao metric, which also holds for more general costs.
7.17 Entropic approximation of $\infty $ optimal transport problems
Participants: Camilla Brizzi, Guillaume Carlier, Luigi DePascale.
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 (nonsmooth) optimal transport map. We exhibit a tight Hölderbehavior 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 Höldercontinuous 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 pushforward 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 ${\mathbb{R}}^{d}$ , we connect Wasserstein medians to a minimal (multi) flow problem à la Beckmann and a system of PDEs of MongeKantorovichtype, for which we propose a pLaplacian approximation. Our analysis eventually leads to a new numerical method to compute Wasserstein medians, which is based on a DouglasRachford scheme applied to the minimal flow formulation of the problem.
8 Bilateral contracts and grants with industry
Participants: JeanDavid Benamou, Gregoire Loeper.
CIFRE PhD thesis scholarship (Guillaume Chazareix ) with BNP. Main supervisor JeanDavid Benamou , cosupervision with Guillaume Carlier (Inria Mokaplan) and Gregoire Loeper (BNP). This contract is handled by Dauphine University.
9 Partnerships and cooperations
9.1 International research visitors
9.1.1 Visits of international scientists
Other international visits to the team
Participants: Pankaj Gautam.

Status
Postdoc

Institution of origin:
Norvegian University of Science and Technology (NTNU)

Country:
Norway

Dates:
November 13th to November 17th

Context of the visit:
The ERCIM postdoctoral fellowship program requires the students to spend one week in one of the institutions of the ERCIM program (Inria is one of them). Pankaj Gautam has given a lecture, attended several seminars, and discussed with the members of our team.

Mobility program/type of mobility:
research stay and lecture
Participants: Luigi De Pascale.

Status
Researcher

Institution of origin:
University of Florence

Country:
Italy

Dates:
May 2023

Context of the visit:
Luigi De Pascale was invited by Paul Pegon and Guillaume Carlier through the Invited Professors Program of Université ParisDauphine to work on a project on supremal optimal transport. As a regular visitor and collaborator of the team, he interacted with several other members of our team.

Mobility program/type of mobility:
research stay
9.1.2 Visits to international teams
Research stays abroad
 JeanDavid Benamou has visited Pr. Colin Cotter (ICL) during the fall under a Nelder (ICL) fellowship.
 Paul Pegon has visited Pr. Mircea Petrache (PUC, Chile) for two weeks in March 2023
9.2 National initiatives

PRAIRIE chair
: Irène Waldspurger .

ANR CIPRESSI
(20192024) 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.

PDE AI
(20232027) Antonin Chambolle is the main coordinator of the PDEAI 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 Dissemination
10.1 Promoting scientific activities
10.1.1 Scientific events: organisation
 JeanDavid Benamou coorganized a Workshop on Numerical Optimal Transport at FOCM conference, Paris June.
 JeanDavid Benamou coorganized an Ecole des Houches conference on Optimal Transport Theory : Applications to Physics, March.
 Guillaume Carlier coorganized a workshop at Lagrange Center.
 Vincent Duval was the main organizer of the Workshop on OfftheGrid methods for Optimization and Inverse Problems in Imaging ($\approx $ 60 participants) at Institut Henri Poincaré (November 21st and 22nd).
Member of the organizing committees
 Guillaume Carlier coorganized the séminaire Parisien d’Optimisation.
 Vincent Duval coorganized the Julia and Optimization Days 2023 ($\approx $ 140 participants) at Conservatoire National des Arts et Métiers (October 4th, 5th and 6th).
 Vincent Duval coorganizes the monthly seminar Imaging in Paris.
 Antonin Chambolle coorganizes the monthly seminar "Séminaire Parisien d'Optimisation"
 Paul Pegon coorganizes the regular seminar of Calculus of Variations GT CalVa (until June 2023)
10.1.2 Scientific events: selection
Reviewer
 Vincent Duval has reviewed contributions to the GRETSI and SSVM conferences.
 Antonin Chambolle has reviewed contributions for SSVM and AISTATS 24
10.1.3 Journal
Member of the editorial boards
 Vincent Duval is associate editor at the Journal of Mathematical Imaging and Vision (JMIV)

Antonin Chambolle is associate editor at:
 Inverse Problems and Imaging (AIMS)
 Journal of Mathematical Imaging and Vision (JMIV, Springer)
 IPOL (Image Processing On Line),
 ESAIM : Mathematical Modelling and Numerical Analysis(M2AN, Cambridge UP),
 Applied Mathematics and Optimization (AMO, Springer),
 IMA journal of numerical analysis (Oxford)
 Journal of the European Math. Society (JEMS, EMS Publishing).
He is also one of the 4 editors of “Interfaces and Free Boundaries”.
 Irène Waldspurger is associate editor for the IEEE Transactions on Signal Processing.

