RESEARCH CENTRE

Inria Paris Centre

2023
ACTIVITY REPORT

IN PARTNERSHIP WITH:

Université Paris-Dauphine, CNRS

Project-Team

MOKAPLAN

Advances in Numerical Calculus of
Variations

IN COLLABORATION WITH: CEREMADE

DOMAIN

Applied Mathematics, Computation and
Simulation

THEME

Numerical schemes and simulations

Contents

Project-Team MOKAPLAN

1 Team members, visitors, external collaborators

2 Overall objectives

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Static Optimal Transport and Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Diffeomorphisms and Dynamical Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Sparsity in Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 MOKAPLAN unified point of view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Research program

3.1 OT and related variational problems solvers encore et toujours
3.2 Application of OT numerics to non-variational and non convex problems
3.3 Inverse problems with structured priors
3.4 Geometric variational problems, and their interactions with transport

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4 Application domains
4.1 Natural Sciences
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4.2 Signal Processing and inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
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4.3 Social Sciences

5 Highlights of the year

6 New software, platforms, open data

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7 New results

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7.1 Entropic Optimal Transport Solutions of the Semigeostrophic Equations . . . . . . . . . . . . . . . 10
7.2 Wasserstein gradient flow of the Fisher information from a non-smooth convex minimization

viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7.3 Total variation regularization with Wasserstein penalization . . . . . . . . . . . . . . . . . . . . . . . 11
7.4 1D approximation of measures in Wasserstein spaces
. . . . . . . . . . . . . . . . . . . . . . . . . . 11
7.5 Exact recovery of the support of piecewise constant images via total variation regularization . . . 11
7.6 A geometric Laplace method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.7 Convergence rate of general entropic optimal transport costs . . . . . . . . . . . . . . . . . . . . . . 12
7.8 A geometric approach to apriori estimates for optimal transport maps
. . . . . . . . . . . . . . . . 12
7.9 Gradient descent with a general cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.10 Second-order methods for Burer-Monteiro factorization . . . . . . . . . . . . . . . . . . . . . . . . . 13
7.11 Optimization for imaging and machine learning, analysis of inverse problems . . . . . . . . . . . . 13
7.12 Free discontinuity problems, fractures and shape optimization . . . . . . . . . . . . . . . . . . . . . 13
7.13 Interface evolution problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
7.14 Optimal quantization via branched optimal transport distance . . . . . . . . . . . . . . . . . . . . . 14
7.15 From geodesic extrapolation to a variational BDF2 scheme for Wasserstein gradient flows . . . . . 14
7.16 Regularity theory and geometry of unbalanced optimal transport
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7.17 Entropic approximation of ∞ optimal transport problems . . . . . . . . . . . . . . . . . . . . . . . . 15
7.18 Quantitative Stability of the Pushforward Operation by an Optimal Transport Map . . . . . . . . . 15
7.19 Quantitative Stability of Barycenters in the Wasserstein Space . . . . . . . . . . . . . . . . . . . . . 15
7.20 Wasserstein medians: robustness, PDE characterization and numerics . . . . . . . . . . . . . . . . 16

8 Bilateral contracts and grants with industry

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9 Partnerships and cooperations

9.1 International research visitors

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9.1.1 Visits of international scientists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
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9.1.2 Visits to international teams
9.2 National initiatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

10 Dissemination

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10.1 Promoting scientific activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.1.1 Scientific events: organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
10.1.2 Scientific events: selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
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10.1.3 Journal
10.1.4 Invited talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10.1.5 Scientific expertise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10.1.6 Research administration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
10.2 Teaching - Supervision - Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.2.1 Teaching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.2.2 Supervision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.2.3 Juries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

11 Scientific production

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11.1 Major publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
11.2 Publications of the year
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11.3 Cited publications

Project-Team MOKAPLAN

Creation of the Project-Team: 2015 December 01

Keywords

Computer sciences and digital sciences

A5.3. – Image processing and analysis

A5.9. – Signal processing

A6.1.1. – Continuous Modeling (PDE, ODE)

A6.2.1. – Numerical analysis of PDE and ODE

A6.2.6. – Optimization

A6.3.1. – Inverse problems

A8.2.3. – Calculus of variations

A8.12. – Optimal transport

A9. – Artificial intelligence

Other research topics and application domains

B9.5.2. – Mathematics

B9.5.3. – Physics

B9.5.4. – Chemistry

B9.6.3. – Economy, Finance

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1 Team members, visitors, external collaborators

Research Scientists

• Vincent Duval [Team leader, INRIA, Senior Researcher, HDR]

• Jean-David Benamou [INRIA, Senior Researcher, HDR]

• Antonin Chambolle [CNRS, Senior Researcher, HDR]

• Thomas Gallouët [INRIA, Researcher]

• Flavien Leger [INRIA, Researcher]

• Irène Waldspurger [CNRS, Researcher]

Faculty Members

• Guillaume Carlier [DAUPHINE PSL, Professor, HDR]

• Paul Pegon [DAUPHINE PSL, Associate Professor, from Sep 2023]

• Paul Pegon [DAUPHINE PSL, Associate Professor Delegation, until Aug 2023]

• François-Xavier Vialard [UNIV GUSTAVE EIFFEL, Professor Delegation, from Feb 2023 until Jul 2023]

Post-Doctoral Fellow

• Adrien Vacher [INRIA, Post-Doctoral Fellow, from Dec 2023]

PhD Students

• Guillaume Chazareix [BNP, CIFRE]

• Hugo Malamut [DAUPHINE PSL]

• Joao Pinto Anastacio Machado [DAUPHINE PSL]

• Faniriana Rakoto Endor [CNRS, from Oct 2023]

• Erwan Stampfli [UNIV PARIS SACLAY]

• Maxime Sylvestre [DAUPHINE PSL]

Administrative Assistant

• Derya Gok [INRIA]

External Collaborators

• Yann Brenier [CNRS]

• François-Xavier Vialard [UNIV GUSTAVE EIFFEL, from Sep 2023]

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2 Overall objectives

2.1 Introduction

The last decade has witnessed a remarkable convergence between several sub-domains of the calculus of
variations, namely optimal transport (and its many generalizations), infinite dimensional geometry of dif-
feomorphisms groups and inverse problems in imaging (in particular sparsity-based regularization). This
convergence is due to (i) the mathematical objects manipulated in these problems, namely sparse measures
(e.g. coupling in transport, edge location in imaging, displacement fields for diffeomorphisms) and (ii) the
use of similar numerical tools from non-smooth optimization and geometric discretization schemes. Optimal
Transportation, diffeomorphisms and sparsity-based methods are powerful modeling tools, that impact a
rapidly expanding list of scientific applications and call for efficient numerical strategies. Our research program
shows the important part played by the team members in the development of these numerical methods and
their application to challenging problems.

2.2 Static Optimal Transport and Generalizations

Optimal Transport, Old and New. Optimal Mass Transportation is a mathematical research topic which
started two centuries ago with Monge’s work on the “Théorie des déblais et des remblais" (see [104]). This
engineering problem consists in minimizing the transport cost between two given mass densities. In the 40’s,
Kantorovich [113] introduced a powerful linear relaxation and introduced its dual formulation. The Monge-
Kantorovich problem became a specialized research topic in optimization and Kantorovich obtained the 1975
Nobel prize in economics for his contributions to resource allocations problems. Since the seminal discoveries
of Brenier in the 90’s [67], Optimal Transportation has received renewed attention from mathematical analysts
and the Fields Medal awarded in 2010 to C. Villani, who gave important contributions to Optimal Transportation
and wrote the modern reference monographs [137, 138], arrived at a culminating moment for this theory.
Optimal Mass Transportation is today a mature area of mathematical analysis with a constantly growing
range of applications. Optimal Transportation has also received a lot of attention from probabilists (see for
instance the recent survey [117] for an overview of the Schrödinger problem which is a stochastic variant of the
Benamou-Brenier dynamical formulation of optimal transport). The development of numerical methods for
Optimal Transportation and Optimal Transportation related problems is a difficult topic and comparatively
underdeveloped. This research field has experienced a surge of activity in the last five years, with important
contributions of the MOKAPLAN group (see the list of important publications of the team). We describe below a
few of recent and less recent Optimal Transportation concepts and methods which are connected to the future
activities of MOKAPLAN :

Brenier’s theorem [69] characterizes the unique optimal map as the gradient of a convex potential. As such
Optimal Transportation may be interpreted as an infinite dimensional optimisation problem under “convexity
constraint": i.e. the solution of this infinite dimensional optimisation problem is a convex potential. This
connects Optimal Transportation to “convexity constrained" non-linear variational problems such as, for
instance, Newton’s problem of the body of minimal resistance. The value function of the optimal transport
problem is also known to define a distance between source and target densities called the Wasserstein distance
which plays a key role in many applications such as image processing.

