PARMA

PARMA - 2025

2025Activity‌‌ reportProject-TeamPARMA

RNSR:‌ 202424470Y

Research center Inria Saclay Centre at Université‌ Paris-Saclay
In partnership with:CNRS, Université Paris-Saclay
Team‌ name: Particle methods using Monge-Ampère
In collaboration with:‌Laboratoire de mathématiques d'Orsay de l'Université de Paris-Saclay‌ (LMO)

Creation of the Project-Team: 2024 January 01‌

Each year, Inria research teams publish an Activity‌ Report presenting their work and results over the‌ reporting period. These reports follow a common structure,‌ with some optional sections depending on the specific‌ team. They typically begin by outlining the overall‌ objectives and research programme, including the main research‌ themes, goals, and methodological approaches. They also describe‌ the application domains targeted by the team, highlighting‌ the scientific or societal contexts in which their‌ work is situated.

The reports then present the‌ highlights of the year, covering major scientific achievements,‌ software developments, or teaching contributions. When relevant, they‌ include sections on software, platforms, and open data,‌ detailing the tools developed and how they are‌ shared. A substantial part is dedicated to new‌ results, where scientific contributions are described in detail,‌ often with subsections specifying participants and associated keywords.‌

Finally, the Activity Report addresses funding, contracts, partnerships,‌ and collaborations at various levels, from industrial agreements‌ to international cooperations. It also covers dissemination and‌ teaching activities, such as participation in scientific events,‌ outreach, and supervision. The document concludes with a‌ presentation of scientific production, including major publications and‌ those produced during the year.

Keywords

Computer Science‌ and Digital Science

A6.1. Methods in mathematical modeling‌
A6.2. Scientific computing, Numerical Analysis & Optimization
A6.3.‌ Computation-data interaction
A6.5. Mathematical modeling for physical sciences‌
A8.2. Optimization
A8.2.3. Calculus of variations
A8.12. Optimal‌ transport

1‌ Team members, visitors, external collaborators

Research Scientists

Thomas‌ Gallouët [Team leader, INRIA, Researcher‌, HDR]
Yann Brenier [CNRS,‌ Emeritus]
Katharina Eichinger [Inria Saclay,‌ Researcher, from Oct 2025]
Cyril Letrouit‌ [CNRS, Researcher, from Apr 2025‌]
Bruno Levy [INRIA, Senior Researcher‌, HDR]
Andrea Natale [INRIA,‌ Researcher, from Jul 2025]

Faculty Members‌

Bertrand Maury [UNIV PARIS SACLAY, Associate‌ Professor Delegation, until Aug 2025]
Quentin‌ Merigot [UNIV PARIS SACLAY, Professor]‌
Luca Nenna [UNIV PARIS SACLAY, Associate‌ Professor, HDR]

Post-Doctoral Fellows

Alessandro Cosenza‌ [UNIV PARIS SACLAY, Post-Doctoral Fellow,‌ from Nov 2025]
Cyprien Plateau-Holleville [INRIA‌, Post-Doctoral Fellow]
Filippo Quattrocchi [IST‌ AUSTRIA, Post-Doctoral Fellow, from Dec 2025‌]

PhD Students

Elise Bonet Weil [ENPC‌]
Adrien Cances [UNIV PARIS SACLAY]‌
Benjamin Capdeville [ENS PARIS-SACLAY]
William Ford‌ [ECOLE POLY PALAISEAU]
Mattia Garatti [‌UNIV PARIS SACLAY, from Oct 2025]‌
Quentin Giton [UNIV PARIS SACLAY]
Julien‌ Guerin [ENS PARIS-SACLAY]
Erwan Stampfli [UNIV PARIS SACLAY]‌
Louis Tocquec [UNIV‌ PARIS SACLAY]
Christophe‌‌ Vauthier [Université Paris Saclay ]

Technical Staff‌

Sylvain Faure [CNRS‌, Engineer]
Hugo‌‌ Leclerc [CNRS, Engineer]

Administrative Assistant‌

Laetitia Jubely [INRIA‌, from May 2025‌‌]

External Collaborators

Siwan Boufadene [UNIV GUSTAVE‌ EIFFEL, until Sep‌ 2025]
Katharina Eichinger‌‌ [UNIV PARIS SACLAY, until Sep 2025‌]
Anna Korba [‌ENSAE]
Maxime Laborde‌‌ [Université Paris Cité, until Aug 2025‌]

2 Overall objectives‌

The ParMA project focuses‌‌ on the application of OT techniques to the‌ numerical resolution of problems‌ where one wants to‌‌ impose that a point cloud (representing either some‌ data, or the result‌ of a simulation) follows‌‌ a certain probability distribution. This will for instance‌ be done by using‌ the Wasserstein distance as‌‌ a penalization. We will especially focus on the‌ numerical analysis and resolution‌ relying on particle discretization‌‌ (i.e. point clouds). ParMA will be organized around‌ three main objectives:

Numerical‌ analysis of OT and‌‌ solvers (3.1): The numerical analysis of‌ optimal transport (convergence estimates,‌ regularity properties of discrete‌‌ solutions, a posteriori estimates) is still widely open,‌ and needs to be‌ further studied. Applications of‌‌ this theoretical analysis include the development of efficient,‌ robust and large-scale OT‌ solvers, and a better‌‌ mathematical understanding of the numerical schemes that we‌ will develop in the‌ next two objectives.
OT‌‌ for fluid dynamics (3.3): On the‌ one hand it is‌ known since the work‌‌ of Brenier that optimal transport can be used‌ to impose incompressibility in‌ fluid dynamics 46.‌‌ This is illustrated in Figure 1. On‌ the other hand many‌ dissipative PDE/system of PDE‌‌ can be interpreted as gradient flows in the‌ Wasserstein space. Our aim‌ within the ParMA team‌‌ is to use these ideas to design, analyze‌ and implement new particle‌ schemes for more general‌‌ evolution equations from fluid dynamics, biology and social‌ sciences. Benefits of this‌ approach include interface tracking‌‌ and adaptive discretization; other benefits are discussed in‌ Section 3.2.
OT‌ for inverse problems (‌‌3.3): Wasserstein distances have been used as‌ a data fidelity term‌ in various problems involving‌‌ inverse problems where the signal generated by the‌ model can be interpreted‌ as a distribution of‌‌ mass. For instance it is the case in‌ seismic imaging 73,‌ where it has been‌‌ observed that OT distances tend to convexify the‌ energy landscape, enlarge the‌ basin of attraction of‌‌ the optimal solution and make it possible to‌ estimate the underground from‌ a very crude guess‌‌ (in comparison to what is obtained with a‌ $L^{2}$ distance). Understanding‌ both qualitatively and quantitatively‌‌ this "convexification" property of OT is a long‌ term goal of ParMA‌. We also plan‌‌ to use Wasserstein distances as data fidelity terms‌ in other applicative contexts.‌
OT for Social Sciences‌‌ Some prelimary attempts have‌ been made to model the repartition of inhabitants‌ in an urban area as a constrained optimization‌ problem for a measure. In collaboration with geographers,‌ we plan to extend those ideas to better‌ account for social tendencies.

2.1 Scientific and application‌ context

2.1.1 Optimal transport

The problem of optimal‌ transport (OT), introduced by Gaspard Monge in 1871,‌ was motivated by engineering applications: find the most‌ economical way to transport a certain amount of‌ sand from a quarry to a construction site.‌ The modern theory of optimal transport has been‌ initiated by Leonid Kantorovich in the 1940s, via‌ a convex relaxation of Monge's problem. The Monge-Kantorovich‌ theory can be used to define notions of‌ distance between probability measures over the Euclidean space,‌ often called Wasserstein distances. Unlike other notions —‌ such as total variations, relative entropies, etc. —‌ the Wasserstein distances fully encode the geometry of‌ the underlying space, making them ideal to compare‌ probability measures in a geometric context. This approach‌ has been used and revisited by many authors‌ from the 1980s, who connected it to various‌ domains of mathematics, physics (fluid mechanics, quantum chemistry),‌ economics (principal-agent problem, finance), mathematical learning (loss functions,‌ statistics on point clouds), etc. Optimal transport is‌ now more than ever a vivid topic, with‌ more than 9200 articles published in 2021 containing‌ the words "optimal transport" (Source: Google Scholar).

2.1.2‌ Computational optimal transport

In the 2000s, the theory‌ of optimal transport was already mature and applied‌ within mathematics, but also in theoretical physics or‌ in economy. However, numerical applications were essentially limited‌ to 1-dimensional problems because of prohibitive cost of‌ existing algorithms, whose complexity is more than quadratic‌ in the size of the data in dimension‌ larger than two. Numerous numerical methods have been‌ introduced to date, such as the Benamou-Brenier algorithm‌ 40, but two in particular have changed‌ this state of affairs:

Entropic regularization of the‌ optimal transport problem, which can be solved by‌ Sinkhorn-Knopp's algorithm. This idea has been introduced in‌ optimal transport by Alfred Galichon 57 (in Economy)‌ and Marco Cuturi 54 (in Machine Learning).
Semi-discrete‌ formulation of optimal transport, introduced by Cullen in‌ 1984 53, which involves solving asymmetric problems‌ between a probability density $ρ$ and a probability‌ measure with a finite support $μ$ . The‌ first effective implementation comes from the work of‌ Q. Mérigot 69, using computational geometry techniques‌ and a multiscale strategy.

Thanks to these two‌ approaches, which have quite distinct domains of applicability,‌ solving an optimal transport problem in small dimension‌ with reasonable sized data (e.g. $N = {10‌}^{6}$ in $d = 2, 3$ )‌ and for a "geometric" cost function has become‌ a formality. Optimal transport can therefore be used,‌ with the limitations set out above, as a‌ brick within a more complex system. Applications requiring‌ the resolution of optimal transport problems within an‌ iterative algorithm include for example the resolution of inverse problems, machine learning,‌ or the construction of‌ numerical schemes for some‌‌ evolution equations.

3 Research program

3.1‌‌ Optimal transport: models, numerical analysis and solvers

3.1.1‌ Models

Participants: Yann Brenier‌, Thomas Gallouët,‌‌ Quentin Merigot, Bertrand Maury, Luca Nenna‌.

The team is‌ interested in extending OT‌‌ to multiple applications. The main ones are as‌ follows:

Unbalanced and partial‌ OT. These metrics aim‌‌ to compare measures with different masses 64.‌ They can be very‌ pertinent for example when‌‌ dealing with incomplete datum or PDE for which‌ involve variation of mass‌ (e.g. with a reaction‌‌ term). These methods share a lot of property‌ with classical OT 58‌. A particular case‌‌ of unbalanced optimal transport is the Moreau-Yosida regularization‌ of the Wasserstein distance,‌ which allows to define‌‌ a notion of $ϵ$ -entropy to a point‌ cloud, and which plays‌ an important role in‌‌ numerical schemes for fluids dynamics (Section 3.2).‌
Linearized OT is a‌ way to lift up‌‌ the Wasserstein space in a classical weighted ${L‌}^{2}$ Hilbert space, allowing‌ one to apply the‌‌ "linear" statistical toolbox to families of probability measures.‌ Q. Mérigot and A.‌ Delalande recently proved that‌‌ this injection is bi-Hölder 70, thus showing‌ a coarse relationship between‌ this metric and the‌‌ original OT metric. This work is specialised in‌ the quadratic case, and‌ calls for natural extensions‌‌ to other OT models.
Unequal dimensional Optimal Transport.‌ Another natural extension of‌ classical OT arises in‌‌ Economy in the case in which one wants‌ to compare two populations‌ having a different amount‌‌ of characteristics. This translates into an optimal transport‌ problem between measures defined‌ on space with different‌‌ dimension. This kind of problem has been recently‌ studied in 51,‌ 68 and further developed‌‌ in the framework of variational problems involving an‌ OT term (for instance,‌ the Cournot-Nash equilibria) in‌‌ 75. In particular we will focus on‌ developing new metrics on‌ the space of probability‌‌ measures by means of unequal dimensional transport and‌ study the link with‌ a special instance of‌‌ the linearised optimal transport LOT.
Gromov-Wasserstein. This variant‌ of OT has been‌ introduced in order to‌‌ take into count inner‌ metric of the data structure. We aim to‌ leverage the properties of this metric for the‌ modelisation of geographic behaviour, for example we can‌ try to use them in order to classify‌ different cities. A long term objective here is‌ to provide tools to help decision making in‌ urban planning. This research is a part of‌ the PEPR project "Maths vives" and done in‌ collaboration with Geographers.

3.1.2 Numerical analysis

Participants: Thomas‌ Gallouët, Quentin Merigot, Luca Nenna.‌

Convergence speed.

The numerical analysis of optimal transport‌ is still in its infancy. The convergence of‌ the solutions of the discretized optimal transport problem‌ towards solutions of the continuous problem is usually‌ obtained by a compactness argument, which is therefore‌ not quantitative. For many applications, it is necessary‌ to have a more explicit control of the‌ distance between the optimal transport maps for different‌ data (e.g. discrete and continuous), depending on the‌ distance between the data. This problem is difficult‌ even if we restrict ourselves to optimal transport‌ for the quadratic cost (mainly because the Monge-Ampère‌ equations appearing in transport optimal is not strongly‌ elliptic). Recent results have been obtained by members‌ of the team, in the quadratic case 70‌, but many questions are still open: extension‌ to other costs, improved exponents, a posteriori analysis,‌ etc. We also expect applications to the design‌ and analysis of fast multi-scale algorithms for optimal‌ transport.