Guillaume Carlier
is associate editor at:
 Journal de l'Ecole Polytechnique
 Applied Mathematics an Optimization
 J. Math. Analysis and Appl.
 J. Dynamics and Games
Reviewer  reviewing activities
 Vincent Duval has reviewed contributions to the journal of Foundations of Computational Mathematics (FoCM), Journal of NonSmooth Analysis and Optimization (JNSAO) and SIAM Journal on Imaging Sciences (SIIMS). He has also written reviews for the AMS: Mathematical Reviews.
 Antonin Chambolle is a reviewer for many journals including Arch. Rational Mech. Anal., Journal of the European Math. Society, Math. Programming, Calc. Var. PDE., etc.
 Irène Waldspurger has reviewed contributions to the Compterendus de l'académie des sciences (CRAS), to Applied and Computational Harmonic Analysis (ACHA), to Foundations of Computational Mathematics (FoCM) and to the Journal de mathématiques pures et appliquées (JMPA).
 Paul Pegon has reviewed contributions to ESAIM: Mathematical Modelling and Numerical Analysis (M2AN) and ESAIM: Control, Optimisation and Calculus of Variations (COCV).
 Flavien Leger has reviewed contributions to Journal of Convex Analysis and Advances in Mathematics.
 Thomas Gallouët has reviewed contributions to Analysis & PDE and other journals.
10.1.4 Invited talks
 Vincent Duval gave invited talks at the SIGOPT conference (Cottbus, Germany), Applied Inverse Problems (AIP) conference (Göttingen, Germany) and the colloquium 30 years of mathematics for optical imaging in Marseille. He was also invited at the local seminar "Modélisation, Analyse et Calcul Scientifique" in Université de Lyon.
 Antonin Chambolle was an invited speaker at ICIAM 2023 (Tokyo) (August 2023), and an invited speaker at the 3rd AlpsAdriatic Inverse Problems workshop, July 2023. He was also invited speaker at the conference in honor of the 60th birthday of Prof. Luigi Ambrosio, ETH, Zurich, Sept 2023.
 Flavien Leger gave invited talks at the "Machine Learning and Signal Processing seminar" (ENS Lyon), the "Geometric Analysis Seminar" (Iowa State University), the "Analysis and Applied Mathematics Seminar Series" (Bocconi University), the "PGMO Days" (EDF Lab, Saclay) and the "Séminaire Parisien d’Optimisation" (Institut Henri Poincaré).
 Irène Waldspurger gave an invited talk at the conference for Stéphane Mallat's 60th birthday, "A multiscale tour of harmonic analysis and machine learning".
 Paul Pegon gave invited talks at the "Analysis and Geometry seminar" of the Pontifical Catholic University of Chile (Santiago, Chile), at the June 2023 meeting of the "GdR Calva" (Université ParisCité) and the "PGMO Days" (EDF Lab, Saclay).
 Guillaume Carlier gave talks at the GT transport optimal, Orsay; the Analysis and Applied Mathematics seminar, Bocconi, Milan; The Financial math. seminar, ETH Zurich; PGMO Days, session transport optimal; workshop Optimal Transport and the Calculus of Variations, ICMS Edinburgh.
 Thomas Gallouët gave talks at the worshop on Optimal Transport and the Calculus of Varia tions, Edinburgh, Scotland. The FoCM workshop on Opptimal Transport in Paris, France. The conference on Optimal Transport Theory And Applications to Physics, Centre de physique des Houches, France. A seminar for the journée de rentrée of the ANEDP team of the Laboratoire de Mathématique d’Orsay, Université de Paris Saclay, Orsay.
10.1.5 Scientific expertise
 Vincent Duval was a member of the selection committee for a Chaire de Professeur Junior (CPJ) at Sorbonne Université.
 Antonin Chambolle was the head of the hiring comittee for a "maitre de conférence" in nonlinear analysis at Univ. ParisDauphine.
 Irène Waldspurger was a member of the selection committee for a "maître de conférence" position at Université Côte d'Azur.
 Thomas Gallouët was the vicepresident of the hiring comittee for a "maitre de conférence" in nonlinear analysis at Univ. ParisDauphine.
10.1.6 Research administration
 Antonin Chambolle represents France in the IFIP TC7 group system modeling and optimization,
 Antonin Chambolle is member of the scientific comittee and of the board of the PGMO (programme Gaspard Monge pour l'Optimisation et la Recherche Opérationnelle) (FMJH  EDF).
 Irène Waldspurger is a member of the SMAIMODE group.
 Vincent Duval was a member of the Comité de suivi doctoral (CSD) until June, 2023.
 Vincent Duval was a member of the Commission Emplois Scientifiques (CES) 2023 of the Paris research center.
 Vincent Duval has been "Délégué Scientifique Adjoint" (DSA) since September, 1st.
10.2 Teaching  Supervision  Juries
10.2.1 Teaching
 JeanDavid Benamou has given a serie of Lectures on Optimal Transport at Imperial College London. October and November.
 Master: Antonin Chambolle Optimisation Continue, 24h, 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, 9h, niveau M2, Université PSL/ENS, FR
 Licence : Irène Waldspurger , Prérentrée raisonnement, 31,2 h équivalent TD, niveau L1, Université ParisDauphine, FR