Monge-Ampère Methods. A formal substitution of the optimal transport map as the gradient of a convex
potential in the mass conservation constraint (a Jacobian equation) gives a non-linear Monge-Ampère equation.
Caffarelli [73] used this result to extend the regularity theory for the Monge-Ampère equation. In the last ten
years, it also motivated new research on numerical solvers for non-linear degenerate Elliptic equations [96]
[121] [59] [60] and the references therein. Geometric approaches based on Laguerre diagrams and discrete data
[124] have also been developed. Monge-Ampère based Optimal Transportation solvers have recently given the
first linear cost computations of Optimal Transportation (smooth) maps.

Generalizations of OT.
In recent years, the classical Optimal Transportation problem has been extended in
several directions. First, different ground costs measuring the “physical" displacement have been considered.
In particular, well posedness for a large class of convex and concave costs has been established by McCann
and Gangbo [103]. Optimal Transportation techniques have been applied for example to a Coulomb ground
cost in Quantum chemistry in relation with Density Functional theory [90]. Given the densities of electrons

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Figure 1: Example of color transfer between two images, computed using the method developed in [57], see
also [133]. The image framed in red and blue are the input images. Top and middle row: adjusted image where
the color of the transported histogram has been imposed. Bottom row: geodesic (displacement) interpolation
between the histogram of the chrominance of the image.

Optimal Transportation models the potential energy and their relative positions. For more than more than 2
electrons (and therefore more than 2 densities) the natural extension of Optimal Transportation is the so called
Multi-marginal Optimal Transport (see [128] and the references therein). Another instance of multi-marginal
Optimal Transportation arises in the so-called Wasserstein barycenter problem between an arbitrary number of
densities [44]. An interesting overview of this emerging new field of optimal transport and its applications can
be found in the recent survey of Ghoussoub and Pass [129].

Numerical Applications of Optimal Transportation. Optimal transport has found many applications, starting
from its relation with several physical models such as the semi-geostrophic equations in meteorology [108, 93,
92, 55, 120], mesh adaptation [119], the reconstruction of the early mass distribution of the Universe [101, 70] in
Astrophysics, and the numerical optimisation of reflectors following the Optimal Transportation interpretation
of Oliker [74] and Wang [139]. Extensions of OT such as multi-marginal transport has potential applications in
Density Functional Theory , Generalized solution of Euler equations [68] (DFT) and in statistics and finance [53,
102] . . . . Recently, there has been a spread of interest in applications of OT methods in imaging sciences [63],
statistics [61] and machine learning [94]. This is largely due to the emergence of fast numerical schemes
to approximate the transportation distance and its generalizations, see for instance [57]. Figure 1 shows an
example of application of OT to color transfer. Figure 2 shows an example of application in computer graphics
to interpolate between input shapes.

2.3 Diffeomorphisms and Dynamical Transport

Dynamical transport. While the optimal transport problem, in its original formulation, is a static problem
(no time evolution is considered), it makes sense in many applications to rather consider time evolution. This is
relevant for instance in applications to fluid dynamics or in medical images to perform registration of organs
and model tumor growth.

In this perspective, the optimal transport in Euclidean space corresponds to an evolution where each
particule of mass evolves in straight line. This interpretation corresponds to the Computational Fluid Dynamic
(CFD) formulation proposed by Brenier and Benamou in [54]. These solutions are time curves in the space
of densities and geodesics for the Wasserstein distance. The CFD formulation relaxes the non-linear mass
conservation constraint into a time dependent continuity equation, the cost function remains convex but is
highly non smooth. A remarkable feature of this dynamical formulation is that it can be re-cast as a convex but
non smooth optimization problem. This convex dynamical formulation finds many non-trivial extensions and
applications, see for instance [56]. The CFD formulation also appears to be a limit case of Mean Fields games

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Figure 2: Example of barycenter between shapes computed using optimal transport barycenters of the uniform
densities inside the 3 extremal shapes, computed as detailed in [133]. Note that the barycenters are not in
general uniform distributions, and we display them as the surface defined by a suitable level-set of the density.

(MFGs), a large class of economic models introduced by Lasry and Lions [115] leading to a system coupling
an Hamilton-Jacobi with a Fokker-Planck equation. In contrast, the Monge case where the ground cost is the
euclidan distance leads to a static system of PDEs [64].

Gradient Flows for the Wasserstein Distance.
Another extension is, instead of considering geodesic for
transportation metric (i.e. minimizing the Wasserstein distance to a target measure), to make the density evolve
in order to minimize some functional. Computing the steepest descent direction with respect to the Wasserstein
distance defines a so-called Wasserstein gradient flow, also known as JKO gradient flows after its authors [112].
This is a popular tool to study a large class of non-linear diffusion equations. Two interesting examples are
the Keller-Segel system for chemotaxis [111, 82] and a model of congested crowd motion proposed by Maury,
Santambrogio and Roudneff-Chupin [123]. From the numerical point of view, these schemes are understood
to be the natural analogue of implicit scheme for linear parabolic equations. The resolution is however costly
as it involves taking the derivative in the Wasserstein sense of the relevant energy, which in turn requires the
resolution of a large scale convex but non-smooth minimization.

Geodesic on infinite dimensional Riemannian spaces. To tackle more complicated warping problems, such
as those encountered in medical image analysis, one unfortunately has to drop the convexity of the functional
involved in defining the gradient flow. This gradient flow can either be understood as defining a geodesic on
the (infinite dimensional) group of diffeomorphisms [52], or on a (infinite dimensional) space of curves or
surfaces [140]. The de-facto standard to define, analyze and compute these geodesics is the “Large Deformation
Diffeomorphic Metric Mapping” (LDDMM) framework of Trouvé, Younes, Holm and co-authors [52, 107].
While in the CFD formulation of optimal transport, the metric on infinitesimal deformations is just the L2 norm
(measure according to the density being transported), in LDDMM, one needs to use a stronger regularizing
metric, such as Sobolev-like norms or reproducing kernel Hilbert spaces (RKHS). This enables a control over
the smoothness of the deformation which is crucial for many applications. The price to pay is the need to solve
a non-convex optimization problem through geodesic shooting method [125], which requires to integrate
backward and forward the geodesic ODE. The resulting strong Riemannian geodesic structure on spaces of
diffeomorphisms or shapes is also pivotal to allow us to perform statistical analysis on the tangent space, to
define mean shapes and perform dimensionality reduction when analyzing large collection of input shapes (e.g.
to study evolution of a diseases in time or the variation across patients) [75].

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Figure 3: Examples of displacement interpolation (geodesic for optimal transport) according to a non-Euclidean
Riemannian metric (the mass is constrained to move inside a maze) between to input Gaussian distributions.
Note that the maze is dynamic: its topology change over time, the mass being “trapped” at time t = 1/3.

2.4 Sparsity in Imaging

Sparse ℓ1 regularization. Beside image warping and registration in medical image analysis, a key problem
in nearly all imaging applications is the reconstruction of high quality data from low resolution observations.
This field, commonly referred to as “inverse problems”, is very often concerned with the precise location of
features such as point sources (modeled as Dirac masses) or sharp contours of objects (modeled as gradients
being Dirac masses along curves). The underlying intuition behind these ideas is the so-called sparsity model
(either of the data itself, its gradient, or other more complicated representations such as wavelets, curvelets,
bandlets [122] and learned representation [141]).

The huge interest in these ideas started mostly from the introduction of convex methods to serve as proxy
for these sparse regularizations. The most well known is the ℓ1 norm introduced independently in imaging by
Donoho and co-workers under the name “Basis Pursuit” [87] and in statistics by Tibshirani [134] under the
name “Lasso”. A more recent resurgence of this interest dates back to 10 years ago with the introduction of the
so-called “compressed sensing” acquisition techniques [76], which make use of randomized forward operators
and ℓ1-type reconstruction.

Regularization over measure spaces. However, the theoretical analysis of sparse reconstructions involving
real-life acquisition operators (such as those found in seismic imaging, neuro-imaging, astro-physical imaging,
etc.) is still mostly an open problem. A recent research direction, triggered by a paper of Candès and Fernandez-
Granda [78], is to study directly the infinite dimensional problem of reconstruction of sparse measures (i.e.
sum of Dirac masses) using the total variation of measures (not to be mistaken for the total variation of 2-D
functions). Several works [77, 98, 97] have used this framework to provide theoretical performance guarantees
by basically studying how the distance between neighboring spikes impacts noise stability.