Discrete regularity

For many of our targeted‌ applications, it is important to understand quantitatively the‌ regularity of solution of semi-discrete optimal transport problems.‌ The question is be to find a discrete‌ analogue of the regularity theory for the optimal‌ transport (continuity method, or $𝒞^{1}$ regularity à‌ la Caffarelli), which would imply for instance that‌ when the point cloud is close to a‌ probability density, then the Laguerre cells are not‌ too anisotropic. Should this be true, it would‌ also have consequences in the analysis of the‌ particle schemes for fluid dynamics proposed in §‌3.2, but would also provide a better‌ understanding of the behaviour of Newton's algorithm for‌ semi-discrete OT.

3.1.3 Solvers

Participants: Sylvain Faure,‌ Hugo Leclerc, Bruno Levy, Quentin Merigot‌, Luca Nenna.

Algorithms

Despite the recent‌ progress on the algorithmic aspects of optimal transport,‌ many questions are still not understood: most of‌ the existing complexity analysis are in a worst-case‌ setting, and thus do not explain at all‌ the practical good behaviour of OT solvers. More‌ generally, we expect that the progress in the‌ numerical analysis of OT will lead to the‌ design of more efficient solvers, using multi-scale and‌ preconditioning strategies (in semi-discrete OT) or $ϵ$ -scaling‌ techniques (in the context of entropy-regularized OT).

Implementation‌ of a semi-discrete solver

Technical work has been‌ done to be able to generate power diagrams‌ and associated integrals in fast and memory efficient‌ way. Power diagrams are at the center of the calculations to obtain‌ Kantorovich potentials for optimal‌ transport in semi-discrete settings.‌‌ Among other things, the aim is to be‌ able to use the‌ capabilities of GPUs and‌‌ SIMD instructions on CPUs (parallel instructions), and then‌ to be able to‌ scale up to cluster-based‌‌ calculations. The first results are very encouraging since‌ some cases show significantly‌ lower computation times, naturally‌‌ exploiting the parallelism of contemporary architectures, and above‌ all, for an almost‌ negligible memory occupation (compared‌‌ to what was previously required).

These properties allow‌ us to seriously consider‌ the development of new‌‌ classes of iterative algorithms adapted to different semi-discrete‌ OT problems, which would‌ be very fast and‌‌ inexpensive in terms of memory occupation. Some first‌ proposals to improve the‌ robustness or the convergence‌‌ speed (adapted preconditioners) have shown promising results. Moreover,‌ extensions to larger numbers‌ of dimensions have started‌‌ and will be followed up in the context‌ of the project. All‌ generic developments are deposited‌‌ under open source licenses, associated with rigorous documentation‌ and testing procedures.

Milestones‌

black 4 years Objective:‌‌ An efficient and user friendly Semi-Discrete OT solver‌ is yet to exist.‌ This is the objective‌‌ of Sdot (solver written in C++) and PySdot‌ (interface with Python), developed‌ by H. Leclerc. It‌‌ will have to be accessible for non-specialists and‌ non mathematicians users and‌ able to solve of‌‌ wide range of problems. This software is open‌ source, and is currently‌ used by some colleagues‌‌ but the API is not stable yet. To‌ develop our strategy on‌ these questions we hope‌‌ to benefit from the experience and expertise of‌ Inria.

Task (GEOT-NP) in‌ the context of geography‌‌ (urban planning), we propose to develop conceptual and‌ numerical tools to analyse‌ and compare different areas‌‌ represented as measured metric spaces (population densities and‌ public transportation network). We‌ aim in particular at‌‌ developing an adapted metric, in the spirit of‌ Gromov Wassertsin distance, which‌ would incorporate information on‌‌ the paths which are really used (based on‌ origin destination matrices), which‌ is not the case‌‌ for the existing GW distance.
Task (API-V1) improve‌ the Python API to‌ make it more pythonic,‌‌ and to allow an easy combination with existing‌ machine learning libraries (Scikit-Learn,‌ PyTorch, etc.), especially allowing‌‌ seamless use of automatic differentiation.
Task (API-D) Write‌ a detailed documentation for‌ Task (API-V1).
Task‌‌ (API-V2) Create bindings/interface for other languages (julia, R).‌

8-12 years Objective :‌ black

Task (GEOT-V) Valorization‌‌ of the interactions developed in Task (GEOT-NP) .‌
Task (API-VN) Develop a‌ very fast parallel SD-solver‌‌ for high resolution problems in 2D/3D, exploiting the‌ capabilities of modern hardware‌ (GPU, SIMD).

3.2 OT-based‌‌ particle methods for fluids mechanics

Participants: Yann Brenier‌, Sylvain Faure,‌ Thomas Gallouët, Bruno‌‌ Levy, Hugo Leclerc, Bertrand Maury,‌ Quentin Merigot.

A‌ number of evolution problems,‌‌ appearing in physics (porous media, incompressible Euler, isentropic‌ Euler), in biology (chemotaxis)‌ or human sciences (crowd‌‌ movements) can formally be‌ described as the evolution of a large system‌ of particles. We consider the lagrangian variant of‌ these problems, where one is looking for the‌ evolution of a displacement field by a dissipative‌ or a conservative evolution:

Dissipative equations are associated‌ with the notion of gradient flows in the‌ Wasserstein space. This idea was originally proposed by‌ Otto in the late 90s 63, 76‌ and systematically studied in the seminal book by‌ Ambrosio-Gigli-Savare 39. Many equations enter this framework,‌ modeling porous media 77, crowd movements, chemiotaxy,‌ etc. The gradient flow formulation is used for‌ several purpose: analysis (uniqueness, convergence to equilibrium), modeling‌ (rate of dissipation of some energies) and construction‌ of numerical schemes.
Conservative equations are associated with‌ geometric methods for hydrodynamics, starting form Arnold’s interpretation‌ of the incompressible inviscid Euler's equations as geodesics‌ in the space of volume-preserving diffeomorphisms. This has‌ been revisited, in the context of optimal transport,‌ by Brenier: by his polar factorization theorem, the‌ computation of the distance between a map from‌ the domain to itself and the space of‌ volume-preserving diffeomorphisms is equivalent to an OT problem.‌

In combination with semi-discrete optimal transport, these ideas‌ lead to Lagrangian numerical schemes for Euler's equation‌ for incompressible and compressible fluids, and for many‌ gradient flows with a suitable energy functional 59‌, 60. Many other evolution problems can‌ be formulated in this framework, perhaps with some‌ adaptation (fluid-structure interactions, problems involving surface tension, and/or‌ interaction between particles, etc.). Our goal within the‌ ParMA project is to derive systematically particle discretizations‌ of these forms, using optimal transport or variants‌ thereof (e.g. partial or unbalanced optimal transport). In‌ particular, we will adapt this strategy to the‌ crowd motion models proposed by B. Maury 67‌ and to pressureless Euler equation 66, which‌ both rely on partial optimal transport. This approach‌ offers many advantages over existing numerical methods:

Semi-discrete‌ OT automatically associates a moving mesh to the‌ particle discretization, allowing one to track precisely the‌ interfaces between particles, e.g. between two phases of‌ a fluid (see Fig. 1).
These discretizations‌ respect the structure of the equations, which is‌ therefore recovered at the discrete level some properties‌ of the equation, such as mass preservation and‌ positivity, or convergence to the equilibrium estimates coming‌ from the gradient flow structure.
These particle approach‌ allows us to simulate equations with an explosive‌ behaviour, which would be difficult to describe finely‌ using fixed-mesh methods. The OT approach makes it‌ easy to refine/coarsen the solution in a principled‌ way, simply by looking at the shape of‌ Laguerre cells.

Our approach bears some similarity with‌ smoothed particle hydrodynamics (SPH) discretizations, where the interaction‌ forces amongst the particles are computed by reconstructing‌ the fluid density through convolution with a fixed‌ kernel, and which have been widely used in‌ the context of the discretization of fluid models.‌ However, these methods are prone to unstability and are difficult to analyze‌ from a mathematical viewpoints:‌ very few convergence results‌‌ exist, and recent works show that the choice‌ of the kernel is‌ both non-trivial and crucial‌‌ to obtain convergence.

Milestones

black 4 years Objective:‌

Task (INS-P) Incompressible Navier-Stokes‌ (INS). We already know‌‌ how to implement a SD-OT scheme for INS.‌ The proof is yet‌ to be done. In‌‌ order to adapt our proof done for Incompressible‌ Euler one need to‌ deal with the dissipation‌‌ term. This term is not easy to handle‌ our strategy will be‌ to desymetrize the remaining‌‌ non quadratic term.
Task (INSF-PN) The following step‌ will be to add‌ more physics to this‌‌ model, for example surface tension.
Task (ORD2-PN) We‌ aim to develop a‌ space-time order 2 numerical‌‌ scheme for a wide class of gradient flow.‌ This will be done‌ thanks to a metric‌‌ formulation of BDF-2 scheme and appropriate notion of‌ Wasserstein extrapolation studied in‌ Task (Ext-PN).
Task‌‌ (EDGF-PN) We study the gradient flow where the‌ interaction energy is given‌ by the distance and‌‌ especially the long time beahviour of this motion.‌ This problem is non-convex‌ but seems to converge‌‌ to a steady state anyway. Here we want‌ to leverage some harmonic‌ analysis properties. This task‌‌ is the subject of the Phd of Siwan‌ Boufadene.
Task Muskat-P In‌ is Phd thesis Erwan‌‌ Stämpfli is studying the Muskat problem. In particular‌ using modulatd energy methods‌ he proves the convergence‌‌ of multiphase flows towards the Muskat problem in‌ dimension 1.

8-12 years‌ Objective : black

Task‌‌ (SG-PN) The semi-geostrophic model is related to optimal‌ transport. SD models are‌ well adapted in order‌‌ to approximate its solutions. Proving the convergence of‌ the scheme is more‌ challenging especially in the‌‌ 2D approximation. A first step in this direction‌ is the Task (Quant-P)‌ describe below.
Task (Inter-PN)‌‌ Another driving direction is to add interactions term‌ is these model. The‌ relative entropy method is‌‌ well adapted for controlling these term and works‌ in the mean field‌ limit community.

3.3 OT‌‌ for inverse problems

3.3.1 OT as a data-fidelity‌ term

Participants: Thomas Gallouët‌, Quentin Merigot,‌‌ Sylvain Faure.

In this context optimal transport‌ is used as a‌ way to evaluate the‌‌ distance from a measure to given data. Let‌ us mention two class‌ of problem following this‌‌ approach.

Constraint minimisation problem. The first one is‌ given by the minimisation‌ of a functional under‌‌ some fidelity data attachment given by an OT‌ problem. The construction of‌ numerical approximations of Brenier's‌‌ generalized solutions for incompressible Euler, done by Q.‌ Mérigot and J. M.‌ Mirebeau 71, is‌‌ a typical example of such problems where the‌ kinetic energy of a‌ large number of trajectories‌‌ are minimized under incompressiblity constraints enforced with OT.‌ Changing the minimization of‌ the kinetic energy of‌‌ the velocity field by the square of the‌ acceleration leads to the‌ definition of Wasserstein splines‌‌ as introduced By J.D.‌ Benamou, T. Gallouët and F.X. Vialard 44,‌ providing a way to construct smooth interpolations between‌ probability measures. We will pursue the work in‌ these directions. This type of approaches is also‌ popular in imaging, with contributions from K. Bredies,‌ B. Schmitzer, H. Lavenant, etc.
Parametric minimization. We‌ are interested in another class of problem where‌ one wants to minimize the Wasserstein distance between‌ the data ( $ρ$ ) and a measure‌ ( $μ_{θ})$ obtained for example either‌ through a parametric model or by a more‌ complex simulation. The problems can be written as‌ follows, where $θ$ lives in some parameter space:‌

$min_{θ} W_{p}^{p} (μ_{θ‌}, ρ) .$ 1

A typical problem‌ falling into this category is the Wasserstein generative‌ adversarial networks (WGAN), which seeks to construct a‌ model able to generate data statistically compatible with‌ a dataset We have several ongoing collaborations around‌ problems of the form (1). First,‌ with astrophysicists from the Observatoire de Paris (R.‌ Mohayaee), around the early universe reconstruction problem 47‌ and now with more general problems which can‌ be put under the form (1).‌ Another ongoing collaboration on this type of problem‌ is the reconstruction of a model of the‌ underground using full waveform inversion techniques, where optimal‌ transport has already proven to be a valuable‌ tool to avoid cycle skipping 72. Finally,‌ non-imaging optics problems are solved using the formulation‌ (1), and is the object of‌ a maturation project with the Linksium SATT (ANIDOLIX‌ project, with J.B. Keck, Q. Mérigot and B.‌ Thibert).

One of the main difficulty of some‌ of these problems is that their Eulerian formulation,‌ while convex can be very high dimensional. The‌ particle (or Lagrangian) discretizations are lower dimensional but‌ non-convex. However, we hope to leverage some traces‌ of convexity properties present at the Eulerian level‌ in order to prove quantitative estimates on the‌ local minimizers as done in 70.

3.3.2‌ Multi-marginal OT

Participants: Quentin Merigot, Luca Nenna‌.