 Master : Irène Waldspurger , Optimization for Machine Learning, 9h, niveau M2, Université PSL/ENS, FR
 Master : Irène Waldspurger , Introduction à la géométrie différentielle et aux équations différentielles, 29,25 h équivalent TD, niveau M1, Université Paris Dauphine, FR
 Master : Irène Waldspurger , Nonconvex inverse problems, 27 h d'équivalent TD, niveau M2, Université Paris Dauphine, FR
 Licence : Guillaume Carlier , algebre 1, L1 78h, Dauphine, FR
 Master : Guillaume Carlier Variational and transport methods in economics, M2 Masef, 27h, Dauphine, FR
 Agregation : Thomas Gallouët , Optimisation, Analyse numérique, 48h équivalent TD, niveau M2, Université d'Orsay), FR
 Guillaume Carlier : Licence Algèbre 1, Dauphine 70h, M2 Masef: Variatioanl and transport problems in economics, 18h
 Flavien Léger : Graduate course, two lectures in ‘math+econ+code’ masterclass on equilibrium transport and matching models in economics, NYU Paris. 5h.
10.2.2 Supervision
 PhD in progress: JoaoMiguel Machado , Transport optimal et structures géométriques, 01/10/2021, Cosupervised by Vincent Duval and Antonin Chambolle .
 PhD in progress : Chazareix Guillaume 1/08/2021, Non Linear Parabolic equations and Volatility Calibration. Cosupervised by JeanDavid Benamou and Grégoire Loeper .
 PhD in progress : Hugo Malamut 1/09/2022, Régularisation Entropique et Transport Optimal Généralisé. Cosupervised by JeanDavid Benamou and Guillaume Carlier .
 PhD in progress : Maxime Sylvestre 01/09/2002 on Hybrid methods fot Optimal Transport. Supervised by Guillaume Carlier and Alfred Galichon .
 PhD: Faniriana Rakoto Endor started a PhD on BurerMonteiro methods, under the supervision of Antonin Chambolle and Irène Waldspurger .
 Postdoc: Adrien Vacher started a postdoc in December 2023 under the supervision of Flavien Léger .
 Phd in progress: Erwan Stämplfi , 01/09/2021, on singular limit for multiphase flow, co supervised with Yann Brenier
 Phd in progress: Siwan Boufadene , 01/09/2022, on energy distance flow, co supervised with FrancoisXavier Vialard
10.2.3 Juries
 JeanDavid Benamou participated to the PhD Juries of Rodrigue Lelotte, Adrien Seguret.
 Vincent Duval was a referee for the PhD manuscripts of Adrien Frigerio (Université de Dijon, September) and ThuLe Tran (Université de Rennes, December)
 Vincent Duval was an examiner in the PhD committee of Bastien Laville (Inria SophiaAntipolis, September).
 Irène Waldspurger was an examiner in the PhD committees of Adrien Vacher (Université Gustave Eiffel) and Gaspar Rochette (ENS Paris).
 Guillaume Carlier was a member of the PhD comittee of Raphael Prunier (Sorbonne Université); Quentin Jacquet (Ecole Polytechnique).
 Guillaume Carlier was a member of the HdR comittee of Thibaut Le Gouic (Université de AixMarseille).
11 Scientific production
11.1 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 States2022HAL
 2 articleIterative Bregman Projections for Regularized Transportation Problems.SIAM Journal on Scientific Computing2372015, A1111A1138HALDOI
 3 articleSecond order models for optimal transport and cubic splines on the Wasserstein space.Foundations of Computational MathematicsOctober 2019HALDOI
 4 articleOn Representer Theorems and Convex Regularization.SIAM Journal on Optimization292May 2019, 1260–1281HALDOI
 5 articleA variational finite volume scheme for Wasserstein gradient flows.Numerische Mathematik14632020, pp 437  480HALDOI
 6 articleConvergence of Entropic Schemes for Optimal Transport and Gradient Flows.SIAM Journal on Mathematical Analysis492April 2017HALDOI
 7 miscConvergence rate of general entropic optimal transport costs.June 2022HAL
 8 miscGradient descent with a general cost.December 2023HAL
 9 miscA geometric Laplace method.December 2022HALback to text
 10 articlePhase retrieval with random Gaussian sensing vectors by alternating projections.IEEE Transactions on Information Theory6452018, 33013312HAL
 11 articleRank optimality for the BurerMonteiro factorization.SIAM Journal on Optimization3032020, 25772602HALDOI
11.2 Publications of the year
International journals
 12 articleA mean field model for the interactions between firms on the markets of their inputs.Mathematics and Financial EconomicsMarch 2023HALDOI
 13 articleIsing systems, measures on the sphere, and zonoids.Tunisian Journal of Mathematics2023HALback to text
 14 articleA free discontinuity approach to optimal profiles in stokes flows.Annales de l'Institut Henri Poincaré C, Analyse non linéaire2023HALback to textback to text
 15 articleConvergence rate of general entropic optimal transport costs.Calculus of Variations and Partial Differential Equations624May 2023, 116HALDOIback to text
 16 articleA general compactness theorem in <i>G(S)BD</i>.Indiana University Mathematics Journal2023HALback to textback to textback to text