Low complexity regularization and partial smoothness.
In image processing, one of the most popular
methods is the total variation regularization [132, 71]. It favors low-complexity images that are piecewise
constant, see Figure 4 for some examples on how to solve some image processing problems. Beside applications
in image processing, sparsity-related ideas also had a deep impact in statistics [134] and machine learning [47].
As a typical example, for applications to recommendation systems, it makes sense to consider sparsity of the
singular values of matrices, which can be relaxed using the so-called nuclear norm (a.k.a. trace norm) [48].
The underlying methodology is to make use of low-complexity regularization models, which turns out to be

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Segmentation input

output

Zooming input

output

Figure 4: Two example of application of the total variation regularization of functions. Left: image segmentation
into homogeneous color regions. Right: image zooming (increasing the number of pixels while keeping the
edges sharp).

equivalent to the use of partly-smooth regularization functionals [118, 136] enforcing the solution to belong to
a low-dimensional manifold.

2.5 MOKAPLAN unified point of view

The dynamical formulation of optimal transport creates a link between optimal transport and geodesics on
diffeomorphisms groups. This formal link has at least two strong implications that MOKAPLAN will elaborate on:
(i) the development of novel models that bridge the gap between these two fields ; (ii) the introduction of novel
fast numerical solvers based on ideas from both non-smooth optimization techniques and Bregman metrics.
In a similar line of ideas, we believe a unified approach is needed to tackle both sparse regularization
in imaging and various generalized OT problems. Both require to solve related non-smooth and large scale
optimization problems. Ideas from proximal optimization has proved crucial to address problems in both
fields (see for instance [54, 131]). Transportation metrics are also the correct way to compare and regularize
variational problems that arise in image processing (see for instance the Radon inversion method proposed
in [57]) and machine learning (see [94]).

3 Research program

Since its creation, the Mokaplan team has made important contributions in Optimal Transport both on the
theoretical and the numerical side, together with applications such as fluid mechanics, the simulation biological
systems, machine learning. We have also contributed to to the field of inverse problems in signal and image
processing (super-resolution, nonconvex low rank matrix recovery). In 2022, the team was renewed with the
following research program which broadens our spectrum and addresses exciting new problems.

3.1 OT and related variational problems solvers encore et toujours

Participants:

Flavien Léger , Jean-David Benamou , Guillaume Carlier , Thomas Gallouët ,
François-Xavier Vialard , Guillaume Chazareix , Adrien Vacher , Paul Pegon.

Asymptotic analysis of entropic OT for a small entropic parameter is well understood for regular data on com-
pact manifolds and standard quadratic ground cost [88], the team will extend this study to more general
settings and also establish rigorous asymptotic estimates for the transports maps. This is important to
provide a sound theoretical background to efficient and useful debiasing approaches like Sinkhorn Diver-
gences [99]. Guillaume Carlier, Paul Pegon and Luca Tamanini are investigating speed of convergence
and quantitative stability results under general conditions on the cost (so that optimal maps may not be
continuous or even fail to exist). Some sharp bounds have already been obtained, the next challenging
goal is to extend the Laplace method to a nonsmooth setting and understand what entropic OT really
selects when there are several optimal OT plans.

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High dimensional - Curse of dimensionality We will continue to investigate the computation or approxima-
tion of high-dimensional OT losses and the associated transports [43] in particular in relation with their
use in ML. In particular for Wasserstein 2 metric but also the repulsive Density Functional theory cost
[100].

Back-and-forth The back-and-forth method [110, 109] is a state-of-the-art solver to compute optimal transport
with convex costs and 2-Wasserstein gradient flows on grids. Based on simple but new ideas it has great
potential to be useful for related problems. We plan to investigate: OT on point clouds in low dimension,
the principal-agent problem in economics and more generally optimization under convex constraints [114,
126].

Transport and diffusion The diffusion induced by the entropic regularization is fixed and now well understood.
For recent variations of the OT problem (Martingale OT, Weak OT see [49]) the diffusion becomes an
explicit constraint or the control itself [105]. The entropic regularisation of these problems can then be
understood as metric/ground cost learning [79] (see also [130]) and offers a tractable numerical method.

Wasserstein Hamiltonian systems We started to investigate the use of modern OT solvers for the SG equation
[91, 55] Semi-Discrete and entropic regularization. This is a special instance Hamiltonian Systems in the
sense of [45]. with an OT component in the Energy.

Nonlinear fourth-order diffusion equations such as thin-films or (the more involved) DLSS quantum drift
equations are WGF. Such WGF are challenging both in terms of mathematical analysis (lack of maximum
principle...) and of numerics. They are currently investigated by Jean-David Benamou, Guillaume Carlier
in collaboration with Daniel Matthes. Note also that Mokaplan already contributed to a related topic
through the TV-JKO scheme [80].

Lagrangian approaches for fluid mechanics More generally we want to extend the design and implementa-
tion of Lagrangian numerical scheme for a large class of problem coming from fluids mechanics (WHS or
WGF) using semi-discrete OT or entropic regularization. We will also take a special attention to link this
approaches with problems in machine/statistical learning. To achieve this part of the project we will join
forces with colleagues in Orsay University: Y. Brenier, H. Leclerc, Q. Mérigot, L. Nenna.

∞

L

optimal transport is a variant of OT where we want to minimize the maximal displacement of the transport
plan, instead of the average distance. Following the seminal work of [86], and more recent developments
[95], Guillaume Carlier, Paul Pegon and Luigi De Pascale are working on the description of restrictable
solutions (which are cyclically ∞-monotone) through some potential maps, in the spirit of Mange-
Kantorovich potentials provided by a duality theory. Some progress has been made to partially describe
cyclically quasi-motonone maps (related in some sense to cyclically ∞-monotone maps), through quasi-
convex potentials.

3.2 Application of OT numerics to non-variational and non convex problems

Participants:

Flavien Léger , Guillaume Carlier , Jean-David Benamou , François-
Xavier Vialard.

Market design Z-mappings form a theory of non-variational problems initiated in the ’70s but that has been
for the most part overlooked by mathematicians. We are developing a new theory of the algorithms
associated with convergent regular splitting of Z-mappings. Various well-established algorithms for
matching models can be grouped under this point of view (Sinkhorn, Gale–Shapley, Bertsekas’ auction)
and this new perspective has the potential to unlock new convergence results, rates and accelerated
methods.

Non Convex inverse problems The PhD [142] provided a first exploration of Unbalanced Sinkhorn Divergence
in this context. Given enough resources, a branch of PySit, a public domain software to test misfit

8

functions in the context of Seismic imaging will be created and will allow to test other signal processing
strategies in Full Waveform Inversion. Likewise the numerical method tested for 1D reflectors in [58]
coule be develloped further (in particular in 2D).

Equilibrium and transport Equilibrium in labor markets can often be expressed in terms of the Kantorovich
duality. In the context of urban modelling or spatial pricing, this observation can be fruitfully used to
compute equilibrium prices or densities as fixed points of operators involving OT, this was used in [51]
and [50]. Quentin Petit, Guillaume Carlier and Yves Achdou are currently developing a (non-variational)
new semi-discrete model for the structure of cities with applications to tele-working.

Non-convex Principal-Agent problems Guillaume Carlier, Xavier Dupuis, Jean-Charles Rochet and John Thanas-

soulis are developing a new saddle-point approach to non-convex multidimensional screening problems
arising in regulation (Barron-Myerson) and taxation (Mirrlees).

3.3 Inverse problems with structured priors

Participants:

Irène Waldspurger , Antonin Chambolle , Vincent Duval , Robert Tovey ,
Romain Petit .

Off-the-grid reconstruction of complex objects Whereas, very recently, some methods were proposed for
the reconstruction of curves and piecewise constant images on a continuous domain ([66] and [81]),
those are mostly proofs of concept, and there is still some work to make them competitive in real
applications. As they are much more complex than point source reconstruction methods, there is room
for improvements (parametrization, introduction of several atoms. . . ). In particular, we are currently
working on an improvement of the algorithm [66] for inverse problems in imaging which involve Optimal
Transport as a regularizer (see [135] for preliminary results). Moreover, we need to better understand their
convergence and the robustness of such methods, using sensitivity analysis.

Correctness guarantees for Burer-Monteiro methods Burer-Monteiro methods work well in practice and are
therefore widely used, but existing correctness guarantees [65] hold under unrealistic assumptions only.
In the long term, we aim at proposing new guarantees, which would be slightly weaker but would hold in
settings more relevant to practice. A first step is to understand the “average” behavior of Burer-Monteiro
methods, when applied to random problems, and could be the subject of a PhD thesis.

3.4 Geometric variational problems, and their interactions with transport

Participants:

Vincent Duval , Paul Pegon , Antonin Chambolle , Joao-Miguel Machado .

Approximation of measures with geometric constraints Optimal Transport is a powerful tool to compare and
approximate densities, but its interaction with geometric constraints is still not well understood. In
applications such as optimal design of structures, one aims at approximating an optimal pattern while
taking into account fabrication constraints [62]. In Magnetic Resonance Imaging (MRI), one tries to
sample the Fourier transform of the unknown image according to an optimal density but the acquisition
device can only proceed along curves with bounded speed and bounded curvature [116]. Our goal is
to understand how OT interacts with energy terms which involve, e.g. the length, the perimeter or the
curvature of the support... We want to understand the regularity of the solutions and to quantify the
approximation error. Moreover, we want to design numerical methods for the resolution of such problems,
with guaranteed performance.