Multi-marginal optimal transport is a challenging variant‌ of OT, where one seeks a coupling between‌ more than two probability measures minimizing some cost‌ functionals. These kind of problems arise naturally in‌ physics such as the computation of electronic density‌ in quantum chemistry 49, 52 (Density Functional‌ Theory) or in fluid dynamics, Brenier’s generalized solutions‌ of incompressible Euler equations 48, as well‌ as computation of barycenters in the Wasserstein space‌ 34 (with its many applications to machine learning‌ and imaging analysis), and other applications in economics‌ (matching for teams 50) and risk estimation,‌ as we detail below. Understanding the structure, regularity‌ and sparsity properties (namely the existence of Monge‌ minimizers) of optimal plans for multi-marginal transport problems‌ is a very active and challenging area of‌ research. Fast numerical solvers yet are still to‌ be found to address these typically very high-dimensional problems. For instance, a‌ crude discretization of each‌ of a multimarginal OT‌‌ problems with 6 marginals and using 100 Dirac‌ masses per marginal would‌ yield a convex optimization‌‌ problem with $10^{12}$ unknowns! This makes this‌ convex formulation of the‌ problem practically intractable.

Application:‌‌ Risk estimation

We want to study the modeling‌ of risk in an‌ industrial setting, where the‌‌ distribution of several risk factors is known but‌ the joint probability distribution‌ is unknown. For safety‌‌ reasons, one sometimes needs to have a pessimistic‌ estimation of the risk,‌ i.e. finding a coupling‌‌ between the distribution of risk factors that maximize‌ a certain measure of‌ risk. An example of‌‌ such a problems for evaluating the risk of‌ an industrial facility close‌ to a river and‌‌ protected by a dyke can be found in‌ 62: one knows‌ how to evaluate a‌‌ risk (such as the level of water in‌ the river, compared to‌ the height of the‌‌ dyke) depending on a few variables (e.g. river‌ width, maximal annual flowrate,‌ etc.). What is unknown‌‌ however, is the coupling between these variables. Natural‌ questions in the context‌ of risk estimation are‌‌ the following: (1) What is the coupling (i.e.‌ the correlation) between the‌ variables that yields the‌‌ worst-case scenario, e.g. the highest mean risk ?‌ (2) Which coupling gives‌ the highest mean risk‌‌ among the e.g. 1% riskiest events ? Answering‌ these questions leads to‌ an high dimensional multi-marginal‌‌ (or partial multi-marginal) optimal transport:

min_{γ} {\int​​​‌}_{ℝ^{d_{1}} \times﻿﻿﻿‌ \dots \times ℝ^{{d﻿‌​‌}_{ℓ}}} c ({x﻿​​﻿}_{1}, \dots,​​​‌ x_{ℓ}) d﻿﻿﻿‌ γ (x_{1﻿‌​‌}, \dots, {x﻿​​﻿}_{ℓ}) .

2‌

In the above problem,‌ $c$ is a cost‌‌ function, and the minimum is taken over $γ‌ \in ℳ^{+} (‌ ℝ^{d_{1}} \times‌‌ \dots \times ℝ^{{d}_{ℓ}})$ a non-negative‌ measures of mass $α‌$ over $ℝ^{d_{1‌‌}} \times \dots \times {ℝ}^{d_{ℓ}}$ , whose‌ $i$ th marginal ${Π‌}_{i #} γ$ (that‌‌ is the projection of $γ$ on $ℝ^{{d‌}_{i}}$ ) is upper‌ bounded by a prescribed‌‌ probability measure $μ_{i}$ on $ℝ^{d_{i‌}}$ . The case $α‌ = 1$ corresponds to‌‌ the multi-marginal optimal transport problem.

Application: density functional‌ theory

Another interesting application‌ of multi-marginal transport arises‌‌ quite naturally in the framework of Density Functional‌ Theory: it has‌ been recently realized by‌‌ mathematicians 52, 49 that the central problem‌ of finding the lowest‌ Coulomb energy of $N‌‌$ -particle probabilities at given first marginal (a.k.a. charge‌ density) $ρ$ is indeed‌ multi-marginal optimal transport problem‌‌ similar to (2) where the cost‌ is now the Coulomb‌ potential, $μ_{i} =‌‌ ρ$ for all $i$ and $α = 1‌$ .

Objectives:

The main‌ objective on the theoretical‌‌ side will be then to characterize the dimension‌ of the support of‌ the solution of (‌‌2) in general‌ context. This information could allow to further improve‌ some existing numerical methods such as the Sinkhorn‌ algorithm based on the entropic regularization of Optimal‌ Transport 41, 42, 43 and the‌ ones arising from the approximation of optimal transport‌ problems with marginal moments constraints 38, 37‌.

Milestones

black 4 years Objective:

Task (Ext-PN)‌ A task falling in the category is the‌ definition of a Wasserstein geodesic extrapolation. This is‌ not obvious due to the presence of possible‌ shocks appearing while extending a Wasserstein geodesic. We‌ will propose different notions of extrapolation and numerical‌ scheme in order to compute them. The SD‌ approach seems particularly promising to tackle the metric‌ extrapolation. This discretization leads to a non-convex approximation‌ of a convex problem.
Task (Quant-P) In this‌ task we investigate the convergence of the gradient‌ flow for the quantization problem. We use here‌ a relative entropy method and think of this‌ task as a toy model for more intricate‌ dynamics such as Tasks (Quant-P) and (SG-PN).‌
Task (BQuant-P) This task is similar as (Quant-P)‌ but this time we aim to quantize the‌ barycenter of a fixed number of measures. This‌ generalization is not straightforward since the dynamic of‌ the energy is less well understood.
Task (Risk-Pa)‌ Characterize the dimension of the support of the‌ solution of MMOT in a general context. We‌ plan to rely on and to extend the‌ work of B. Pass 78, which gives‌ upper bounds on this dimension under technical assumptions‌ on the cost.
(Risk-Pb) Study the behaviour of‌ a simple particle discretization of the problem, where‌ the solution is a sum of a finite‌ number of Dirac masses whose positions are free‌ to move. The constraints could be enforced via‌ penalization, using optimal transport data attachment terms. This‌ discretization makes the discrete problem non-convex, but one‌ could hope to study the convergence of global‌ minimizers. Notice that a coarser discretization, thanks to‌ a better understanding of the structure of the‌ solution, will make the numerical method competitive with‌ respect to the actual ones: in the entropic‌ case the dimension of the support of the‌ solution is naturally high dimensional since the regularization‌ effect, meaning that a finer discretization is needed.‌
Task (Risk-N) Implement this algorithm and experiment it‌ numerically on some applications as the risk estimation‌ problem presented in 62. These three Risk‌ tasks are the core of A. Cancès Phd‌ thesis.
Task (MMOT-DFT-N) In quantum chemistry, the MMOT‌ problem models the electron-electron repulsion, but numerical simulations‌ to compute the solution to MMOT can be‌ afforded only for a very small number of‌ electrons/marginals. Here we aim at generalize the numerical‌ methods to the case of grand canonical optimal‌ transport introduced in 55, a useful generalization‌ of MMOT which let us to decompose a‌ molecule in sub-domain (with a less number of‌ elctrons/marginals) and compute the electronic configuration in each of them.

8-12 years‌ Objective : black

Task‌ (MMOT-Prosp) Since multi-marginal optimal‌‌ transport arises in many domains, and especially in‌ machine learning, imaging‌ science and computational quantum‌‌ chemistry, we expect that the theoretical and‌ numerical tools developed during‌ the project will find‌‌ many other applications and help us to further‌ improve interactions with other‌ domain as the already‌‌ existing ones with chemists (e.g. L. Nenna already‌ works with Paola Gori-Giorgi,‌ a renowned theoretical chemist‌‌ based in Amsterdam) or with researchers from EDF‌ working on risk measures.‌

3.4 Optimal transport for‌‌ Social Sciences

Participants: Sylvain Faure, Bertrand Maury‌.

In the context‌ of the Géomaths project‌‌ (PEPR Maths VivES), some research axes are connected‌ to Optimal Transportation. In‌ particular, we aims at‌‌ anylizing (population) density distributions in terms of Wasserstein‌ distance to the uniform‌ density. The knowledge of‌‌ the optimal transport plan will give some information‌ concerning the most predominant‌ scales of the zone,‌‌ and possibly predominant directions. Another research track follows‌ some preliminary steps taken‌ by F. Santambrogio and‌‌ G. Buttazzo. It consists in formulating the question‌ of population settlement in‌ terms of optimal transportation.‌‌ A population represented by a density aims at‌ optimizing its utility by‌ minimizing a functional which‌‌ encodes various tendencies. We shall develop an incremental‌ versiion of this approach,‌ in order to account‌‌ for the fact that a city dos not‌ develop from scratch, but‌ rather by successive phases‌‌ of migration and construction. We also plan to‌ explore the possibility to‌ account for populaitons with‌‌ various tendencies (people are not interchangeable).

3.5 Optimal‌ transport for Cosmology

Participants:‌ Quentin Leclerc, Bruno‌‌ Levy, Quentin Merigot.

Modern Cosmology is‌ still confronted with two‌ fundamental questions:

the rotation‌‌ curves of galaxies imply dark matter, the‌ nature of which is‌ yet to be understood;‌‌
explaining the accelerated expansion of the Universe requires‌ a mysterious dark energy‌.

The situation today‌‌ seems favorable: increasingly accurate acquisition data is available‌ (DESI, Euclid, Gaia ...),‌ and computer power has‌‌ continued to make considerable progress. To gain understanding‌ on the evolution of‌ the Universe throughout its‌‌ history, one can try to reconstruct the movement‌ of all galaxies by‌ solving an inverse problem‌‌ (Peeble's action method). This problem was characterized as‌ an optimal transport problem‌ by Firsch, Matarrese, Sobolevskii‌‌ and Mohayaee, and solved numerically using Bertsekas's algorithm‌ for discrete optimal transport‌ on problems of moderate‌‌ size (hundred thousands to millions Dirac masses).

We‌ develop new semi-discrete optimal‌ transport algorithms, as well‌‌ as computational methods to construct the involved Laguerre‌ diagrams, in order to‌ scale up to sizes‌‌ of $10^{6} - 10^{9}$ Dirac masses.‌ This makes it possible‌ to reconstruct the trajectories‌‌ of galaxies, and enhance a subtle signal in‌ the initial condition (Baryonic‌ Acoustic Oscillations), characteristic of‌‌ the early history of the Universe. This also‌ makes it possible to‌ conduct large scale numerical‌‌ simulations for modified laws‌ of gravity (Brenier-Monge-Ampère 65).

4 Application domains‌

4.1 Geography-Social science

In the context of the‌ Géomaths project (PEPR Maths VivES), some research axes‌ are connected to Optimal Transportation. In particular, we‌ aims at anylizing (population) density distributions in terms‌ of Wasserstein distance to the uniform density. The‌ knowledge of the optimal transport plan will give‌ some information concerning the most predominant scales of‌ the zone, and possibly predominant directions. Another research‌ track follows some preliminary steps taken by F.‌ Santambrogio and G. Buttazzo. It consists in formulating‌ the question of population settlement in terms of‌ optimal transportation. A population represented by a density‌ aims at optimizing its utility by minimizing a‌ functional which encodes various tendencies. We shall develop‌ an incremental versiion of this approach, in order‌ to account for the fact that a city‌ dos not develop from scratch, but rather by‌ successive phases of migration and construction. We also‌ plan to explore the possibility to account for‌ populaitons with various tendencies (people are not interchangeable).‌

4.2 Medicine

Positron emission tomography is a common‌ imaging technique, a medical scintillography technique used in‌ nuclear medicine. A radiopharmaceutical - a radioisotope attached‌ to a drug - is injected into the‌ body as a tracer. When the radiopharmaceutical undergoes‌ beta plus decay, a positron is emitted, and‌ when the positron interacts with an ordinary electron,‌ the two particles annihilate and two gamma rays‌ are emitted in opposite directions. In the future,‌ we plan to study the reconstruction of 3D‌ PET-scan images from raw data (listmod i.e. positron‌ emission events) using an optimal transport approach.

4.3‌ Optics

Anidolic or non-imaging optics is a branch‌ of optics that aims to design devices (lenses‌ or mirrors) that reflect or refract light emitted‌ from a source toward a target, following a‌ prescribed intensity distribution and support geometry. Typically, anidolic‌ optics problems where the source is ideal (point-like‌ or collimated) and the target is in the‌ far field are formulated as optimal transport problems‌ or Monge-Ampère equations. On the other hand, problems‌ where the target is in the near field‌ are formulated as a generated Jacobian equation. Semi-discrete‌ optimal transport solvers are especially well suited for‌ these family of problems.

4.4 Crowd motion

If‌ you have people-counting sensors that measure the flow‌ of people into and out of a location,‌ you can estimate the time people spend there,‌ using an optimal transport approach. Two applications have‌ already been realized: the calculation of waiting times‌ for company restaurants, and the calculation of attendance‌ times for visitors or worshippers at Notre-Dame de‌ Paris.

4.5 Astrophysics

Some inverse problems in astrophysics‌ have a natural connection with optimal transport theory,‌ such as Peeble's action problem, that reconstructs the‌ motion of galaxies back in time. It is‌ stated as an action minimization problem, equivalent to‌ an optimal transport problem (Firsch, Matarrese, Sobolevskii, Mohayaee,‌ Brenier). Efficient solvers for optimal transport open the door to new computational‌ tools, that will make‌ it possible to extract‌‌ knowledge about the history of the Universe from‌ observational data.