 17 articleEquilibrium Configurations For Nonhomogeneous Linearly Elastic Materials With Surface Discontinuities.Annali della Scuola Normale Superiore di Pisa, Classe di ScienzeXXIV3September 2023, 15751610HALDOIback to textback to text
 18 articleMinimizing Movements for Anisotropic and Inhomogeneous Mean Curvature Flows.Advances in Calculus of Variation2023HALDOIback to text
International peerreviewed conferences
 19 inproceedingsLearned Discretization Schemes for the SecondOrder Total Generalized Variation.Lecture Notes in Computer Science.9th International Conference on Scale Space and Variational Methods in Computer Vision. SSVM 202314009Lecture Notes in Computer ScienceSanta Margherita di Pula (Cagliari), ItalySpringer International PublishingMay 2023, 484497HALDOIback to text
 20 inproceedingsThe Total VariationWasserstein Problem: a new derivation of the EulerLagrange equations.Geometric Science of Information. GSI 2023.Geometric Science of Information. GSI 202314071Lecture Notes in Computer ScienceSaint Malo (FR), FranceSpringer Nature SwitzerlandJune 2023, 610619HALDOIback to text
 21 inproceedingsA relaxed proximal gradient descent algorithm for convergent plugandplay with proximal denoiser.International Conference on Scale Space and Variational Methods in Computer Vision (SSVM'23)14009LNCS  Lecture Notes in Computer ScienceSanta Margherita di Pula, ItalySpringerMay 2023HALDOIback to text
 22 inproceedingsExplicit Diffusion of Gaussian Mixture Model Based Image Priors.Lecture Notes in Computer Science9th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 202314009Scale Space and Variational Methods in Computer Vision. SSVM 2023Santa Margherita di Pula (Cagliari), ItalySpringer International PublishingMay 2023, 315HALDOIback to text
National peerreviewed Conferences
 23 inproceedingsRégularisation par la variation totale pour l'identification du support d'images constantes par morceaux.GRETSI 2023  XXIXème Colloque Francophone de Traitement du Signal et des ImagesGrenoble, FranceAugust 2023HALback to text
Reports & preprints
 24 misc Is interpolation benign for random forest regression? February 2023 HAL
 25 miscEntropic Optimal Transport Solutions of the Semigeostrophic Equations.January 2023HALback to text
 26 miscOn the global convergence of Wasserstein gradient flow of the Coulomb discrepancy.November 2023HAL
 27 miscWasserstein gradient flow of the Fisher information from a nonsmooth convex minimization viewpoint.August 2023HALback to text
 28 miscWasserstein medians: robustness, PDE characterization and numerics.July 2023HAL
 29 miscQuantitative Stability of Barycenters in the Wasserstein Space.March 2023HAL
 30 miscExact recovery of the support of piecewise constant images via total variation regularization.July 2023HALback to text
 31 miscStochastic Primal Dual Hybrid Gradient Algorithm with Adaptive StepSizes.January 2023HALback to text
 32 misc1D approximation of measures in Wasserstein spaces.April 2023HALback to text
 33 miscExistence of minimizers in the variational approach to brittle fracture.November 2023HALback to textback to text
 34 miscStability of optimal shapes and convergence of thresholding algorithms in linear and spectral optimal control problems.June 2023HALback to text
 35 miscStability of optimal shapes and convergence of thresholding algorithms in linear and spectral optimal control problems: Supplementary material.June 2023HALback to text
 36 miscL1Gradient Flow of Convex Functionals.October 2023HALback to text
 37 miscInclusion and estimates for the jumps of minimizers in variational denoising.December 2023HALback to text
 38 miscFrom geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows.October 2023HAL
 39 miscConvergent plugandplay with proximal denoiser and unconstrained regularization parameter.November 2023HALback to text
 40 miscConvergence rate of entropyregularized multimarginal optimal transport costs.July 2023HALback to text
 41 miscOptimal Quantization with Branched Optimal Transport distances.September 2023HALback to text
 42 miscModelbased Clustering with Missing Not At Random Data.February 2023HAL
 43 miscParameter tuning and model selection in optimal transport with semidual Brenier formulation.January 2023HALback to text
11.3 Cited publications
 44 articleBarycenters in the Wasserstein space.SIAM J. Math. Anal.4322011, 904924back to text
 45 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, 1853back to text
 46 articleFinite element exterior calculus, homological techniques, and applications.Acta Numerica152006, 1–155DOIback to text
 47 articleConsistency of Trace Norm Minimization.J. Mach. Learn. Res.9June 2008, 10191048URL: http://dl.acm.org/citation.cfm?id=1390681.1390716back to text
 48 articleConsistency of the Group Lasso and Multiple Kernel Learning.J. Mach. Learn. Res.9June 2008, 11791225URL: http://dl.acm.org/citation.cfm?id=1390681.1390721back to text
 49 miscApplications of weak transport theory.2020back to text