9

Discretization of singular measures Beyond the (B)Lasso and the total variation (possibly off-the-grid), nu-
merically solving branched transportation problems requires the ability to faithfully discretize and rep-
resent 1-dimensional structures in the space. The research program of A. Chambolle consists in part in
developing the numerical analysis of variational problems involving singular measures, such as lower-
dimensional currents or free surfaces. We will explore both phase-field methods (with P. Pegon, V. Duval)
[83, 127] which easily represent non-convex problems, but lack precision, and (with V. Duval) precise
discretizations of convex problems, based either on finite elements (and relying to the FEM discrete
exterior calculus [46], cf [84] for the case of the total variation), or on finite differences and possibly a
clever design of dual constraints as studied in [89, 85] again for the total variation.

Transport problems with metric optimization In urban planning models, one looks at building a network (of
roads, metro or train lines, etc.) so as to minimize a transport cost between two distributions, penalized
by the cost for building the network, usually its length. A typical transport cost is Monge cost M Kω with a
metric ω = ωΣ which is modified as a fraction of the euclidean metric on the network Σ. We would like to
consider general problems involving a construction cost to generate a conductance field σ (having in
mind 1-dimensional integral of some function of σ), and a transport cost depending on this conductance
field. The afore-mentioned case studied in [72] falls into this category, as well as classical branched
transport. The biologically-inspired network evolution model of [106] seems to provide such an energy
in the vanishing diffusivity limit, with a cost for building a 1-dimensional permeability tensor and an
L2 congested transport cost with associated resistivity metric ; such a cost seems particularly relevant
to model urban planning. Finally, we would like to design numerical methods to solve such problems,
taking advantage of the separable structure of the whole cost.

4 Application domains

4.1 Natural Sciences

FreeForm Optics, Fluid Mechanics (Incompressible Euler, Semi-Geostrophic equations), Quantum Chemistry
(Density Functional Theory), Statistical Physics (Schroedinger problem), Porous Media.

4.2 Signal Processing and inverse problems

Full Waveform Inversion (Geophysics), Super-resolution microscopy (Biology), Satellite imaging (Meteorology)

4.3 Social Sciences

Mean-field games, spatial economics, principal-agent models, taxation, nonlinear pricing.

5 Highlights of the year

Antonin Chambolle gave a plenary talk at the International Congress on Industrial and Applied Mathematics
(ICIAM) 2023.

6 New software, platforms, open data

• Antonin Chambolle has implemented a new version of the fast and exact proximal operator of the Graph
Total variation, built upon Boykov and Kolmogorov’s maxflow-v3.04 algorithm (available on his web
page and soon on plmlab), this is used to implement efficient methods for computing crystalline mean
curvature flows.

7 New results

7.1 Entropic Optimal Transport Solutions of the Semigeostrophic Equations

10

Participants:

Jean-David Benamou, Colin Cotter, Hugo Malamut.

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

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

imization viewpoint

Participants:

Jean-David Benamou, Guillaume Carlier, Daniel Matthes.

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

7.3 Total variation regularization with Wasserstein penalization

Participants:

Antonin Chambolle, Vincent Duval, Joao-Miguel Machado.

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

7.4 1D approximation of measures in Wasserstein spaces

Participants:

Antonin Chambolle, Vincent Duval, Joao-Miguel Machado.

We propose in [33] a variational approach to approximate measures with measures uniformly distributed
over a 1 dimensional set. The problem consists in minimizing a Wasserstein distance as a data term with a
regularization given by the length of the support. As it is challenging to prove existence of solutions to this
problem, we propose a relaxed formulation, which always admits a solution. In the sequel we show that if the
ambient space is R 2 , under technical assumptions, any solution to the relaxed problem is a solution to the
original one. Finally we prove that any optimal solution to the relaxed problem, and hence also to the original,
is Ahlfors regular.

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

ization

Participants:

Yohann De Castro, Vincent Duval, Romain Petit.

11

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

7.6 A geometric Laplace method

Participants:

Flavien Léger, François-Xavier Vialard.

A classical tool for approximating integrals is the Laplace method. The first-order, as well as the higher-order
Laplace formula is most often written in coordinates without any geometrical interpretation. In [9], motivated
by a situation arising, among others, in optimal transport, we give a geometric formulation of the first-order
term of the Laplace method. The central tool is the Kim–McCann Riemannian metric which was introduced
in the field of optimal transportation. Our main result expresses the first-order term with standard geometric
objects such as volume forms, Laplacians, covariant derivatives and scalar curvatures of two different metrics
arising naturally in the Kim–McCann framework. Passing by, we give an explicitly quantified version of the
Laplace formula, as well as examples of applications.

7.7 Convergence rate of general entropic optimal transport costs

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

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

7.8 A geometric approach to apriori estimates for optimal transport maps

Participants:

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

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

12

7.9 Gradient descent with a general cost

Participants:

Flavien Leger, Pierre-Cyril Aubin.

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

7.10 Second-order methods for Burer-Monteiro factorization

Participants:

Florentin Goyens, Clément Royer, Irène Waldspurger.

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

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

Participants:

Antonin Chambolle.

In [32], is analysed a stochastic primal-dual hybrid gradient for large-scale inverse problems (with application
mostly to medical imaging), which was proposed some years ago by A. Chambolle and collaborators. The
new result describes how the parameters can be modified/updated at each iteration in a way which still
ensures (almost sure) convergence, and proposes some heuristic rules which fit into the framework and
effectively improve the rate of convergence in practical experiments. The proceeding [21], in collaboration with
U. Bordeaux, shows some convergence guarantees for a particular implementation of a “plug-and-play” image
restoration method, where the regularizer for inverse problems is based on a denoising neural network. A more
developed journal version has been submitted [39].

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

In a different direction, the proceeding [19], also with TU Graz, considers the issue of parameters learning
for a better discretization of variational regularizers allowing for singularities (the “total-generalized-variation”
of Bredies, Kunisch and Pock). A theoretical analysis of this model and of more standard total-variation
regularization models is found in the new preprint [34], which introduces a novel approach (and much simpler
than the previous ones) for studying the stability of the discontinuity sets in elementary denoising models.

7.12 Free discontinuity problems, fractures and shape optimization

Participants:

Antonin Chambolle.

13

The publications [14, 17, 16, 31] are related to “free discontinuity problems” in materials science, with application
to fracture growth or shape optimisation. In [17, 16] we discuss compactness for functionals which appear in
the variational approach to fracture, in particular [16] is a new, very general, and in some sense more natural
proof of compactness with respect to the previous results. The preprint [31] was submitted to the proceedings
of the ICIAM conference, and it contains in an appendix a relatively simple presentation (in a simpler case)
of the proof of Poincaré / Poincaré-Korn inequalities with small jump set developped in the past 10 years by
A. Chambolle.

An application to shape optimization (of an object dragged in a Stokes flow) is presented in [14], while [35,

36] address other type of shape optimization problems in a more classical framework.

7.13 Interface evolution problems

Participants:

Antonin Chambolle.

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

A different dynamics of interfaces, based on L1 gradient flow instead of L2-type, is studied in [37].

7.14 Optimal quantization via branched optimal transport distance

Participants:

Paul Pegon , Mircea Petrache.

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

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

flows

Participants:

Thomas Gallouët, Andrea Natale, Gabriele Todeschi.

We introduce a time discretization for Wasserstein gradient flows based on the classical Backward Differen-
tiation 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.

14

7.16 Regularity theory and geometry of unbalanced optimal transport

Participants:

Thomas Gallouët, Roberta Ghezzi, Francois-Xavier Vialard.

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

7.17 Entropic approximation of ∞ optimal transport problems

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

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

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

Participants: Guillaume Carlier , Alex Delalande, Quentin Mérigot.

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

7.19 Quantitative Stability of Barycenters in the Wasserstein Space

Participants: Guillaume Carlier , Alex Delalande, Quentin Mérigot.

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

15

7.20 Wasserstein medians: robustness, PDE characterization and numerics

Participants: Guillaume Carlier , Enis Chenchene, Katharina Eichinger.

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

8 Bilateral contracts and grants with industry

Participants:

Jean-David Benamou, Gregoire Loeper.

CIFRE PhD thesis scholarship (Guillaume Chazareix) with BNP. Main supervisor Jean-David Benamou,
co-supervision with Guillaume Carlier (Inria Mokaplan) and Gregoire Loeper (BNP). This contract is handled by
Dauphine University.

9 Partnerships and cooperations

9.1 International research visitors

9.1.1 Visits of international scientists
Other international visits to the team

Participants:

Pankaj Gautam.