5 Social‌ and environmental responsibility

5.1‌‌ Footprint of research activities

The team always prioritizes‌ train travel for trips‌ within Europe, and a‌‌ large majority of its members refuse to travel‌ overseas at all or‌ for short periods, in‌‌ line with the spirit of the Labo 1.5‌ project.

6 Highlights‌ of the year

6.1‌‌ Awards

Bertrand Maury , Chair fondamental senior (Institut‌ Universitaire de France).
Luca‌ Nenna , Chair fondamental‌‌ junior (Institut Universitaire de France).
Cyril Letrouit ,‌ Cours Peccot Collège de‌ France.
Andrea Natale obtained‌‌ an ANR JCJC.
Quentin Merigot is half-time professor‌ at ENS.

7 Latest‌ software developments, platforms, open‌‌ data

Sylvain Faure and Bertrand Maury developed 33‌ for crowd motion simulations.‌

7.1 New platforms

7.1.1‌‌ SDOT

The INRIA team has enabled us to‌ develop a new computing‌ platform for optimal transportation‌‌ in the semi-discrete setting (sdot). This‌ platform is unique in‌ the sense that the‌‌ work involved in making optimal transportation accessible to‌ the widest possible communities‌ is currently almost exclusively‌‌ focused in the discrete-discrete setting (e.g. with packages‌ like POT or Geomloss‌ which are excellent but‌‌ are specialized toward discrete-discrete problems), which is not‌ the case of Sdot.‌

We previously had a‌‌ package named pysdot (pysdot) which served‌ as a test bed‌ for the API and‌‌ the implementation choices. The design of Sdot was‌ an opportunity to set‌ things straight, with a‌‌ much clearer and more flexible API, and to‌ catch some fantastic openings‌ for execution speed and‌‌ memory usage (parallelism, GPUs, etc...).

We started the‌ documentation and the communication‌ processes to ensure the‌‌ widest adoption. This package is already used outside‌ the team and the‌ laboratory, but given the‌‌ possibilities it offers, we hope it will reach‌ a significantly larger number‌ of people

For the‌‌ near future, we would like to add compatibility‌ to other frameworks (like‌ PyTorch, ...) and other‌‌ languages.

7.1.2 CroMoSim

Participants: Sylvain Faure, Bertrand‌ Maury.

The initial‌ aim of cromosim was‌‌ to enable reproducibility of the results present in‌ the book “Crowds in‌ equations: an introduction to‌‌ the microscopic modeling of crowds” published in 2018‌ by Bertrand Maury and‌ Sylvain Faure, and to‌‌ provide a starting point for people wishing to‌ model crowd movements. The‌ aim was to propose‌‌ several mathematical models, as well as common tools‌ enabling the use of‌ complex building plans. The‌‌ software is currently available as a package written‌ in Python, which can‌ be installed via PyPI‌‌ (pip). The complete code is available on Github‌ (cromosim), and‌ several examples of use‌‌ are available. Cromosim enables calculations to be made‌ on "real" building plans,‌ by defining a color‌‌ code characterizing the destinations of individuals, walls or‌ furniture... Several microscopic mathematical‌ models of crowd movements‌‌ can thus be used:‌ social force or granular-inspired models, cellular automata, compartment‌ models. The initial target audience was mainly students‌ and researchers, as all the "ingredients" of the‌ models are accessible and modifiable. The commercial softwares‌ used today, for example by engineering firms, are‌ complex and includes numerous parameters that can influence‌ the calculation of an evacuation time. What's more,‌ they are essentially based on Helbing's microscopic model‌ (social forces) or on cellular automata (individuals evolving‌ in boxes). As a result, there is an‌ interest in a non-black-box open-source software package, optimized‌ for simulating dense crowds in real geometry, with‌ well-documented parameterization and enabling results obtained with several‌ mathematical models to be compared. In the coming‌ years, we hope to build up a community‌ of users by developing a web application around‌ this python module.

7.1.3 PET KinetiX

Participants: Florent‌ Besson, Sylvain Faure.

PET KinetiX is‌ a software package for nuclear physicists and clinicians.‌ It has been partly developed thanks to funding‌ from CNRS Innovation (Prematuration and RISE programs) and‌ is currently benefiting from a BPI BFTLaB grant‌ to help prepare its commercialization. It is currently‌ being tested in four hospital centers. The software's‌ functions and initial results are described in 45‌.

7.1.4 GEOGRAM

Participants: Hugo Leclerc, Bruno‌ Levy, Quentin Merigot.

GEOGRAM is a‌ programming library with geometric algorithms. It has geometry-processing‌ functionalities such as reconstruction, remeshing, parameterization, intersections, constructive‌ solid geometry.

It has low-level functionalities such as‌ Delaunay triangulation in 2D and in 3D, highly‌ efficient parallel Delaunay triangulation in 3D used for‌ cosmology, exact predicates, and efficient numerical solvers.

It‌ has efficient solvers for semi-discrete optimal transport in‌ 2D and in 3D.

8 New results‌

Participants: Anna Korba, Quentin Mérigot, Christophe‌ Vauthier.

In 31 the authors investigate the‌ properties of the Sliced Wasserstein Distance (SW) when‌ employed as an objective functional. The SW metric‌ has gained significant interest in the optimal transport‌ and machine learning literature, due to its ability‌ to capture intricate geometric properties of probability distributions‌ while remaining computationally tractable, making it a valuable‌ tool for various applications, including generative modeling and‌ domain adaptation. Our study aims to provide a‌ rigorous analysis of the critical points arising from‌ the optimization of the SW objective. By computing‌ explicit perturbations, we establish that stable critical points‌ of SW cannot concentrate on segments. This stability‌ analysis is crucial for understanding the behaviour of‌ optimization algorithms for models trained using the SW‌ objective. Furthermore, we investigate the properties of the‌ SW objective, shedding light on the existence and‌ convergence behavior of critical points. We illustrate our‌ theoretical results through numerical experiments.

Participants: Boris Thibert‌, Anatole Gallouët, Quentin Mérigot.

The‌ stability of solutions to optimal transport problems under‌ variation of the measures is fundamental from a‌ mathematical viewpoint: it is closely related to the‌ convergence of numerical approaches to solve optimal transport problems and justifies many‌ of the applications of‌ optimal transport. In this‌‌ article, we introduce the notion of strong c-concavity,‌ and we show that‌ it plays an important‌‌ role for proving stability results in optimal transport‌ for general cost functions‌ c. We then introduce‌‌ a differential criterion for proving that a function‌ is strongly c-concave, under‌ an hypothesis on the‌‌ cost introduced originally by Ma-Trudinger-Wang for establishing regularity‌ of optimal transport maps.‌ Finally, we provide two‌‌ examples where this stability result can be applied,‌ for cost functions taking‌ value +∞ on the‌‌ sphere: the reflector problem and the Gaussian curvature‌ measure prescription problem. This‌ probleme is investigated in‌‌ In 12.

Participants: Cyril Letrouit.

In‌ 19 the authors prove‌ quantum ergodicity and quantum‌‌ mixing for sequences of finite Schreier graphs converging‌ to an infinite Cayley‌ graph whose adjacency operator‌‌ has absolutely continuous spectrum. Under Benjamini-Schramm convergence (or‌ strong convergence in distribution),‌ we show that correlations‌‌ between eigenvectors at distinct energies vanish asymptotically when‌ tested against a broad‌ class of local observables.‌‌ Our results apply to all orthonormal eigenbases and‌ do not require tree-like‌ structure or periodicity of‌‌ the limiting graph, unlike previous approaches based on‌ non-backtracking operators or Floquet‌ theory. The proof introduces‌‌ a new framework for quantum ergodicity, based on‌ trace identities, resolvent approximations‌ and representation-theoretic techniques and‌‌ extends to certain families of non-regular graphs. We‌ illustrate the assumptions and‌ consequences of our theorems‌‌ on Schreier graphs arising from free products of‌ groups, right-angled Coxeter groups‌ and lifts of a‌‌ fixed base graph.

Participants: Bruno Lévy, Quentin‌ Mérigot, Hugo Leclerc‌.

Bruno Lévy, Quentin‌‌ Mérigot and Hugo Leclerc continued working on algorithms‌ for large scale semi-discrete‌ optimal transport, and developped‌‌ a new algorithm that scales-up to $10^{8‌}$ points and beyond. The‌ new algorithm is based‌‌ on a generic abstract distributed Delaunay algorithm, that‌ can be instanced into‌ either a parallel implementation‌‌ for multicore machines, or a distributed version running‌ on PC clusters. The‌ new algorithm is experimented‌‌ on a very difficult scenario from computational cosmology,‌ featuring a semi-discrete problem‌ with $3 x {10‌‌}^{8}$ points, with variations of density up to‌ 5 orders of magnitude‌ (in the Universe, some‌‌ regions of space are nearly empty of any‌ matter, whereas some other‌ zones become extremely dense‌‌ due to gravitational collapse that creates supermassive clusters‌ of galaxy). The algorithm‌ opens doors to new‌‌ analyses in computational cosmology, explored in cooperation with‌ our partners (Roya Mohayaee,‌ Institut d'Astrophysique de Paris).‌‌ The corresponding article was published in Journal of‌ Computational Physics DOI [preprint]‌.

Participants: Bruno Lévy‌‌.

Bruno Lévy worked on the low-level machinery‌ used by semi-discrete optimal‌ transport, and proposed a‌‌ new framework for robustly computing intersections between meshes.‌ The algorithm is based‌ on a set of‌‌ predicates, written with arbitrary-precision arithmetics, together with symbolic‌ perturbations to evade from‌ the degenerate configurations. The‌‌ algorithm was tested on‌ a large database of models (Thingi10K). It will‌ become the basic components for new semi-discrete optimal‌ transport algorithms to be developped in the future.‌ The article was published in ACM Transactions on‌ Graphics. [DOI][preprint].

8.1 Grand-canonical optimal transport‌

Participants: Simone Di Marino, Mathieu Lewin,‌ Luca Nenna.

In 10 We study a‌ generalization of the multi-marginal optimal transport problem, which‌ has no fixed number of marginals N and‌ is inspired of statistical mechanics. It consists in‌ optimizing a linear combination of the costs for‌ all the possible N’s, while fixing a certain‌ linear combination of the corresponding marginals. This in‌ particular helped to find a counter-example to a‌ longstanding conjecture in mathematical physics.

8.2 Convergence rates‌ for regularized unbalanced optimal transport: the discrete case‌

Participants: Luca Nenna, Paul Pegon, Louis‌ Tocquec.

Unbalanced optimal transport (UOT) is a‌ natural extension of optimal transport (OT) allowing comparison‌ between measures of different masses. It arises naturally‌ in machine learning by offering a robustness against‌ outliers. The aim of 30 is to provide‌ convergence rates of the regularized transport cost and‌ plans towards their original solution when both measures‌ are weighted sums of Dirac masses.

8.3 Characterizing‌ and computing solutions to regularized semi-discrete optimal transport‌ via an ordinary differential equation

Participants: Luca Nenna‌, Daniyar Omarov, Brendan Pass.

In‌ 29 we investigate the semi-discrete optimal transport (OT)‌ problem with entropic regularization. We characterize the solution‌ using a governing, well-posed ordinary differential equation (ODE).‌ This naturally yields an algorithm to solve the‌ problem numerically, which we prove has desirable properties,‌ notably including global strong convexity of a value‌ function whose Hessian must be inverted in the‌ numerical scheme. Extensive numerical experiments are conducted to‌ validate our approach. We compare the solutions obtained‌ using the ODE method with those derived from‌ Newton's method. Our results demonstrate that the proposed‌ algorithm is competitive for problems involving the squared‌ Euclidean distance and exhibits superior performance when applied‌ to various powers of the Euclidean distance. Finally,‌ we note that the ODE approach yields an‌ estimate on the rate of convergence of the‌ solution as the regularization parameter vanishes, for a‌ generic cost function

8.4 Large deviations for sticky-reflecting‌ Brownian motion with boundary diffusion

Participants: Jean-Baptiste Casteras‌, Leonard Monsaingeon, Luca Nenna.

In‌ 22 we study a Schilder-type large deviation principle‌ for sticky-reflected Brownian motion with boundary diffusion, both‌ at the static and sample path level in‌ the short-time limit. A sharp transition for the‌ rate function occurs, depending on whether the tangential‌ boundary diffusion is faster or slower than in‌ the interior of the domain. The resulting intrinsic‌ distance naturally gives rise to a novel optimal‌ transport model, where motion and kinetic energy are‌ treated differently in the interior and along the‌ boundary.

8.5 Stability of optimal transport maps

Participants:‌ Cyril Letrouit, Quentin Mérigot.

In 27, we establish quantitative‌ stability bounds for the‌ quadratic optimal transport map‌‌ $T_{μ}$ between a fixed probability density $ρ‌$ and a probability measure‌ $μ$ on $ℝ^{d‌‌}$ . Under general assumptions on $ρ$ , we‌ prove that the map‌ $μ \mapsto T_{μ‌‌}$ is bi-Hölder continuous, with dimension-free Hölder exponents. The‌ linearized optimal transport metric‌ is therefore bi-Hölder equivalent‌‌ to the 2-Wasserstein distance, which justifies its use‌ in applications.