 50 unpublishedA spatial Pareto exchange economy problem.December 2021, working paper or preprintHALback to text
 51 articleA mean field game model for the evolution of cities.Journal of Dynamics and Games2021HALback to text
 52 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.aaback to textback to text
 53 articleModelindependent bounds for option prices mass transport approach.Finance and Stochastics1732013, 477501URL: http://dx.doi.org/10.1007/s0078001302058DOIback to text
 54 articleA computational fluid mechanics solution to the MongeKantorovich mass transfer problem.Numer. Math.8432000, 375393URL: http://dx.doi.org/10.1007/s002110050002DOIback to textback to text
 55 articleWeak existence for the semigeostrophic equations formulated as a coupled MongeAmpère/transport problem.SIAM J. Appl. Math.5851998, 14501461back to textback to text
 56 articleAugmented Lagrangian algorithms for variational problems with divergence constraints.JOTA2015back to text
 57 articleIterative Bregman Projections for Regularized Transportation Problems.SIAM J. Sci. Comp.to appear2015back to textback to textback to text
 58 articlePoint Source Regularization of the Finite Source Reflector Problem.Journal of Computational PhysicsMay 2022HALback to text
 59 articleNumerical solution of the optimal transportation problem using the MongeAmpere equation.Journal of Computational Physics2602014, 107126back to text
 60 articleTwo numerical methods for the elliptic MongeAmpère equation.M2AN Math. Model. Numer. Anal.4442010, 737758back to text
 61 articleConsistent estimation of a population barycenter in the Wasserstein space.Preprint arXiv:1212.25622012back to text
 62 articleAdditive manufacturing scanning paths optimization using shape optimization tools.Struct. Multidiscip. Optim.6162020, 24372466URL: https://doi.org/10.1007/s00158020026143DOIback to text
 63 articleSliced and Radon Wasserstein Barycenters of Measures.Journal of Mathematical Imaging and Vision5112015, 2245URL: http://hal.archivesouvertes.fr/hal00881872/back to text
 64 articleCharacterization of optimal shapes and masses through MongeKantorovich equation.J. Eur. Math. Soc. (JEMS)322001, 139168URL: http://dx.doi.org/10.1007/s100970000027DOIback to text
 65 articleDeterministic guarantees for BurerMonteiro factorizations of smooth semidefinite programs.preprinthttps://arxiv.org/abs/1804.020082018back to text
 66 articleA generalized conditional gradient method for dynamic inverse problems with optimal transport regularization.arXiv preprint arXiv:2012.117062020back to textback to text
 67 articleDécomposition polaire et réarrangement monotone des champs de vecteurs.C. R. Acad. Sci. Paris Sér. I Math.305191987, 805808back to text
 68 articleReconstruction of the early universe as a convex optimization problem.Mon. Not. Roy. Astron. Soc.3462003, 501524URL: http://arxiv.org/pdf/astroph/0304214.pdfback to text
 69 articleGeneralized solutions and hydrostatic approximation of the Euler equations.Phys. D23714172008, 19821988URL: http://dx.doi.org/10.1016/j.physd.2008.02.026DOIback to text
 70 articlePolar factorization and monotone rearrangement of vectorvalued functions.Comm. Pure Appl. Math.4441991, 375417URL: http://dx.doi.org/10.1002/cpa.3160440402DOIback to text
 71 articleA guide to the TV zoo.LevelSet and PDEbased Reconstruction Methods, Springer2013back to text
 72 bookOptimal Urban Networks via Mass Transportation.1961Lecture Notes in MathematicsBerlin, HeidelbergSpringer Berlin Heidelberg2009, URL: http://link.springer.com/10.1007/9783540857990DOIback to text
 73 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/226/03233DOIback to text
 74 articleThe regularity of mappings with a convex potential.J. Amer. Math. Soc.511992, 99104URL: http://dx.doi.org/10.2307/2152752DOIback to text
 75 articleComputational Analysis of LDDMM for Brain Mapping.Frontiers in Neuroscience72013back to text
 76 articleSuperResolution from Noisy Data.Journal of Fourier Analysis and Applications1962013, 12291254back to text
 77 articleTowards a Mathematical Theory of SuperResolution.Communications on Pure and Applied Mathematics6762014, 906956back to text
 78 articleAn Introduction to Compressive Sensing.IEEE Signal Processing Magazine2522008, 2130back to text
 79 unpublishedSISTA: Learning Optimal Transport Costs under Sparsity Constraints.October 2020, working paper or preprintHALback to text
 80 articleOn the total variation Wasserstein gradient flow and the TVJKO scheme.ESAIM: Control, Optimisation and Calculus of Variations2019HALback to text
 81 articleTowards Offthegrid Algorithms for Total Variation Regularized Inverse Problems.Journal of Mathematical Imaging and VisionJuly 2022HALDOIback to text
 82 articleKinetic models for chemotaxis and their driftdiffusion limits.Monatsh. Math.142122004, 123141URL: http://dx.doi.org/10.1007/s0060500402347DOIback to text