Status Postdoc

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

Country: Norway

Dates: November 13th to November 17th

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

Mobility program/type of mobility: research stay and lecture

Participants:

Luigi De Pascale.

Status Researcher

16

Institution of origin: University of Florence

Country: Italy

Dates: May 2023

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

Mobility program/type of mobility: research stay

9.1.2 Visits to international teams

Research stays abroad

• Jean-David Benamou has visited Pr. Colin Cotter (ICL) during the fall under a Nelder (ICL) fellowship.

• Paul Pegon has visited Pr. Mircea Petrache (PUC, Chile) for two weeks in March 2023

9.2 National initiatives

PRAIRIE chair : Irène Waldspurger.

ANR CIPRESSI (2019-2024) is a JCJC grant (149k€) carried by Vincent Duval. Its aim is to develop off-the-grid

methods for inverse problems involving the reconstruction of complex objects.

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

10 Dissemination

10.1 Promoting scientific activities

10.1.1 Scientific events: organisation

• Jean-David Benamou co-organized a Workshop on Numerical Optimal Transport at FOCM conference,

Paris June.

• Jean-David Benamou co-organized an Ecole des Houches conference on Optimal Transport Theory :

Applications to Physics, March.

• Guillaume Carlier co-organized a workshop at Lagrange Center.

• Vincent Duval was the main organizer of the Workshop on Off-the-Grid methods for Optimization and
Inverse Problems in Imaging (≈ 60 participants) at Institut Henri Poincaré (November 21st and 22nd).

Member of the organizing committees

• Guillaume Carlier co-organized the séminaire Parisien d’Optimisation.

• Vincent Duval co-organized the Julia and Optimization Days 2023 (≈ 140 participants) at Conservatoire

National des Arts et Métiers (October 4th, 5th and 6th).

• Vincent Duval co-organizes the monthly seminar Imaging in Paris.

• Antonin Chambolle co-organizes the monthly seminar "Séminaire Parisien d’Optimisation"

• Paul Pegon co-organizes the regular seminar of Calculus of Variations GT CalVa (until June 2023)

17

10.1.2 Scientific events: selection

Reviewer

• Vincent Duval has reviewed contributions to the GRETSI and SSVM conferences.

• Antonin Chambolle has reviewed contributions for SSVM and AISTATS 24

10.1.3 Journal

Member of the editorial boards

• Vincent Duval is associate editor at the Journal of Mathematical Imaging and Vision (JMIV)

• Antonin Chambolle is associate editor at:

– Inverse Problems and Imaging (AIMS)

– Journal of Mathematical Imaging and Vision (JMIV, Springer)

– IPOL (Image Processing On Line),

– ESAIM : Mathematical Modelling and Numerical Analysis(M2AN, Cambridge UP),

– Applied Mathematics and Optimization (AMO, Springer),

– IMA journal of numerical analysis (Oxford)

– Journal of the European Math. Society (JEMS, EMS Publishing).

He is also one of the 4 editors of “Interfaces and Free Boundaries”.

• Irène Waldspurger is associate editor for the IEEE Transactions on Signal Processing.

• Guillaume Carlier is associate editor at:

– Journal de l’Ecole Polytechnique

– Applied Mathematics an Optimization

– J. Math. Analysis and Appl.

– J. Dynamics and Games

Reviewer - reviewing activities

• Vincent Duval has reviewed contributions to the journal of Foundations of Computational Mathematics
(FoCM), Journal of NonSmooth Analysis and Optimization (JNSAO) and SIAM Journal on Imaging Sciences
(SIIMS). He has also written reviews for the AMS: Mathematical Reviews.

• Antonin Chambolle is a reviewer for many journals including Arch. Rational Mech. Anal., Journal of the

European Math. Society, Math. Programming, Calc. Var. PDE., etc.

• Irène Waldspurger has reviewed contributions to the Compte-rendus de l’académie des sciences (CRAS),
to Applied and Computational Harmonic Analysis (ACHA), to Foundations of Computational Mathematics
(FoCM) and to the Journal de mathématiques pures et appliquées (JMPA).

• Paul Pegon has reviewed contributions to ESAIM: Mathematical Modelling and Numerical Analysis

(M2AN) and ESAIM: Control, Optimisation and Calculus of Variations (COCV).

• Flavien Leger has reviewed contributions to Journal of Convex Analysis and Advances in Mathematics.

• Thomas Gallouët has reviewed contributions to Analysis & PDE and other journals.

18

10.1.4 Invited talks

• Vincent Duval gave invited talks at the SIGOPT conference (Cottbus, Germany), Applied Inverse Problems
(AIP) conference (Göttingen, Germany) and the colloquium 30 years of mathematics for optical imaging
in Marseille. He was also invited at the local seminar "Modélisation, Analyse et Calcul Scientifique" in
Université de Lyon.

• Antonin Chambolle was an invited speaker at ICIAM 2023 (Tokyo) (August 2023), and an invited speaker at
the 3rd Alps-Adriatic Inverse Problems workshop, July 2023. He was also invited speaker at the conference
in honor of the 60th birthday of Prof. Luigi Ambrosio, ETH, Zurich, Sept 2023.

• Flavien Leger gave invited talks at the "Machine Learning and Signal Processing seminar" (ENS Lyon),
the "Geometric Analysis Seminar" (Iowa State University), the "Analysis and Applied Mathematics Sem-
inar Series" (Bocconi University), the "PGMO Days" (EDF Lab, Saclay) and the "Séminaire Parisien
d’Optimisation" (Institut Henri Poincaré).

• Irène Waldspurger gave an invited talk at the conference for Stéphane Mallat’s 60th birthday, "A multiscale

tour of harmonic analysis and machine learning".

• Paul Pegon gave invited talks at the "Analysis and Geometry seminar" of the Pontifical Catholic University
of Chile (Santiago, Chile), at the June 2023 meeting of the "GdR Calva" (Université Paris-Cité) and the
"PGMO Days" (EDF Lab, Saclay).

• Guillaume Carlier gave talks at the GT transport optimal, Orsay; the Analysis and Applied Mathematics
seminar, Bocconi, Milan; The Financial math. seminar, ETH Zurich; PGMO Days, session transport
optimal; workshop Optimal Transport and the Calculus of Variations, ICMS Edinburgh.

• Thomas Gallouët gave talks at the worshop on Optimal Transport and the Calculus of Varia- tions,
Edinburgh, Scotland. The FoCM workshop on Opptimal Transport in Paris, France. The conference
on Optimal Transport Theory And Applications to Physics, Centre de physique des Houches, France.
A seminar for the journée de rentrée of the ANEDP team of the Laboratoire de Mathématique d’Orsay,
Université de Paris- Saclay, Orsay.

10.1.5 Scientific expertise

• Vincent Duval was a member of the selection committee for a Chaire de Professeur Junior (CPJ) at

Sorbonne Université.

• Antonin Chambolle was the head of the hiring comittee for a "maitre de conférence" in nonlinear analysis

at Univ. Paris-Dauphine.

• Irène Waldspurger was a member of the selection committee for a "maître de conférence" position at

Université Côte d’Azur.

• Thomas Gallouët was the vice-president of the hiring comittee for a "maitre de conférence" in nonlinear

analysis at Univ. Paris-Dauphine.

10.1.6 Research administration

• Antonin Chambolle represents France in the IFIP TC7 group system modeling and optimization,

• Antonin Chambolle is member of the scientific comittee and of the board of the PGMO (programme

Gaspard Monge pour l’Optimisation et la Recherche Opérationnelle) (FMJH - EDF).

• Irène Waldspurger is a member of the SMAI-MODE group.

• Vincent Duval was a member of the Comité de suivi doctoral (CSD) until June, 2023.

• Vincent Duval was a member of the Commission Emplois Scientifiques (CES) 2023 of the Paris research

center.

• Vincent Duval has been "Délégué Scientifique Adjoint" (DSA) since September, 1st.

19

10.2 Teaching - Supervision - Juries

10.2.1 Teaching

• Jean-David Benamou has given a serie of Lectures on Optimal Transport at Imperial College London.

October and November.