We show‌ this property in the‌‌ following cases: (i) for any log-concave density $ρ‌$ with full support in‌ $ℝ^{d}$ , and‌‌ any log-bounded perturbation thereof; (ii) for $ρ$ bounded‌ away from 0 and‌ $+ \infty$ on a‌‌ John domain (e.g., on a bounded Lipschitz domain),‌ while the only previously‌ known result of this‌‌ type assumed convexity of the domain; (iii) for‌ some important families of‌ probability densities on bounded‌‌ domains which decay or blow-up polynomially near the‌ boundary. Concerning the sharpness‌ of point (ii), we‌‌ also provide examples of non-John domains for which‌ the Brenier potentials do‌ not satisfy any Hölder‌‌ stability estimate.

Our proofs rely on local variance‌ inequalities for the Brenier‌ potentials in small convex‌‌ subsets of the support of $ρ$ , which‌ are glued together to‌ deduce a global variance‌‌ inequality. This gluing argument is based on two‌ different strategies of independent‌ interest: one of them‌‌ leverages the properties of the Whitney decomposition in‌ bounded domains, the other‌ one relies on spectral‌‌ graph theory.

Participants: Cyril Letrouit, Quentin Mérigot‌.

In 26,‌ we prove quantitative bounds‌‌ on the stability of optimal transport maps and‌ Kantorovich potentials from a‌ fixed source measure ρ‌‌ under variations of the target measure $μ$ ,‌ when the cost function‌ is the squared Riemannian‌‌ distance on a Riemannian manifold. Previous works were‌ restricted to subsets of‌ Euclidean spaces, or made‌‌ specific assumptions either on the manifold, or on‌ the regularity of the‌ transport maps. Our proof‌‌ techniques combine entropy-regularized optimal transport with spectral and‌ integral-geometric techniques. As some‌ of the arguments do‌‌ not rely on the Riemannian structure, our work‌ also paves the way‌ towards understanding stability of‌‌ optimal transport in more general geometric spaces.

8.6‌ Sliced-Wasserstein distances

Participants: Quentin‌ Mérigot, Christophe Vauthier‌‌.

Sliced Wasserstein distances are widely used in‌ practice as a computationally‌ efficient alternative to Wasserstein‌‌ distances in high dimensions. In 21, motivated‌ by theoretical foundations of‌ this alternative, we prove‌‌ quantitative estimates between the sliced 1-Wasserstein distance and‌ the 1-Wasserstein distance. We‌ construct a concrete example‌‌ to demonstrate the exponents in the estimate is‌ sharp. We also provide‌ a general analysis for‌‌ the case where slicing involves projections onto k-planes‌ and not just lines.‌

In 17, we‌‌ investigate the properties of the Sliced Wasserstein Distance‌ (SW) when employed as‌ an objective functional. The‌‌ SW metric has gained significant interest in the‌ optimal transport and machine‌ learning literature, due to‌‌ its ability to capture‌ intricate geometric properties of probability distributions while remaining‌ computationally tractable, making it a valuable tool for‌ various applications, including generative modeling and domain adaptation.‌ Our study aims to provide a rigorous analysis‌ of the critical points arising from the optimization‌ of the SW objective. By computing explicit perturbations,‌ we establish that stable critical points of SW‌ cannot concentrate on segments. This stability analysis is‌ crucial for understanding the behaviour of optimization algorithms‌ for models trained using the SW objective. Furthermore,‌ we investigate the properties of the SW objective,‌ shedding light on the existence and convergence behavior‌ of critical points. We illustrate our theoretical results‌ through numerical experiments.

8.7 Crowd motion

Participants: Sylvain‌ Faure, Bertrand Maury.

In 35 we‌ study the so-called Faster is Slower (FIS) eﬀect‌ which is observed in some particular real-life or‌ experimental situations. In the context of an evacuation‌ process, it expresses that increasing the speed (or,‌ more generally, the competitiveness) of individuals may induce‌ a reduction of the flow through the exit‌ door. We propose here a parameter-free model to‌ reproduce and investigate this eﬀect (more precisely its‌ backward “Slower is Faster” equivalent). In spite of‌ its non-smooth character, which makes it diﬃcult to‌ analyze, this granular approach is based on very‌ basic ingredients in terms of behavior. In its‌ native, purely asocial version, individuals are represented by‌ hard-discs, each of which has a desired velocity,‌ and the actual velocity is built as the‌ projection of this field on the set of‌ admissible velocities (which respect the non-overlapping constraints). We‌ implement the slowereﬀect by introducing here an extra‌ step to account for the fact that individuals‌ refrain from pushing, and therefore tend to reduce‌ their desired velocity accounting for the velocities of‌ people upfront. The present paper has two objectives:‌ establish the relevance of this model by showing‌ that it satisfactorily reproduces various empirical eﬀects in‌ highly crowded evacuations with various levels of competitiveness,‌ and explore how it can be implemented to‌ recover and explain the FIS eﬀect. In this‌ spirit, we confront this Inhibition-Based (IB) model to‌ experimental data, focusing on the Faster is Slower‌ eﬀect. We show in particular that this approach‌ makes it possible to accurately recover the eﬀect‌ of competitiveness upon power-law distributions of tim lapses‌ which have been experimentally observed. We also study‌ the eﬀect of mixed behaviors, by introducing a‌ two-population model using both approaches. We investigate in‌ particular the eﬀect upon evacuation eﬃciency of the‌ ratio between competitive agents and non-competitive ones. In‌ a similar context, we investigate the role of‌ an obstacle placed upstream the exit upon evacuation‌ eﬃciency.

8.8 Kinetic modeling for nuclear medicine

Participants:‌ Florent Besson, Sylvain Faure.

The purpose‌ of 45 was to announce the birth of‌ PET KinetiX software to the nuclear medicine community.‌ Mathematically, the aim is to identify the parameters‌ of several biological models at each point of the image (several millions),‌ the difficulty being to‌ perform these calculations quickly‌‌ enough for clinical use. Optimal Transport is not‌ part of this work,‌ but it will be‌‌ of interest in the future: for reconstructing the‌ data used, and for‌ comparing the results obtained‌‌ for different groups of patients. This work therefore‌ concerns the kinetic modeling‌ which represents the ultimate‌‌ foundations of Positron Emission Tomography (PET) quantitative imaging,‌ a unique opportunity to‌ better characterize the diseases‌‌ or prevent the reduction of drugs development. Primarily‌ designed for research, parametric‌ imaging based on PET‌‌ kinetic modeling may become a reality in future‌ clinical practice, enhanced by‌ the technical abilities of‌‌ the latest generation of commercially available PET systems.‌ In the era of‌ precision medicine, such paradigm‌‌ shift should be promoted, regardless of the PET‌ system. In order to‌ anticipate and stimulate this‌‌ emerging clinical paradigm shift, we developed a constructor-independent‌ software package, called PET‌ KinetiX, allowing a faster‌‌ and easier computation of parametric images from any‌ 4D PET DICOM series,‌ at the whole field‌‌ of view level. The PET KinetiX package is‌ currently a plug-in for‌ Osirix DICOM viewer. The‌‌ package provides a suite of five PET kinetic‌ models: Patlak, Logan, 1-tissue‌ compartment model, 2-tissue compartment‌‌ model, and first pass blood flow. After uploading‌ the 4D-PET DICOM series‌ into Osirix, the image‌‌ processing requires very few steps: the choice of‌ the kinetic model and‌ the definition of an‌‌ input function. After a 2-min process, the PET‌ parametric and error maps‌ of the chosen model‌‌ are automatically estimated voxel-wise and written in DICOM‌ format. The software benefits‌ from the graphical user‌‌ interface of Osirix, making it user-friendly. Compared to‌ PMOD-PKIN (version 4.4) on‌ twelve 18F-FDG PET dynamic‌‌ datasets, PET KinetiX provided an absolute bias of‌ 0.1% (0.05–0.25) and 5.8%‌ (3.3–12.3) for KiPatlak‌‌ and Ki2TCM, respectively. Several clinical research illustrative‌ cases acquired on different‌ hybrid PET systems (standard‌‌ or extended axial fields of view, PET/CT, and‌ PET/MRI), with different acquisition‌ schemes (single-bed single-pass or‌‌ multi-bed multipass), are also provided. PET KinetiX is‌ a very fast and‌ efficient independent research software‌‌ that helps molecular imaging users easily and quickly‌ produce 3D PET parametric‌ images from any reconstructed‌‌ 4D-PET data acquired on standard or large PET‌ systems.

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

Participants: Thomas‌ Gallouët, Andrea Natale‌, Gabriele Todeschi.‌‌

In 61 we introduce a time discretization for‌ Wasserstein gradient flows based‌ on the classical Backward‌‌ Differentiation Formula of order two. The main building‌ block of the scheme‌ is the notion of‌‌ geodesic extrapolation in the Wasserstein space, which in‌ general is not uniquely‌ defined. We propose several‌‌ possible definitions for such an operation, and we‌ prove convergence of the‌ resulting scheme to the‌‌ limit PDE, in the case of the Fokker-Planck‌ equation. For a specific‌ choice of extrapolation we‌‌ also prove a more‌ general result, that is convergence towards EVI flows.‌ Finally, we propose a variational finite volume discretization‌ of the scheme which numerically achieves second order‌ accuracy in both space and time.

8.10 Metric‌ extrapolation in the Wasserstein space

Participants: Thomas Gallouët‌, Andrea Natale, Gabriele Todeschi.

In‌ 13 we study a variational problem providing a‌ way to extend for all times minimizing geodesics‌ connecting two given probability measures, in the Wasserstein‌ space. This is simply obtained by allowing for‌ negative coefficients in the classical variational characterization of‌ Wasserstein barycenters. We show that this problem admits‌ two equivalent convex formulations: the first can be‌ seen as a particular instance of Toland duality‌ and the second is a barycentric optimal transport‌ problem. We propose an efficient numerical scheme to‌ solve this latter formulation based on entropic regularization‌ and a variant of Sinkhorn algorithm.

8.11 Regularity‌ theory and geometry of unbalanced optimal transport

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

In 11 using the dual formulation only,‌ we show that the regularity of unbalanced optimal‌ transport also called entropy-transport inherits from the regularity‌ of standard optimal transport. We provide detailed examples‌ of Riemannian manifolds and costs for which unbalanced‌ optimal transport is this http URL all entropy-transport‌ formulations, Wasserstein-Fisher-Rao (WFR) metric, also called Hellinger-Kantorovich, stands‌ out since it admits a dynamic formulation, which‌ extends the Benamou-Brenier formulation of optimal transport. After‌ demonstrating the equivalence between dynamic and static formulations‌ on a closed Riemannian manifold, we prove a‌ polar factorization theorem, similar to the one due‌ to Brenier and Mc-Cann. As a byproduct, we‌ formulate the Monge-Ampère equation associated with WFR metric,‌ which also holds for more general costs. Last,‌ we study the link between c-convex functions for‌ the cost induced by the WFR metric and‌ the cost on the cone. The main result‌ is that the weak Ma-Trudinger-Wang condition on the‌ cone implies the same condition on the manifold‌ for the cost induced by WFR.

8.12 Monge‌ Ampère gravity: from the large deviation principle to‌ cosmological simulations through optimal transport

Participants: Bruno Lévy‌, Yann Brenier, Roya Mohayaee.

In‌ 65 we study Monge-Ampère gravity (MAG) as an‌ effective theory of cosmological structure formation through optimal‌ transport theory. MAG is based on the Monge-Ampère‌ equation, a nonlinear version of the Poisson equation,‌ that relates the Hessian determinant of the potential‌ to the density field. We explain how MAG‌ emerges from a conditioned system of independent and‌ indistinguishable Brownian particles, through the large deviation principle,‌ in the continuum limit. To numerically explore this‌ highly non-linear theory, we develop a novel N-body‌ simulation method based on semi-discrete optimal transport. Our‌ results obtained from the very first N-body simulation‌ of Monge-Ampère gravity with over 100 millions particles‌ show that on large scales, Monge-Ampère gravity is‌ similar to the Newtonian gravity but favours the‌ formation of anisotropic structures such as filaments. At small scales, MAG has‌ a weaker clustering and‌ is screened in high-density‌‌ regions. Although here we study the Monge-Ampère gravity‌ as an effective rather‌ than a fundamental theory,‌‌ our novel highly-performant optimal transport algorithm can be‌ used to run high-resolution‌ simulations of a large‌‌ class of modified theories of gravity, such as‌ Galileons, in which the‌ equations of motion are‌‌ second-order and of Monge-Ampère type.

8.13 Faster is‌ Slower effect for evacuation‌ processes: a granular standpoint‌‌

Participants: Sylvain Faure, Bertrand Maury.