83
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/2018027DOIback to text  84 articleCrouzeixRaviart approximation of the total variation on simplicial meshes.J. Math. Imaging Vision62672020, 872899URL: https://doi.org/10.1007/s10851019009393DOIback to text
 85 articleLearning consistent discretizations of the total variation.SIAM J. Imaging Sci.1422021, 778813URL: https://doi.org/10.1137/20M1377199DOIback to text

86
articleThe
$$ Wasserstein Distance: Local Solutions and Existence of Optimal Transport Maps.401January 2008, 120URL: https://epubs.siam.org/doi/10.1137/07069938XDOIback to text  87 articleAtomic decomposition by basis pursuit.SIAM journal on scientific computing2011999, 3361back to text
 88 inproceedingsFaster Wasserstein Distance Estimation with the Sinkhorn Divergence.Neural Information Processing SystemsAdvances in Neural Information Processing SystemsVancouver, CanadaDecember 2020HALback to text
 89 articleDiscrete total variation: new definition and minimization.SIAM J. Imaging Sci.1032017, 12581290URL: https://doi.org/10.1137/16M1075247back to text
 90 articleDensity Functional Theory and Optimal Transportation with Coulomb Cost.Communications on Pure and Applied Mathematics6642013, 548599URL: http://dx.doi.org/10.1002/cpa.21437DOIback to text
 91 bookA Mathematical Theory of LargeScale Atmosphere/Ocean Flow.Imperial College Press2006, URL: https://books.google.fr/books?id=JxBqDQAAQBAJback to text
 92 articleThe semigeostrophic equations discretized in reference and dual variables.Arch. Ration. Mech. Anal.18522007, 341363URL: http://dx.doi.org/10.1007/s0020500600406DOIback to text
 93 articleGeneralised Lagrangian solutions for atmospheric and oceanic flows.SIAM J. Appl. Math.5111991, 2031back to text
 94 inproceedingsSinkhorn Distances: Lightspeed Computation of Optimal Transport.Proc. NIPS2013, 22922300back to textback to text