• Master: Antonin Chambolle Optimisation Continue, 24h, niveau M2, Université Paris Dauphine-PSL, FR

• Master : Vincent Duval, Problèmes Inverses, 22,5 h équivalent TD, niveau M1, Université PSL/Mines

ParisTech, FR

• Master : Vincent Duval, Optimization for Machine Learning, 9h, niveau M2, Université PSL/ENS, FR

• Licence : Irène Waldspurger, Pré-rentrée raisonnement, 31,2 h équivalent TD, niveau L1, Université

Paris-Dauphine, FR

• Master : Irène Waldspurger, Optimization for Machine Learning, 9h, niveau M2, Université PSL/ENS, FR

• Master : Irène Waldspurger, Introduction à la géométrie différentielle et aux équations différentielles,

29,25 h équivalent TD, niveau M1, Université Paris Dauphine, FR

• Master : Irène Waldspurger, Non-convex inverse problems, 27 h d’équivalent TD, niveau M2, Université

Paris Dauphine, FR

• Licence : Guillaume Carlier, algebre 1, L1 78h, Dauphine, FR

• Master : Guillaume Carlier Variational and transport methods in economics, M2 Masef, 27h, Dauphine,

FR

• Agregation : Thomas Gallouët, Optimisation, Analyse numérique, 48h équivalent TD, niveau M2, Uni-

versité d’Orsay), FR

• Guillaume Carlier: Licence Algèbre 1, Dauphine 70h, M2 Masef: Variatioanl and transport problems in

economics, 18h

• Flavien Léger: Graduate course, two lectures in ‘math+econ+code’ masterclass on equilibrium transport

and matching models in economics, NYU Paris. 5h.

10.2.2 Supervision

• PhD in progress: Joao-Miguel Machado, Transport optimal et structures géométriques, 01/10/2021,

Co-supervised by Vincent Duval and Antonin Chambolle.

• PhD in progress : Chazareix Guillaume 1/08/2021, Non Linear Parabolic equations and Volatility Calibra-

tion. Co-supervised by Jean-David Benamou and Grégoire Loeper.

• PhD in progress : Hugo Malamut 1/09/2022, Régularisation Entropique et Transport Optimal Généralisé.

Co-supervised by Jean-David Benamou and Guillaume Carlier.

• PhD in progress : Maxime Sylvestre 01/09/2002 on Hybrid methods fot Optimal Transport. Supervised by

Guillaume Carlier and Alfred Galichon.

• PhD: Faniriana Rakoto Endor started a PhD on Burer-Monteiro methods, under the supervision of

Antonin Chambolle and Irène Waldspurger.

• Postdoc: Adrien Vacher started a postdoc in December 2023 under the supervision of Flavien Léger.

• Phd in progress: Erwan Stämplfi, 01/09/2021, on singular limit for multiphase flow, co supervised with

Yann Brenier

• Phd in progress: Siwan Boufadene, 01/09/2022, on energy distance flow, co supervised with Francois-

Xavier Vialard

20

10.2.3 Juries

• Jean-David Benamou participated to the PhD Juries of Rodrigue Lelotte, Adrien Seguret.

• Vincent Duval was a referee for the PhD manuscripts of Adrien Frigerio (Université de Dijon, September)

and Thu-Le Tran (Université de Rennes, December)

• Vincent Duval was an examiner in the PhD committee of Bastien Laville (Inria Sophia-Antipolis, Septem-

ber).

• Irène Waldspurger was an examiner in the PhD committees of Adrien Vacher (Université Gustave Eiffel)

and Gaspar Rochette (ENS Paris).

• Guillaume Carlier was a member of the PhD comittee of Raphael Prunier (Sorbonne Université); Quentin

Jacquet (Ecole Polytechnique).

• Guillaume Carlier was a member of the HdR comittee of Thibaut Le Gouic (Université de Aix-Marseille).

11 Scientific production

11.1 Major publications

[1] P.-C. Aubin-Frankowski, A. Korba and F. Léger. ‘Mirror Descent with Relative Smoothness in Measure
Spaces, with application to Sinkhorn and EM’. In: NeurIPS 2022 - Thirty-sixth Conference on Neural
Information Processing Systems. New Orleans, United States, 2022. URL: https://hal.science/hal-
03811583.

[2]

[3]

J.-D. Benamou, G. Carlier, M. Cuturi, L. Nenna and G. Peyré. ‘Iterative Bregman Projections for Regular-
ized Transportation Problems’. In: SIAM Journal on Scientific Computing 2.37 (2015), A1111–A1138. DOI:
10.1137/141000439. URL: https://hal.science/hal-01096124.

J.-D. Benamou, T. Gallouët and F.-X. Vialard. ‘Second order models for optimal transport and cubic
splines on the Wasserstein space’. In: Foundations of Computational Mathematics (Oct. 2019). DOI:
10.1007/s10208-019-09425-z. URL: https://hal.science/hal-01682107.

[4] C. Boyer, A. Chambolle, Y. de Castro, V. Duval, F. de Gournay and P. Weiss. ‘On Representer Theorems
and Convex Regularization’. In: SIAM Journal on Optimization 29.2 (9th May 2019), pp. 1260–1281. DOI:
10.1137/18M1200750. URL: https://hal.archives-ouvertes.fr/hal-01823135.

[5] C. Cancès, T. Gallouët and G. Todeschi. ‘A variational finite volume scheme for Wasserstein gradient
flows’. In: Numerische Mathematik 146.3 (2020), pp 437 –480. DOI: 10.1007/s00211-020-01153-9.
URL: https://hal.science/hal-02189050.

[6] G. Carlier, V. Duval, G. Peyré and B. Schmitzer. ‘Convergence of Entropic Schemes for Optimal Transport
and Gradient Flows’. In: SIAM Journal on Mathematical Analysis 49.2 (18th Apr. 2017). DOI: 10.1137/15
M1050264. URL: https://hal.science/hal-01246086.

[7] G. Carlier, P. Pegon and L. Tamanini. Convergence rate of general entropic optimal transport costs. 7th June

2022. URL: https://hal.archives-ouvertes.fr/hal-03689945.

[8] F. Léger and P.-C. Aubin-Frankowski. Gradient descent with a general cost. 14th Dec. 2023. URL: https:

//hal.science/hal-04344054.

[9] F. Léger and F.-X. Vialard. A geometric Laplace method. 22nd Dec. 2022. URL: https://hal.science/h

al-03911149.

[10]

[11]

I. Waldspurger. ‘Phase retrieval with random Gaussian sensing vectors by alternating projections’. In:
IEEE Transactions on Information Theory 64.5 (2018), pp. 3301–3312. URL: https://hal.science/hal
-01645081.

I. Waldspurger and A. Waters. ‘Rank optimality for the Burer-Monteiro factorization’. In: SIAM Journal
on Optimization 30.3 (2020), pp. 2577–2602. DOI: 10.1137/19M1255318. URL: https://hal.science
/hal-01958814.

21

11.2 Publications of the year

International journals

[12] Y. Achdou, G. Carlier, Q. Petit and D. Tonon. ‘A mean field model for the interactions between firms on
the markets of their inputs’. In: Mathematics and Financial Economics (Mar. 2023). DOI: 10.1007/s115
79-023-00333-z. URL: https://hal.science/hal-03720158.

[13] A. Braides and A. Chambolle. ‘Ising systems, measures on the sphere, and zonoids’. In: Tunisian Journal

of Mathematics (2023). URL: https://hal.science/hal-04320040.

[14] D. Bucur, A. Chambolle, A. Giacomini and M. Nahon. ‘A free discontinuity approach to optimal profiles
in stokes flows’. In: Annales de l’Institut Henri Poincaré C, Analyse non linéaire (2023). URL: https://cn
rs.hal.science/hal-03920819.

[15] G. Carlier, P. Pegon and L. Tamanini. ‘Convergence rate of general entropic optimal transport costs’. In:
Calculus of Variations and Partial Differential Equations 62.4 (May 2023), p. 116. DOI: 10.1007/s00526-
023-02455-0. URL: https://hal.science/hal-03689945.

[16] A. Chambolle and V. Crismale. ‘A general compactness theorem in <i>G(S)BD</i>’. In: Indiana University

Mathematics Journal (2023). URL: https://hal.science/hal-03807668.

[17] A. Chambolle and V. Crismale. ‘Equilibrium Configurations For Nonhomogeneous Linearly Elastic
Materials With Surface Discontinuities’. In: Annali della Scuola Normale Superiore di Pisa, Classe di
Scienze XXIV.3 (29th Sept. 2023), pp. 1575–1610. DOI: 10.2422/2036-2145.202006_002. URL: https:
//hal.science/hal-02667936.

[18] A. Chambolle, D. de Gennaro and M. Morini. ‘Minimizing Movements for Anisotropic and Inhomogen-
eous Mean Curvature Flows’. In: Advances in Calculus of Variation (2023). DOI: 10.1515/acv-2022-01
02. URL: https://hal.science/hal-03894146.

International peer-reviewed conferences

[19] L. Bogensperger, A. Chambolle, A. Effland and T. Pock. ‘Learned Discretization Schemes for the Second-
Order Total Generalized Variation’. In: Lecture Notes in Computer Science. 9th International Conference
on Scale Space and Variational Methods in Computer Vision. SSVM 2023. Vol. 14009. Lecture Notes in
Computer Science. Santa Margherita di Pula (Cagliari), Italy: Springer International Publishing, 10th May
2023, pp. 484–497. DOI: 10.1007/978-3-031-31975-4_37. URL: https://hal.science/hal-04239
970.