In‌ 35 we studied the‌ so-called Faster is Slower‌‌ (FIS) effect which is observed in some particular‌ real-life or experimental situations.‌ In the context of‌‌ an evacuation process, it expresses that increasing the‌ speed (or, more generally,‌ the competitive- ness) of‌‌ individuals may induce a reduction of the flow‌ through the exit door.‌ We propose here a‌‌ parameter-free model to reproduce and investigate this effect‌ (more precisely its backward‌ “Slower is Faster” equivalent).‌‌ In spite of its non-smooth character, which makes‌ it difficult to analyze,‌ this gran- ular approach‌‌ is based on very basic ingredients in terms‌ of behavior. In its‌ native, purely asocial version,‌‌ individuals are represented by hard-discs, each of which‌ has a desired velocity,‌ and the actual velocity‌‌ is built as the projection of this field‌ on the set of‌ admissible velocities (which respect‌‌ the non-overlapping constraints). We implement the slower effect‌ by introducing here an‌ extra step to account‌‌ for the fact that individuals refrain from pushing,‌ and therefore tend to‌ reduce their desired velocity‌‌ accounting for the velocities of people upfront. The‌ present paper has two‌ objectives: estab- lish the‌‌ relevance of this model by showing that it‌ satisfactorily reproduces various empirical‌ effects in highly crowded‌‌ evacuations with various levels of competitiveness, and explore‌ how it can be‌ implemented to recover and‌‌ explain the FIS effect. In this spirit, we‌ confront this Inhibition-Based (IB)‌ model to experimental data,‌‌ focusing on the Faster is Slower effect. We‌ show in particular that‌ this approach makes it‌‌ possible to accurately recover the effect of competitiveness‌ upon power-law distributions of‌ time lapses which have‌‌ been experimentally observed. We also study the effect‌ of mixed behaviors, by‌ introducing a two-population model‌‌ using both approaches. Weinvestigate in particular the effect‌ upon evacuation efficiency of‌ the ratio be- tween‌‌ competitive agents and non-competitive ones. In a similar‌ context, we investigate the‌ role of an obstacle‌‌ placed upstream the exit upon evacuation efficiency.

8.14‌ Semi-discrete convex order and‌ Laguerre tessellation fitting

Participants:‌‌ David Bourne, Thomas Gallouët, Quentin Mérigot‌, Andrea Natale.‌

In 20 we study‌‌ the problem of reconstructing a Laguerre tessellation with‌ prescribed cell volumes from‌ the barycenters of its‌‌ cells. We show that this problem can be‌ reformulated as a Wasserstein‌ projection onto the convex‌‌ set of discrete measures dominated in convex order‌ by an absolutely continuous‌ measure. We provide a‌‌ complete characterization of this‌ set and exploit this to construct a regularized‌ projection problem that can be solved efficiently and‌ yields an approximation of the desired reconstruction. The‌ same method can also be applied to fit‌ a Laguerre tessellation to an arbitrary set of‌ barycenters. We give a concrete application of this‌ in materials science, of fitting a Laguerre tessellation‌ to an electron backscatter diffraction (EBSD) image of‌ a steel. Interestingly, our regularized problem can also‌ be reinterpreted as a semi-discrete Wasserstein metric extrapolation‌ problem.

8.15 Stability of Wasserstein projections in convex‌ order via metric extrapolation

Participants: Jakwang Kim,‌ Young-Heon Kim, Andrea Natale.

In 25‌ we provide a precise characterization of the link‌ between backward/forward Wasserstein projections in convex order and‌ the recently introduced metric extrapolation problem. They use‌ such a link to derive new quantitative stability‌ estimates for both problems.

8.16 Infinitely many saturated‌ travelling waves for a degenerate Fisher-KPP equation not‌ in divergence form

Participants: Matthieu Alfaro, Maxime‌ Herda, Andrea Natale.

In 36 we‌ consider an epidemic model with distributed-contacts consisting in‌ a very degenerate Fisher-KPP equation with a diffusion‌ term that is not in divergence form, and‌ which arises when the contact kernel concentrates. They‌ make an exhaustive study of its travelling waves.‌ For every admissible speed, there exist not only‌ a unique non-saturated (smooth) wave but also infinitely‌ many saturated (sharp) ones. Furthermore their tails may‌ differ from what is usually expected. These results‌ are thus in sharp contrast with their counterparts‌ on related models.

8.17 Gradient flows of interacting‌ Laguerre cells as discrete porous media flows

Participants:‌ Andrea Natale.

In 74 we study a‌ class of discrete models in which a collection‌ of particles evolves in time following the gradient‌ flow of an energy depending on the cell‌ areas of an associated Laguerre (i.e. a weighted‌ Voronoi) tessellation. He considers the high number of‌ cell limit of such systems and, using a‌ modulated energy argument, he proves convergence towards smooth‌ solutions of nonlinear diffusion PDEs of porous medium‌ type.

8.18 A coupling approach to Lipschitz transport‌ maps

Participants: Giovanni Conforti, Katharina Eichinger.‌

In 24, we propose a probabilistic approach‌ to bound the (dimension-free) Lipschitz constant of the‌ Langevin flow map on Rd introduced by Kim‌ and Milman. As example of application, we construct‌ Lipschitz maps from a uniformly log-concave probability measure‌ to log-Lipschitz perturbations as in 56. Our‌ proof is based on coupling techniques applied to‌ the stochastic representation of the family of vector‌ fields inducing the transport map. This method is‌ robust enough to relax the uniform convexity to‌ a weak asymptotic convexity condition and to remove‌ the bound on the third derivative of the‌ potential of the source measure.

8.19 Propagation of‌ weak log-concavity along generalised heat flows via Hamilton-Jacobi‌ equations

Participants: Louis-Pierre Chaintron, Giovanni Conforti,‌ Katharina Eichinger.

A well-known consequence of the Prékopa-Leindler inequality is the‌ preservation of logconcavity by‌ the heat semigroup. Unfortunately,‌‌ this property does not hold for more general‌ semigroups. In 23,‌ we exhibit a slightly‌‌ weaker notion of log-concavity that can be propagated‌ along generalised heat semigroups.‌ As a consequence, we‌‌ obtain logsemiconcavity properties for the ground state of‌ Schrödinger operators for non-convex‌ potentials, as well as‌‌ propagation of functional inequalities along generalised heat flows.‌ We then investigate the‌ preservation of weak log-concavity‌‌ by conditioning and marginalisation, following the seminal works‌ of Brascamp and Lieb.‌ To our knowledge, our‌‌ results are the first of this type in‌ non log-concave settings. We‌ eventually study generation of‌‌ log-concavity by parabolic regularisation and prove novel two-sided‌ log-Hessian estimates for the‌ fundamental solution of parabolic‌‌ equations with unbounded coefficients, which can be made‌ uniform in time. These‌ properties are obtained as‌‌ a consequence of new propagation of weak convexity‌ results for quadratic Hamilton-Jacobi-Bellman‌ (HJB) equations. The proofs‌‌ rely on a stochastic control interpretation combined with‌ a second order analysis‌ of reflection coupling along‌‌ HJB characteristics.

9 Partnerships and cooperations

9.1 International‌ initiatives

9.1.1 Inria associate‌ team not involved in‌‌ an IIL or an international program

Participants: Thomas‌ Gallouët, Luca Nenna‌, Katharina Eichinger,‌‌ Andrea Natale, all the students.

Karma‌

Title:
Optimal Transport geometry‌ and asymptotics
Duration:
2024‌‌ -> 2026
Coordinator:
Brendan Pass (pass@ualberta.ca)
Partners:
- University‌ of British Columbia Vancouver‌ (Canada)
Inria contact:
Thomas‌‌ Gallouët
Summary:
This research project aims to join‌ the forces of two‌ major teams working on‌‌ Optimal transport worldwide. Sharing our experiences we will‌ tackle several interesting and‌ challenging questions related to‌‌ Optimal transport. It is composed of three main‌ axes. The first direction‌ is to better understand‌‌ the geometry inherits from the ground cost function.‌ The second direction focuses‌ on the very dynamic‌‌ research field concerning the asymptotic of Entropy OT‌ when the regularization parameter‌ goes to zero. The‌‌ third direction is the study of extension of‌ Optimal Transport such as‌ Multi-marginal Optimal Transport Problems,‌‌ an important subject for theParMA, as well as‌ the weak Optimal transport‌ in which falls for‌‌ exemple the martingale Optimal Transport a domain of‌ expertise of the Canadians‌ partners.

9.2 International research‌‌ visitors

9.2.1 Visits of international scientists

Other international‌ visits to the team‌

Leonard Monsaingeon

Status
Researcher‌‌
Institution of origin: University of Lisbon
Country: Portugal‌
Dates: December 13-19, 2025‌
Context of the visit:‌‌
Work week+ Phd Jury
Mobility program/type of mobility:‌
Research stay

Dmitry Vorotnikov‌

Status
Associate professor
Institution‌‌ of origin:
University of Coimbra
Country:
Portugal
Dates:‌
December 13-19, 2025
Context‌ of the visit:
Work‌‌ week+ Phd Jury
Mobility program/type of mobility:
Research‌ stay

9.2.2 Visits to‌ international teams

Research stays‌‌ abroad

Benjamin Capdeville

Visited institution: University of British‌ Columbia
Country:
Canada
Dates:‌
January 6-21, 2025
Context‌‌ of the visit:
Project within the KarMA collaboration‌ with Young-Heon Kim and‌ Soumik Pal (University of‌‌ Washington, Seattle): "On the‌ rate of convergence of mirror Langevin diffusions"
Mobility‌ program/type of mobility:
Research stay

Andrea Natale

Visited‌ institution:
University of British Columbia
Country:
Canada
Dates:‌
April 4-23, 2025
Context of the visit:
work‌ with Young-Heon Kim to work on Wasserstein projections‌ in convex order.
Mobility program/type of mobility:
Research‌ stay funded by the joint Inria team KarMA‌ (Kantorovich Initiative and ParMA).

Andrea Natale

Visited institution:‌
Herriot-Watt University
Country:
United-Kingdom
Dates:
July 6-12, 2025‌
Context of the visit:
Work with David Bourne‌ to work on Laguerre tessellation fitting problems.
Mobility‌ program/type of mobility:
Research stay funded by the‌ Herriot-Watt University.

9.3 European initiatives

9.3.1 Mathematical Institute‌ for Planet Earth (IMPT)

Andrea Natale is PI‌ with G. Beaunée (INRAE) for the project “Calibration‌ of epidemic models on graphs with Optimal Transport‌ and derivative-free optimization" (2023-2024) financed by the Mathematical‌ Institute for Planet Earth (IMPT).

9.4 National initiatives‌

Quentin Mérigot and Thomas Gallouët are members of‌ the PEPR IA-EDP led by Antonin Chambolle (Inria‌ Mokaplan). Quentin Mérigot is co-PI.
Andrea Natale is‌ PI of the ANR BARYFLOW.
Luca Nenna is‌ PI of the ANR GOTA.
Bruno Levy is‌ the PI of the Inria AEX GEOMERIX.
Quentin‌ Mérigot has an IUF Junior.
Luca Nenna has‌ an IUF Junior.
Bertrand Maury has an IUF‌ senior.

9.5 Regional initiatives

- HCODE and PGMO‌ projects Luca Nenna , Quentin Merigot , Thomas‌ Gallouët .

10 Dissemination

10.1 Promoting scientific activities‌

10.1.1 Scientific events: organisation

Member of the organizing‌ committees

Thomas Gallouët and Andrea Natale co-organized the‌ Workshop “Geometry, duality and convexity in new‌ OT problems” held on November 19-21 at‌ the Institut de Mathématique d'Orsay.
Quentin Mérigot co-organized‌ with Stéphane Gaubert the PGMO invited course of‌ Bo'az Klartag, which gathered 45 participants from academia‌ and industry, in LMO (February 5-6,2026).

10.1.2 Scientific‌ events: selection

Reviewer

Each team member review several‌ papers a year for reference international journals.

10.1.3‌ Journal

Member of the editorial boards

Quentin Mérigot‌ was member of the editorial boards of SIAM‌ J Imaging science, ESAIM ProcS (EiC), ESAIM M2AN‌ and SMF Panoramas et synthèses
Bruno Lévy is‌ a member of the editorial board of Computer‌ and Graphics Journal (since 06/07/2025)

Reviewer - reviewing‌ activities

In 2025, Katharina Eichinger has reviewed for‌ PTRF, EJP, SD.

10.1.4 Invited talks

In 2025‌ Luca Nenna gave the following talks:

TSE Seminar,‌ Toulouse.
International Conference in Basic Science, Benjing (plenary).‌
Analysis semianr, University of Alberta, Edmonton.
Analysis seminar,‌ LMB, Besançon.
MACS seminar, ICJ, Lyon.

In 2025‌ Thomas Gallouët gave the following talks:

PGMO days,‌ invité dans le workshop New trends in Optimal‌ Transport and Applications, Part II.
Conférencier Plénier‌ à la conférence Nonlinear PDEs in Guimarães au‌ Portugal.