95
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/S0022123619300643DOIback to text  96 articleNumerical methods for fully nonlinear elliptic equations of the MongeAmpère type.Comput. Methods Appl. Mech. Engrg.19513162006, 13441386back to text
 97 articleExact Support Recovery for Sparse Spikes Deconvolution.Foundations of Computational Mathematics2014, 141URL: http://dx.doi.org/10.1007/s1020801492286DOIback to text
 98 articleSupport detection in superresolution.Proc. Proceedings of the 10th International Conference on Sampling Theory and Applications2013, 145148back to text
 99 unpublishedInterpolating between Optimal Transport and MMD using Sinkhorn Divergences.October 2018, working paper or preprintHALback to text
 100 articleBreaking the Curse of Dimension in MultiMarginal Kantorovich Optimal Transport on Finite State Spaces.SIAM Journal on Mathematical Analysis5042018, 39964019URL: https://doi.org/10.1137/17M1150025DOIback to text
 101 articleMongeAmpèreKantorovitch (MAK) reconstruction of the eary universe.Nature4172602002back to text
 102 articleA stochastic control approach to NoArbitrage bounds given marginals, with an application to Loopback options.submitted to Annals of Applied Probability2011back to text
 103 articleThe geometry of optimal transportation.Acta Math.17721996, 113161URL: http://dx.doi.org/10.1007/BF02392620DOIback to text
 104 articleGaspard Monge, Le mémoire sur les déblais et les remblais.Image des mathématiques, CNRS2012, URL: http://images.math.cnrs.fr/GaspardMonge,1094.htmlback to text