[20] A. Chambolle, V. Duval and J. M. Machado. ‘The Total Variation-Wasserstein Problem: a new derivation
of the Euler-Lagrange equations’. In: Geometric Science of Information. GSI 2023. Geometric Science
of Information. GSI 2023. Vol. 14071. Lecture Notes in Computer Science. Saint Malo (FR), France:
Springer Nature Switzerland, 1st June 2023, pp. 610–619. DOI: 10.1007/978-3-031-38271-0_61. URL:
https://hal.science/hal-04113284.

[21]

S. Hurault, A. Chambolle, A. Leclaire and N. Papadakis. ‘A relaxed proximal gradient descent algorithm
for convergent plug-and-play with proximal denoiser’. In: International Conference on Scale Space
and Variational Methods in Computer Vision (SSVM’23). Vol. 14009. LNCS - Lecture Notes in Computer
Science. Santa Margherita di Pula, Italy: Springer, 21st May 2023. DOI: 10.1007/978-3-031-31975-4
_29. URL: https://hal.science/hal-03967018.

[22] M. Zach, T. Pock, E. Kobler and A. Chambolle. ‘Explicit Diffusion of Gaussian Mixture Model Based
Image Priors’. In: Lecture Notes in Computer Science. 9th International Conference on Scale Space and
Variational Methods in Computer Vision, SSVM 2023. Vol. 14009. Scale Space and Variational Methods
in Computer Vision. SSVM 2023. Santa Margherita di Pula (Cagliari), Italy: Springer International Pub-
lishing, 10th May 2023, pp. 3–15. DOI: 10.1007/978-3-031-31975-4_1. URL: https://hal.science
/hal-04239963.

22

National peer-reviewed Conferences

[23] Y. de Castro, V. Duval and R. Petit. ‘Régularisation par la variation totale pour l’identification du support
d’images constantes par morceaux’. In: GRETSI 2023 - XXIXème Colloque Francophone de Traitement
du Signal et des Images. Grenoble, France, 28th Aug. 2023. URL: https://inria.hal.science/hal-0
4171535.

Reports & preprints

[24] L. Arnould, C. Boyer and E. Scornet. Is interpolation benign for random forest regression? 9th Feb. 2023.

URL: https://hal.science/hal-03560047.

[25]

J.-D. Benamou, C. Cotter and H. Malamut. Entropic Optimal Transport Solutions of the Semigeostrophic
Equations. 28th Jan. 2023. URL: https://hal.science/hal-03960859.

[26] S. Boufadène and F.-X. Vialard. On the global convergence of Wasserstein gradient flow of the Coulomb

discrepancy. 20th Nov. 2023. URL: https://hal.science/hal-04282762.

[27] G. Carlier, J.-D. Benamou and D. Matthes. Wasserstein gradient flow of the Fisher information from a
non-smooth convex minimization viewpoint. 8th Aug. 2023. URL: https://inria.hal.science/hal-
04178416.

[28] G. Carlier, E. Chenchene and K. Eichinger. Wasserstein medians: robustness, PDE characterization and

numerics. 6th July 2023. URL: https://hal.science/hal-04154602.

[29] G. Carlier, A. Delalande and Q. Merigot. Quantitative Stability of Barycenters in the Wasserstein Space.

8th Mar. 2023. URL: https://hal.science/hal-03781835.

[30] Y. de Castro, V. Duval and R. Petit. Exact recovery of the support of piecewise constant images via total
variation regularization. 10th July 2023. URL: https://inria.hal.science/hal-04171496.

[31] A. Chambolle. Existence of minimizers in the variational approach to brittle fracture. 10th Nov. 2023. URL:

https://hal.science/hal-04279405.

[32] A. Chambolle, C. Delplancke, M. J. Ehrhardt, C.-B. Schönlieb and J. Tang. Stochastic Primal Dual Hybrid
Gradient Algorithm with Adaptive Step-Sizes. 6th Jan. 2023. URL: https://hal.science/hal-039276
44.

[33] A. Chambolle, V. Duval and J. M. Machado. 1D approximation of measures in Wasserstein spaces. 26th Apr.

2023. URL: https://hal.science/hal-04082932.

[34] A. Chambolle and M. Łasica. Inclusion and estimates for the jumps of minimizers in variational denoising.

5th Dec. 2023. URL: https://hal.science/hal-04323807.

[35] A. Chambolle, I. Mazari-Fouquer and Y. Privat. Stability of optimal shapes and convergence of thresholding
algorithms in linear and spectral optimal control problems. 24th June 2023. URL: https://hal.scienc
e/hal-04140177.

[36] A. Chambolle, I. Mazari-Fouquer and Y. Privat. Stability of optimal shapes and convergence of thresholding
algorithms in linear and spectral optimal control problems: Supplementary material. 25th June 2023.
URL: https://hal.science/hal-04140334.

[37] A. Chambolle and M. Novaga. L1-Gradient Flow of Convex Functionals. 12th Oct. 2023. URL: https://h

al.science/hal-03805962.

[38] T. Gallouët, A. Natale and G. Todeschi. From geodesic extrapolation to a variational BDF2 scheme for

Wasserstein gradient flows. 18th Oct. 2023. URL: https://hal.science/hal-03790981.

[39]

S. Hurault, A. Chambolle, A. Leclaire and N. Papadakis. Convergent plug-and-play with proximal denoiser
and unconstrained regularization parameter. 2nd Nov. 2023. URL: https://hal.science/hal-04269
033.

[40] L. Nenna and P. Pegon. Convergence rate of entropy-regularized multi-marginal optimal transport costs.

6th July 2023. URL: https://hal.science/hal-04154453.

[41] P. Pegon and M. Petrache. Optimal Quantization with Branched Optimal Transport distances. 15th Sept.

2023. URL: https://hal.science/hal-04208973.

23

[42] A. Sportisse, M. Marbac, C. Biernacki, C. Boyer, G. Celeux, J. Josse and F. Laporte. Model-based Clustering
with Missing Not At Random Data. 10th Feb. 2023. URL: https://hal.science/hal-03494674.

[43] A. Vacher and F.-X. Vialard. Parameter tuning and model selection in optimal transport with semi-dual

Brenier formulation. 22nd Jan. 2023. URL: https://hal.science/hal-03475455.

11.3 Cited publications

[44] M. Agueh and G. Carlier. ‘Barycenters in the Wasserstein space’. In: SIAM J. Math. Anal. 43.2 (2011),

pp. 904–924.

[45] L. Ambrosio and W. Gangbo. ‘Hamiltonian ODEs in the Wasserstein space of probability measures’.
In: Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of
Mathematical Sciences 61.1 (2008), pp. 18–53.

[46] D. N. Arnold, R. S. Falk and R. Winther. ‘Finite element exterior calculus, homological techniques, and

applications’. In: Acta Numerica 15 (2006), pp. 1–155. DOI: 10.1017/S0962492906210018.

[47] F. R. Bach. ‘Consistency of the Group Lasso and Multiple Kernel Learning’. In: J. Mach. Learn. Res. 9 (June

2008), pp. 1179–1225. URL: http://dl.acm.org/citation.cfm?id=1390681.1390721.

[48] F. R. Bach. ‘Consistency of Trace Norm Minimization’. In: J. Mach. Learn. Res. 9 (June 2008), pp. 1019–

1048. URL: http://dl.acm.org/citation.cfm?id=1390681.1390716.

[49]

J. D. Backhoff-Veraguas and G. Pammer. Applications of weak transport theory. 2020. arXiv: 2003.05338
[math.PR].

[50] X. Bacon, G. G. Carlier and B. Nazaret. ‘A spatial Pareto exchange economy problem’. working paper or

preprint. Dec. 2021. URL: https://hal.science/hal-03480323.

[51] C. Barilla, G. Carlier and J.-M. Lasry. ‘A mean field game model for the evolution of cities’. In: Journal of

Dynamics and Games (2021). URL: https://hal.science/hal-03086616.

[52] M. F. Beg, M. I. Miller, A. Trouvé and L. Younes. ‘Computing Large Deformation Metric Mappings via
Geodesic Flows of Diffeomorphisms’. In: International Journal of Computer Vision 61.2 (Feb. 2005),
pp. 139–157. URL: http://dx.doi.org/10.1023/B:VISI.0000043755.93987.aa.

[53] M. Beiglbock, P. Henry-Labordère and F. Penkner. ‘Model-independent bounds for option prices mass
transport approach’. English. In: Finance and Stochastics 17.3 (2013), pp. 477–501. DOI: 10.1007/s0078
0-013-0205-8. URL: http://dx.doi.org/10.1007/s00780-013-0205-8.