In 2025 Andrea Natale gave the following‌ talks:

Kantorovich Initiative Seminar, University of British Columbia,‌ Vancouver.
Séminaire Parisien d'Optimisation, Paris
Journée de rentrée‌ de l'équipe ANEDP, Université Paris-Saclay, Orsay
Dagstuhl seminar,‌ Leibniz-Zentrum für Informatik, Wadern
GT CalVa seminar, Paris

In 2025 Cyril Letrouit‌ gave the following talks:‌

Séminaires :

Online seminar‌‌ on interactions between geometry and statistics, Grenoble (Physique‌ mathématique),
Paris 13 LAGA‌ (Systèmes dynamiques),
Dauphine (Analyse‌‌ et probabilités),
Séminaire parisien D’optimisation,
Séminaire du LPSM,‌
Rome Tor Vergata (Differential‌ equations),
Collège de France‌‌ (Séminaire de mathématiques appliquées)

Conference talks :

July‌ 2025 Workshop of ANR‌ SuSa, Clermont-Ferrand
July 2025‌‌ Control of PDEs and related topics, Toulouse
June‌ 2025 Hypoellipticity in Lund,‌ Lund University, Sweden
June‌‌ 2025 Meeting of the ANR GeoDSIC, Nantes
May‌ 2025 Optimal control theory‌ and sub-Riemannian geometry in‌‌ Paris, Paris
March 2025 PDE AI annual meeting,‌ Paris

In 2025 Quentin‌ Mérigot gave the following‌‌ talks

March 2025: Online talk at Kantorovich Initiative‌ (Canada/US)
July 2025: mini-course‌ at Festum Pi (La‌‌ Canée)
November 2025: Talk at Bocconi university in‌ Milano
November 2025: Talk‌ at Università di Firenze,‌‌ CalcVar Days

In 2025, Katharina Eichinger gave the‌ following talks

February: Seminar‌ SAMM, Université Paris Panthéon-Sorbonne‌‌
March: Symposium on Mean Field Games, University of‌ Durham
May: Kantorovich Initiative‌ Seminar, University of British‌‌ Columbia
May: Probability seminar, University of Washington
June:‌ Probability seminar, University of‌ Augsburg
July: Seed seminar,‌‌ IHES
September: Annual ÖMG-DMV Meeting, Linz
September: Journée‌ de rentrée de l'équipe‌ ANEDP, Université Paris-Saclay
October:‌‌ Seminar EDPA, Université Claude Bernard Lyon 1
November:‌ Workshop Optimal transport: stochastics,‌ projections, and applications, The‌‌ Fields Institute Toronto
November: PGMO Days, Saclay
November:‌ Rencontre RT Optimisation, Lyon‌

In 2025, Bruno Levy‌‌ gave the following talks

invited talk at the‌ LJLL-Inria seminar (12/15/2025)
participated‌ to the Dagstuhl seminar‌‌ on generalized Voronoi diagrams and Applications (25492). (11/30-12/5/2025)‌
gave an invited talk‌ at CEA (11/12-13/2025)
Cyprien‌‌ Plateau-Holleville and Bruno Lévy gave two presentations at‌ the <a RING meeting‌, two presentations, one‌‌ on fluid simulation with Cyprien Plateau-Holleville, and one‌ on software design (09/16-19/2025)‌
co-organizer of the Premiegrave;res‌‌ rencontres Inria Quadrant (06/18/2025)
keynote at Plénières GDR‌ IG-RV , on optimal‌ transport for computational physics‌‌ (06/02-03/2025).
attended the Choose Europe for Science, grand‌ amphi de la Sorbonne‌ (invited by the President‌‌ de la République) (05/05/2025)
gave a keynote Giving‌ a keynote on semi-discrete‌ optimal transport with ${10‌‌}^{9}$ points and beyond, how and why at‌ journées Transport Optimal du‌ GDR IASIS (02/17/2025)

Conférences‌‌ PARMA (normalement elles sont déjà dans le RA;-)‌
10/06/2025: PARMA team day‌ at Laboratoire de Maths‌‌ d'Orsay.
05/07/2025: Mokaplan/ParMA/Kantorovich common scientific day in Inria‌ Paris

10.2 Teaching -‌ Supervision - Juries -‌‌ Educational and pedagogical outreach

Thomas Gallouët taught 48h‌ at Université Paris-Saclay, including‌ a course on Optimal‌‌ Transport in M2 and Optimization in Master2.

Luca‌ Nenna taught 128h at‌ Université Paris-Saclay, including a‌‌ course in Calculus of Variations in M2, Optimization‌ in M1 and M2.‌

Andrea Natale taught 35h‌‌ at Université Paris-Saclay, including an intensive course in‌ Optimisation (M2), and support‌ sessions in Analysis for‌‌ M2 students.

Quentin Mérigot taught 96h at ENS‌ Paris (Optimisation & optimal‌ transport L3/M1 level) and‌‌ 96h at Université Paris-Saclay‌ (L3 modélisation en analyse + responsabilities)

Katharina Eichinger‌ taught 48h at Université Paris-Saclay, including exercise classes‌ of Integration theory and Differential Calculus.

10.2.1 Supervision‌

Luca Nenna supervised the following PhD students and‌ post-doc:

Adrien Cances (PhD)
Louis Tocquec (PhD)
Elise‌ Bonnet-Weill (PhD)
Mattia Garatti (PhD)
Alessandro Cosenza (Post-Doc)‌

Thomas Gallouët supervised the following PhD students and‌ post-doc:

Katherina Eichinger (Post-doc –> CR in ParMA)‌
Erwan Stämpfli (fourth year, co-direction with Y. brenier)‌
Siwan Boufadene (third year, co-direction with F.X. Vialard)‌
Benjamin Capdeville (first/second year, co-direction with L. Monsaingeon)‌
Quentin Giton (first year/second, co-direction with Z. Kobeissi)‌

10.2.2 Juries

Thomas Gallouët participated to the following‌ juries:

Erwan Stämpfli (Phd Defense, Université Paris-Saclay, co-director)‌
Siwan Boufadene (Phd Defense, Université Gustave Eiffel, co-director)‌
Master 2 juries

Quentin Mérigot participated to the‌ following juries:

Laurent Pfeiffer (HDR, rapporteur)
Averil Aussedat‌ (Thèse Insa Rouen, rapporteur)
Eloi Tanguy (Thèse U‌ Paris-Cité, rapporteur)
Charly Boricaud (Thèse Université Paris-Saclay, président)‌
Maximilian Penka (Thèse TU Munich, rapporteur)
Master 2‌ defenses of M2 Optimization ( $\sim$ 20 defenses)‌

Bruno Levy participated to the following juries:

member‌ of the Inria evaluation committee
scientific director of‌ Program Inria Quadrant, and president of the expert‌ commitee
member of the Inria C3 bonus committee‌ (12/9-11/2025)
participated to the IRMIA++ advisory board‌ (Strasbourg). (11/17-18/2025)
member of the Jury d'admission ISFP‌ Centre Inria Paris (06/06/2025)
member of the Jury‌ CRCN-ISFP in Bordeaux (05/15-16/2025)
member of the Inria‌ Exploratory Actions selection comittee (05/14/2025)
memnber of the‌ admission jury of Telecom Nancy (04/28/2025)

and the‌ following Phd/HdR juries :

member of the PhD‌ thesis committee of Emile Hohnadel (07/01/2025)
member of‌ the HdR commitee of Nicolas Mellado (06/23/2025)

10.3‌ Popularization

10.3.1 Productions (articles, videos, podcasts, serious games,‌ ...)

Cyril Letrouit did a short video on‌ mathematics for AudiMath.

Bruno Lévy wrote an article‌ for TIPE 2025-2026, cycles et boucles, Cosmologie numérique‌ : La Physique, les Mathématiques et l'Informatique unissent‌ leur force pour tenter de résoudre des mystères‌ dans le ciel

11 Scientific production

11.1 Major‌ publications

1 articleF.Fatima Al Reda,‌ S.Sylvain Faure, B. A.Bertrand Antti‌ Maury and E.Etienne Pinsard. Faster is‌ Slower effect for evacuation processes: a granular standpoint‌.Journal of Computational Physics5042024,‌ 112861HAL DOI
2 articleF.Florent Besson‌ and S.Sylvain Faure. PET KinetiX—A Software‌ Solution for PET Parametric Imaging at the Whole‌ Field of View Level.Journal of Imaging‌ Informatics in Medicine372January 2024,‌ 842-850HAL DOI
3 miscL.-P.Louis-Pierre Chaintron‌, G.Giovanni Conforti and K.Katharina Eichinger‌. Propagation of weak log-concavity along generalised heat‌ flows via Hamilton-Jacobi equations.August 2025HAL‌
4 articleS.Simone Di Marino, M.‌Mathieu Lewin and L.Luca Nenna. Grand-Canonical‌ Optimal Transport.Archive for Rational Mechanics and‌ Analysis2492025, art. 12HAL DOI‌
5 articleS.Simone Di Marino, M.Mathieu Lewin and L.‌Luca Nenna. Ground‌ state energy is not‌‌ always convex in the number of electrons.‌Journal of Physical Chemistry‌ A12849November‌‌ 2024, 10697–10706HALDOI
6 articleT.‌Thomas Gallouët, A.‌Andrea Natale and G.‌‌Gabriele Todeschi. From geodesic extrapolation to a‌ variational BDF2 scheme for‌ Wasserstein gradient flows.‌‌Mathematics of Computation932024, 2769-2810HAL‌DOI
7 miscC.‌Cyril Letrouit and Q.‌‌Quentin Mérigot. Gluing methods for quantitative stability‌ of optimal transport maps‌.January 2025HAL‌‌
8 articleB.Bruno Lévy, Y.Yann‌ Brenier and R.Roya‌ Mohayaee. Monge Ampère‌‌ gravity: from the large deviation principle to cosmological‌ simulations through optimal transport‌.Physical Review D‌‌11062024, 063550HAL DOI
9‌ articleL.Luca Nenna‌ and P.Paul Pegon‌‌. Convergence rate of entropy-regularized multi-marginal optimal transport‌ costs.Canadian Journal‌ of Mathematics = Journal‌‌ Canadien de MathématiquesMarch 2024HAL DOI

11.2‌ Publications of the year‌

International journals

10 article‌‌S.Simone Di Marino, M.Mathieu Lewin‌ and L.Luca Nenna‌. Grand-Canonical Optimal Transport‌‌.Archive for Rational Mechanics and Analysis249‌2025, art. 12‌HAL DOI back to‌‌ text
11 articleT.Thomas Gallouët, R.‌Roberta Ghezzi and F.-X.‌François-Xavier Vialard. Regularity‌‌ theory and geometry of unbalanced optimal transport.‌Journal of Functional Analysis‌2897October 2025‌‌, 111042HAL DOIback to text
12‌ articleA.Anatole Gallouët‌, Q.Quentin Mérigot‌‌ and B.Boris Thibert. Strong c-concavity and‌ stability in optimal transport‌.Journal de Mathématiques‌‌ Pures et Appliquées205January 2026, 103773‌HAL DOI back to‌ text
13 articleT.‌‌ O.Thomas O. Gallouët, A.Andrea Natale‌ and G.Gabriele Todeschi‌. Metric extrapolation in‌‌ the Wasserstein space.Calculus of Variations and‌ Partial Differential Equations64‌147May 2025HAL‌‌DOI back to text
14 articleB.Bruno‌ Lévy. Exact Predicates,‌ Exact Constructions and Combinatorics‌‌ for Mesh CSG..ACM Transactions on Graphics‌445July 2025‌, 1-27HAL DOI‌‌
15 articleB.Bruno Lévy, N.Nicolas‌ Ray, Q.Quentin‌ Merigot and H.Hugo‌‌ Leclerc. Large-scale semi-discrete optimal transport with distributed‌ Voronoi diagrams.Journal‌ of Computational PhysicsSeptember‌‌ 2025, 114374HALDOI
16 articleH.‌ A.Hugo A. Martin‌, A.Anne Mangeney‌‌, X.Xiaoping Jia, B. A.Bertrand‌ Antti Maury, A.‌Aline Lefebvre-Lepot, Y.‌‌Yvon Maday and P.Paul Dérand. Ultrasound-induced‌ dense granular flows: a‌ two-time scale modeling.‌‌Journal of Fluid Mechanics1004January 2025,‌ A10HAL DOI

International‌ peer-reviewed conferences

17 inproceedings‌‌C.Christophe Vauthier, A.Anna Korba and‌ Q.Quentin Merigot.‌ Towards Understanding Gradient Dynamics‌‌ of the Sliced-Wasserstein Distance via Critical Point Analysis‌.Proceedings of the‌ 42nd International Conference on‌‌ Machine LearningICML 2025‌ - 42nd International Conference on Machine LearningVancouver‌ (BC), CanadaJuly 2025HAL back to text‌