105
articlePath Dependent Optimal Transport and Model Calibration on Exotic Derivatives.SSRN Electron.~J.Available at
doi:10.2139/ssrn.3302384 01 2018DOIback to text  106 articleNotes on a PDE System for Biological Network Formation.138June 2016, 127155URL: https://www.sciencedirect.com/science/article/pii/S0362546X15004344DOIback to text
 107 articleSoliton dynamics in computational anatomy.NeuroImage232004, S170S178back to text
 108 incollectionThe mathematical theory of frontogenesis.Annual review of fluid mechanics, Vol. 14Palo Alto, CAAnnual Reviews1982, 131151back to text
 109 articleThe backandforth method for Wasserstein gradient flows.ESAIM: Control, Optimisation and Calculus of Variations272021, 28back to text
 110 articleA fast approach to optimal transport: The backandforth method.Numer. Math.1462020, 513544URL: https://doi.org/10.1007/s00211020011548DOIback to text
 111 articleOn explosions of solutions to a system of partial differential equations modelling chemotaxis.Trans. Amer. Math. Soc.32921992, 819824URL: http://dx.doi.org/10.2307/2153966DOIback to text
 112 articleThe variational formulation of the FokkerPlanck equation.SIAM J. Math. Anal.2911998, 117back to text
 113 articleOn the translocation of masses.C. R. (Doklady) Acad. Sci. URSS (N.S.)371942, 199201back to text
 114 articleMinimizing within convex bodies using a convex hull method.SIAM Journal on Optimization162January 2005, 368379HALback to text
 115 articleMean field games.Jpn. J. Math.212007, 229260URL: http://dx.doi.org/10.1007/s1153700706578DOIback to text
 116 articleOptimal Transport Approximation of 2Dimensional Measures.SIAM Journal on Imaging Sciences122January 2019, 762787URL: https://epubs.siam.org/doi/10.1137/18M1193736DOIback to text
 117 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.2014.34.1533DOIback to text
 118 articleActive sets, nonsmoothness, and sensitivity.SIAM Journal on Optimization1332003, 702725back to text
 119 articleOptimal transportation meshfree approximation schemes for Fluid and plastic Flows.Int. J. Numer. Meth. Engng 83:1541579832010, 15411579back to text
 120 articleA fully nonlinear version of the incompressible Euler equations: the semigeostrophic system.SIAM J. Math. Anal.3832006, 795823 (electronic)back to text
 121 articleNumerical solution of the MongeAmpére equation by a Newton's algorithm.C. R. Math. Acad. Sci. Paris34042005, 319324back to text
 122 bookA wavelet tour of signal processing.Elsevier/Academic Press, Amsterdam2009back to text
 123 articleA macroscopic crowd motion model of gradient flow type.Math. Models Methods Appl. Sci.20102010, 17871821URL: http://dx.doi.org/10.1142/S0218202510004799DOIback to text
 124 articleA multiscale approach to optimal transport.Computer Graphics Forum3052011, 15831592back to text
 125 articleGeodesic Shooting for Computational Anatomy.Journal of Mathematical Imaging and Vision242March 2006, 209228URL: http://dx.doi.org/10.1007/s1085100536240back to text
 126 articleAdaptive, 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, 807853HALback to text
 127 articleA ModicaMortola Approximation for Branched Transport and Applications.Archive for Rational Mechanics and Analysis2011July 2011, 115142URL: http://link.springer.com/10.1007/s0020501104026DOIback to text
 128 articleOptimal transport: From moving soil to samesex marriage.CMS Notes452013, 1415back to text
 129 articleUniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem.SIAM Journal on Mathematical Analysis4362011, 27582775back to text
 130 miscRegularized Optimal Transport is Ground Cost Adversarial.2020back to text
 131 articleA Generalized ForwardBackward Splitting.SIAM Journal on Imaging Sciences632013, 11991226URL: http://hal.archivesouvertes.fr/hal00613637/DOIback to text
 132 articleNonlinear total variation based noise removal algorithms.Physica D: Nonlinear Phenomena6011992, 259268URL: http://dx.doi.org/10.1016/01672789(92)90242Fback to text
 133 articleConvolutional Wasserstein Distances: Efficient Optimal Transportation on Geometric Domains.ACM Transaction on Graphics, Proc. SIGGRAPH'15to appear2015back to textback to text
 134 articleRegression shrinkage and selection via the Lasso.Journal of the Royal Statistical Society. Series B. Methodological5811996, 267288back to textback to text
 135 unpublishedDynamical Programming for offthegrid dynamic Inverse Problems.December 2022, working paper or preprintHALback to text
 136 articleModel Selection with Piecewise Regular Gauges.Information and Inferenceto appear2015, URL: http://hal.archivesouvertes.fr/hal00842603/back to text
 137 bookOptimal transport.338Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]Old and newBerlinSpringerVerlag2009, xxii+973URL: http://dx.doi.org/10.1007/9783540710509DOIback to text
 138 bookTopics in optimal transportation.58Graduate Studies in MathematicsAmerican Mathematical Society, Providence, RI2003, xvi+370back to text
 139 articleOn the design of a reflector antenna. II.Calc. Var. Partial Differential Equations2032004, 329341URL: http://dx.doi.org/10.1007/s0052600302394DOIback to text
 140 articleA continuum mechanical approach to geodesics in shape space.International Journal of Computer Vision9332011, 293318back to text
 141 articleSparse representation for computer vision and pattern recognition.Proceedings of the IEEE9862010, 10311044back to text
 142 phdthesisEntropic Unbalanced Optimal Transport: Application to FullWaveform Inversion and Numerical Illustration.Université de ParisDecember 2021HALback to text