Reports & preprints

18 miscL.Léa Bohbot‌, C.Cyril Letrouit, G.Gabriel Peyré‌ and F.-X.François-Xavier Vialard. Token Sample Complexity‌ of Attention.December 2025HAL
19 misc‌C.Charles Bordenave, C.Cyril Letrouit and‌ M.Mostafa Sabri. Quantum mixing on large‌ Schreier graphs.January 2026HAL back to‌ text
20 miscD. P.David P. Bourne‌, T. O.Thomas O. Gallouët, Q.‌Quentin Merigot and A.Andrea Natale. Semi-discrete‌ convex order and Laguerre tessellation fitting.March‌ 2025HAL back to text
21 miscG.‌Guillaume Carlier, A.Alessio Figalli, Q.‌Quentin Mérigot and Y.Yi Wang. Sharp‌ comparisons between sliced and standard 1-Wasserstein distances.‌October 2025HAL back to text
22 misc‌J.-B.Jean-Baptiste Casteras, L.Leonard Monsaingeon and‌ L.Luca Nenna. Large deviations for sticky-reflecting‌ Brownian motion with boundary diffusion.January 2025‌HAL back to text
23 miscL.-P.Louis-Pierre‌ Chaintron, G.Giovanni Conforti and K.Katharina‌ Eichinger. Propagation of weak log-concavity along generalised‌ heat flows via Hamilton-Jacobi equations.August 2025‌HAL back to text
24 miscG.Giovanni‌ Conforti and K.Katharina Eichinger. A coupling‌ approach to Lipschitz transport maps.February 2025‌HAL back to text
25 miscJ.Jakwang‌ Kim, Y.-H.Young-Heon Kim and A.Andrea‌ Natale. Stability of Wasserstein projections in convex‌ order via metric extrapolation.July 2025HAL‌back to text
26 miscJ.Jun Kitagawa‌, C.Cyril Letrouit and Q.Quentin Merigot‌. Stability of optimal transport maps on Riemannian‌ manifolds.April 2025HAL back to text‌
27 miscC.Cyril Letrouit and Q.Quentin‌ Mérigot. Gluing methods for quantitative stability of‌ optimal transport maps.January 2025HAL back‌ to text
28 miscC.Cyril Letrouit.‌ Unstable optimal transport maps.October 2025HAL‌
29 miscL.Luca Nenna, D.Daniyar‌ Omarov and B.Brendan Pass. Characterizing and‌ computing solutions to regularized semi-discrete optimal transport via‌ an ordinary differential equation.April 2025HAL‌back to text
30 miscL.Luca Nenna‌, P.Paul Pegon and L.Louis Tocquec‌. Convergence rates for regularized unbalanced optimal transport:‌ the discrete case.October 2025HAL back‌ to text
31 miscC.Christophe Vauthier,‌ Q.Quentin Merigot and A.Anna Korba.‌ Properties of Wasserstein Gradient Flows for the Sliced-Wasserstein‌ Distance.February 2025HAL back to text‌

Scientific popularization

32 miscB. A.Bertrand Antti‌ Maury and P.Patrick Poncet. Mathematical distances,‌ experienced distances.September 2025HAL

Software

33‌ softwareS.Sylvain Faure and B.Bertrand Maury‌. Crowd Motion Simulation (CroMoSim).2.1March‌ 2025 lic: GNU General Public License version >=3‌.HAL Software HeritageVCS back to text

11.3 Cited publications

34‌ articleM.Martial Agueh‌ and G.Guillaume Carlier‌‌. Barycenters in the Wasserstein space.SIAM‌ Journal on Mathematical Analysis‌4322011,‌‌ 904--924back to text
35 articleF.Fatima‌ Al Reda, S.‌Sylvain Faure, B.‌‌ A.Bertrand Antti Maury and E.Etienne Pinsard‌. Faster is Slower‌ effect for evacuation processes:‌‌ a granular standpoint.Journal of Computational Physics‌5042024, 112861‌HAL DOI back to‌‌ text back to text
36 articleM.Matthieu‌ Alfaro, M.Maxime‌ Herda and A.Andrea‌‌ Natale. Infinitely many saturated travelling waves for‌ a degenerate Fisher-KPP equation‌ not in divergence form‌‌.Journal of Differential Equations4534February‌ 2026HAL DOI back‌ to text
37 article‌‌A.Aurélien Alfonsi, R.Rafaël Coyaud and‌ V.Virginie Ehrlacher.‌ Constrained overdamped Langevin dynamics‌‌ for symmetric multimarginal optimal transportation.Mathematical Models‌ and Methods in Applied‌ Sciences32032022‌‌, 403--455back to text
38 articleA.‌Aurélien Alfonsi, R.‌Rafaël Coyaud, V.‌‌Virginie Ehrlacher and D.Damiano Lombardi. Approximation‌ of optimal transport problems‌ with marginal moments constraints‌‌.Mathematics of Computation903282021,‌ 689--737back to text‌
39 bookL.Luigi‌‌ Ambrosio, N.Nicola Gigli and G.Giuseppe‌ Savaré. Gradient flows:‌ in metric spaces and‌‌ in the space of probability measures.Springer‌ Science & Business Media‌2005back to text‌‌
40 articleJ.-D.Jean-David Benamou and Y.Yann‌ Brenier. A computational‌ fluid mechanics solution to‌‌ the Monge-Kantorovich mass transfer problem.Numerische Mathematik‌8432000,‌ 375--393back to text‌‌
41 articleJ.-D.Jean-David Benamou, G.Guillaume‌ Carlier, M.Marco‌ Cuturi, L.Luca‌‌ Nenna and G.Gabriel Peyré. Iterative Bregman‌ projections for regularized transportation‌ problems.SIAM Journal‌‌ on Scientific Computing3722015, A1111--A1138‌back to text
42‌ incollectionJ.-D.Jean-David Benamou‌‌, G.Guillaume Carlier and L.Luca Nenna‌. A numerical method‌ to solve multi-marginal optimal‌‌ transport problems with Coulomb cost.Splitting Methods‌ in Communication, Imaging, Science,‌ and EngineeringSpringer2016‌‌, 577--601back to text
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PARMA - 2025

PARMA - 2025

2025Activity﻿‌​‌ reportProject-TeamPARMA

Keywords

Computer Science​‌﻿﻿ and Digital Science

Other Research Topics​​﻿﻿ and Application Domains

1​‌﻿﻿ Team members, visitors, external​​﻿﻿ collaborators

Research Scientists

Faculty Members​​​‌

Post-Doctoral Fellows

PhD Students

Technical Staff​​​‌

Administrative Assistant​​​‌

External Collaborators

2 Overall objectives﻿﻿﻿‌

2.1 Scientific and application​‌﻿﻿ context

2.1.1 Optimal transport​​﻿﻿

2.1.2​​​‌ Computational optimal transport

3 Research program

3.1﻿‌​‌ Optimal transport: models, numerical﻿​​﻿ analysis and solvers

3.1.1​​​‌ Models

3.1.2﻿​﻿﻿ Numerical analysis

Convergence speed.

Discrete regularity

3.1.3 Solvers​​﻿﻿

Algorithms

Implementation​​​‌ of a semi-discrete solver﻿​﻿﻿

Milestones﻿﻿﻿‌

3.2 OT-based﻿‌​‌ particle methods for fluids﻿​​﻿ mechanics

Milestones﻿​​﻿

3.3 OT﻿‌​‌ for inverse problems

3.3.1﻿​​﻿ OT as a data-fidelity​​​‌ term

3.3.2​‌﻿﻿ Multi-marginal OT

Application:﻿‌​‌ Risk estimation

Application: density functional​​​‌ theory

Objectives:

Milestones

3.4 Optimal transport for﻿‌​‌ Social Sciences

3.5 Optimal​​​‌ transport for Cosmology

4 Application domains​‌﻿﻿

4.1 Geography-Social science

4.2 Medicine

4.3​‌﻿﻿ Optics

4.4 Crowd motion

4.5 Astrophysics

5 Social﻿﻿﻿‌ and environmental responsibility

5.1﻿‌​‌ Footprint of research activities﻿​​﻿

6 Highlights﻿﻿﻿‌ of the year

6.1﻿‌​‌ Awards

7 Latest﻿﻿﻿‌ software developments, platforms, open﻿‌​‌ data

7.1 New platforms

7.1.1﻿‌​‌ SDOT

7.1.2 CroMoSim

7.1.3﻿​﻿﻿ PET KinetiX

7.1.4 GEOGRAM

8 New results​‌﻿﻿

8.1 Grand-canonical optimal transport​‌﻿﻿

8.2 Convergence rates​‌﻿﻿ for regularized unbalanced optimal​​﻿﻿ transport: the discrete case​​​‌

8.3 Characterizing​‌﻿﻿ and computing solutions to​​﻿﻿ regularized semi-discrete optimal transport​​​‌ via an ordinary differential﻿​﻿﻿ equation

8.4​​﻿﻿ Large deviations for sticky-reflecting​​​‌ Brownian motion with boundary﻿​﻿﻿ diffusion

8.5 Stability of﻿​﻿﻿ optimal transport maps

8.6​​​‌ Sliced-Wasserstein distances

8.7​​﻿﻿ Crowd motion

8.8 Kinetic modeling﻿​﻿﻿ for nuclear medicine

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

8.10 Metric​​​‌ extrapolation in the Wasserstein﻿​﻿﻿ space

8.11 Regularity​​​‌ theory and geometry of﻿​﻿﻿ unbalanced optimal transport

8.12 Monge​‌﻿﻿ Ampère gravity: from the​​﻿﻿ large deviation principle to​​​‌ cosmological simulations through optimal﻿​﻿﻿ transport

8.13 Faster is​​​‌ Slower effect for evacuation﻿﻿﻿‌ processes: a granular standpoint﻿‌​‌

8.14​​​‌ Semi-discrete convex order and﻿﻿﻿‌ Laguerre tessellation fitting

8.15 Stability of﻿​﻿﻿ Wasserstein projections in convex​‌﻿﻿ order via metric extrapolation​​﻿﻿

8.16 Infinitely many saturated​​​‌ travelling waves for a﻿​﻿﻿ degenerate Fisher-KPP equation not​‌﻿﻿ in divergence form

8.17​​﻿﻿ Gradient flows of interacting​​​‌ Laguerre cells as discrete﻿​﻿﻿ porous media flows

8.18 A coupling﻿​﻿﻿ approach to Lipschitz transport​‌﻿﻿ maps

8.19 Propagation of​‌﻿﻿ weak log-concavity along generalised​​﻿﻿ heat flows via Hamilton-Jacobi​​​‌ equations

9 Partnerships﻿​​﻿ and cooperations

9.1 International​​​‌ initiatives

9.1.1 Inria associate﻿﻿﻿‌ team not involved in﻿‌​‌ an IIL or an﻿​​﻿ international program

Karma​​​‌

9.2 International research﻿‌​‌ visitors

2025Activity‌‌ reportProject-TeamPARMA

Computer Science‌ and Digital Science

Other Research Topics and Application Domains

1‌ Team members, visitors, external collaborators

Faculty Members‌

Technical Staff‌

Administrative Assistant‌

2 Overall objectives‌

2.1 Scientific and application‌ context

2.1.1 Optimal transport

2.1.2‌ Computational optimal transport

3.1‌‌ Optimal transport: models, numerical analysis and solvers

3.1.1‌ Models

3.1.2 Numerical analysis

3.1.3 Solvers

Implementation‌ of a semi-discrete solver

Milestones‌

3.2 OT-based‌‌ particle methods for fluids mechanics

Milestones

3.3 OT‌‌ for inverse problems

3.3.1 OT as a data-fidelity‌ term

3.3.2‌ Multi-marginal OT

Application:‌‌ Risk estimation

Application: density functional‌ theory

3.4 Optimal transport for‌‌ Social Sciences

3.5 Optimal‌ transport for Cosmology

4 Application domains‌

4.3‌ Optics

5 Social‌ and environmental responsibility

5.1‌‌ Footprint of research activities

6 Highlights‌ of the year

6.1‌‌ Awards

7 Latest‌ software developments, platforms, open‌‌ data

7.1.1‌‌ SDOT

7.1.3 PET KinetiX

8 New results‌

8.1 Grand-canonical optimal transport‌

8.2 Convergence rates‌ for regularized unbalanced optimal transport: the discrete case‌

8.3 Characterizing‌ and computing solutions to regularized semi-discrete optimal transport‌ via an ordinary differential equation

8.4 Large deviations for sticky-reflecting‌ Brownian motion with boundary diffusion

8.5 Stability of optimal transport maps

8.6‌ Sliced-Wasserstein distances

8.7 Crowd motion

8.8 Kinetic modeling for nuclear medicine

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

8.10 Metric‌ extrapolation in the Wasserstein space

8.11 Regularity‌ theory and geometry of unbalanced optimal transport

8.12 Monge‌ Ampère gravity: from the large deviation principle to‌ cosmological simulations through optimal transport

8.13 Faster is‌ Slower effect for evacuation‌ processes: a granular standpoint‌‌

8.14‌ Semi-discrete convex order and‌ Laguerre tessellation fitting

8.15 Stability of Wasserstein projections in convex‌ order via metric extrapolation

8.16 Infinitely many saturated‌ travelling waves for a degenerate Fisher-KPP equation not‌ in divergence form

8.17 Gradient flows of interacting‌ Laguerre cells as discrete porous media flows

8.18 A coupling approach to Lipschitz transport‌ maps

8.19 Propagation of‌ weak log-concavity along generalised heat flows via Hamilton-Jacobi‌ equations

9 Partnerships and cooperations

9.1 International‌ initiatives

9.1.1 Inria associate‌ team not involved in‌‌ an IIL or an international program

Karma‌

9.2 International research‌‌ visitors

9.2.1 Visits of international scientists

Other international‌ visits to the team‌

Dmitry Vorotnikov‌

9.2.2 Visits to‌ international teams

Research stays‌‌ abroad

9.3 European initiatives

9.3.1 Mathematical Institute‌ for Planet Earth (IMPT)

9.4 National initiatives‌

9.5 Regional initiatives

10 Dissemination

10.1 Promoting scientific activities‌

10.1.1 Scientific events: organisation

Member of the organizing‌ committees

10.1.2 Scientific‌ events: selection

10.1.3‌ Journal

Member of the editorial boards

Reviewer - reviewing‌ activities

10.1.4 Invited talks

10.2 Teaching -‌ Supervision - Juries -‌‌ Educational and pedagogical outreach

10.2.1 Supervision‌