[54]

[55]

[56]

[57]

[58]

[59]

[60]

[61]

J.-D. Benamou and Y. Brenier. ‘A computational fluid mechanics solution to the Monge-Kantorovich
mass transfer problem’. In: Numer. Math. 84.3 (2000), pp. 375–393. DOI: 10.1007/s002110050002. URL:
http://dx.doi.org/10.1007/s002110050002.

J.-D. Benamou and Y. Brenier. ‘Weak existence for the semigeostrophic equations formulated as a
coupled Monge-Ampère/transport problem’. In: SIAM J. Appl. Math. 58.5 (1998), pp. 1450–1461.

J.-D. Benamou and G. Carlier. ‘Augmented Lagrangian algorithms for variational problems with diver-
gence constraints’. In: JOTA (2015).

J.-D. Benamou, G. Carlier, M. Cuturi, L. Nenna and G. Peyré. ‘Iterative Bregman Projections for Regular-
ized Transportation Problems’. In: SIAM J. Sci. Comp. (2015). to appear.

J.-D. Benamou, G. Chazareix, G. Rukhaia and W. L. Ijzerman. ‘Point Source Regularization of the Finite
Source Reflector Problem’. In: Journal of Computational Physics (May 2022). URL: https://inria.hal
.science/hal-03344571.

J.-D. Benamou, B. D. Froese and A. Oberman. ‘Two numerical methods for the elliptic Monge-Ampère
equation’. In: M2AN Math. Model. Numer. Anal. 44.4 (2010), pp. 737–758.

J.-D. Benamou, B. D. Froese and A. Oberman. ‘Numerical solution of the optimal transportation problem
using the Monge–Ampere equation’. In: Journal of Computational Physics 260 (2014), pp. 107–126.

J. Bigot and T. Klein. ‘Consistent estimation of a population barycenter in the Wasserstein space’. In:
Preprint arXiv:1212.2562 (2012).

24

[62] M. Boissier, G. Allaire and C. Tournier. ‘Additive manufacturing scanning paths optimization using
shape optimization tools’. In: Struct. Multidiscip. Optim. 61.6 (2020), pp. 2437–2466. DOI: 10.1007/s00
158-020-02614-3. URL: https://doi.org/10.1007/s00158-020-02614-3.

[63] N. Bonneel, J. Rabin, G. Peyré and H. Pfister. ‘Sliced and Radon Wasserstein Barycenters of Measures’.
In: Journal of Mathematical Imaging and Vision 51.1 (2015), pp. 22–45. URL: http://hal.archives-o
uvertes.fr/hal-00881872/.

[64] G. Bouchitté and G. Buttazzo. ‘Characterization of optimal shapes and masses through Monge-Kantorovich

equation’. In: J. Eur. Math. Soc. (JEMS) 3.2 (2001), pp. 139–168. DOI: 10.1007/s100970000027. URL:
http://dx.doi.org/10.1007/s100970000027.

[65] N. Boumal, V. Voroninski and A. S. Bandeira. ‘Deterministic guarantees for Burer-Monteiro factorizations

of smooth semidefinite programs’. In: preprint (2018). https://arxiv.org/abs/1804.02008.

[66] K. Bredies, M. Carioni, S. Fanzon and F. Romero. ‘A generalized conditional gradient method for dynamic

inverse problems with optimal transport regularization’. In: arXiv preprint arXiv:2012.11706 (2020).

[67] Y. Brenier. ‘Décomposition polaire et réarrangement monotone des champs de vecteurs’. In: C. R. Acad.

Sci. Paris Sér. I Math. 305.19 (1987), pp. 805–808.

[68] Y. Brenier. ‘Generalized solutions and hydrostatic approximation of the Euler equations’. In: Phys. D
237.14-17 (2008), pp. 1982–1988. DOI: 10.1016/j.physd.2008.02.026. URL: http://dx.doi.org/1
0.1016/j.physd.2008.02.026.

[69] Y. Brenier. ‘Polar factorization and monotone rearrangement of vector-valued functions’. In: Comm.
Pure Appl. Math. 44.4 (1991), pp. 375–417. DOI: 10.1002/cpa.3160440402. URL: http://dx.doi.org
/10.1002/cpa.3160440402.

[70] Y. Brenier, U. Frisch, M. Henon, G. Loeper, S. Matarrese, R. Mohayaee and A. Sobolevski. ‘Reconstruction
of the early universe as a convex optimization problem’. In: Mon. Not. Roy. Astron. Soc. 346 (2003),
pp. 501–524. URL: http://arxiv.org/pdf/astro-ph/0304214.pdf.

[71] M. Burger and S. Osher. ‘A guide to the TV zoo’. In: Level-Set and PDE-based Reconstruction Methods,

Springer (2013).

[72] G. Buttazzo, A. Pratelli, E. Stepanov and S. Solimini. Optimal Urban Networks via Mass Transportation.
Vol. 1961. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. DOI:
10.1007/978-3-540-85799-0. URL: http://link.springer.com/10.1007/978-3-540-85799-0
(visited on 11/01/2022).

[73] L. A. Caffarelli. ‘The regularity of mappings with a convex potential’. In: J. Amer. Math. Soc. 5.1 (1992),

pp. 99–104. DOI: 10.2307/2152752. URL: http://dx.doi.org/10.2307/2152752.

[74] L. A. Caffarelli, S. A. Kochengin and V. Oliker. ‘On the numerical solution of the problem of reflector
design with given far-field scattering data’. In: Monge Ampère equation: applications to geometry and
optimization (Deerfield Beach, FL, 1997). Vol. 226. Contemp. Math. Providence, RI: Amer. Math. Soc., 1999,
pp. 13–32. DOI: 10.1090/conm/226/03233. URL: http://dx.doi.org/10.1090/conm/226/03233.

[75] C. CanCeritoglu. ‘Computational Analysis of LDDMM for Brain Mapping’. In: Frontiers in Neuroscience 7

(2013).

[76] E. Candes and M. Wakin. ‘An Introduction to Compressive Sensing’. In: IEEE Signal Processing Magazine

25.2 (2008), pp. 21–30.

[77] E. J. Candès and C. Fernandez-Granda. ‘Super-Resolution from Noisy Data’. In: Journal of Fourier

Analysis and Applications 19.6 (2013), pp. 1229–1254.

[78] E. J. Candès and C. Fernandez-Granda. ‘Towards a Mathematical Theory of Super-Resolution’. In:

Communications on Pure and Applied Mathematics 67.6 (2014), pp. 906–956.

[79] G. Carlier, A. Dupuy, A. Galichon and Y. Sun. ‘SISTA: Learning Optimal Transport Costs under Sparsity
Constraints’. working paper or preprint. Oct. 2020. URL: https://hal.science/hal-03504045.

[80] G. Carlier and C. Poon. ‘On the total variation Wasserstein gradient flow and the TV-JKO scheme’. In:
ESAIM: Control, Optimisation and Calculus of Variations (2019). URL: https://hal.science/hal-01
492343.

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[81] Y. de Castro, V. Duval and R. Petit. ‘Towards Off-the-grid Algorithms for Total Variation Regularized
Inverse Problems’. In: Journal of Mathematical Imaging and Vision (July 2022). DOI: 10.1007/s10851-
022-01115-w. URL: https://inria.hal.science/hal-03406710.

[82] F. A. C. C. Chalub, P. A. Markowich, B. Perthame and C. Schmeiser. ‘Kinetic models for chemotaxis and
their drift-diffusion limits’. In: Monatsh. Math. 142.1-2 (2004), pp. 123–141. DOI: 10.1007/s00605-004-
0234-7. URL: http://dx.doi.org/10.1007/s00605-004-0234-7.

[83] A. Chambolle, L. A. D. Ferrari and B. Merlet. ‘Variational approximation of size-mass energies for k-
dimensional currents’. In: ESAIM Control Optim. Calc. Var. 25 (2019), Paper No. 43, 39. DOI: 10.1051/co
cv/2018027. URL: https://doi.org/10.1051/cocv/2018027.

[84] A. Chambolle and T. Pock. ‘Crouzeix-Raviart approximation of the total variation on simplicial meshes’.
In: J. Math. Imaging Vision 62.6-7 (2020), pp. 872–899. DOI: 10.1007/s10851- 019- 00939- 3. URL:
https://doi.org/10.1007/s10851-019-00939-3.

[85] A. Chambolle and T. Pock. ‘Learning consistent discretizations of the total variation’. In: SIAM J. Imaging
Sci. 14.2 (2021), pp. 778–813. DOI: 10.1137/20M1377199. URL: https://doi.org/10.1137/20M1377
199.

[86] T. Champion, L. De Pascale and P. Juutinen. ‘The ∞-Wasserstein Distance: Local Solutions and Existence
of Optimal Transport Maps’. In: SIAM Journal on Mathematical Analysis 40.1 (1st Jan. 2008), pp. 1–20.
DOI: 10.1137/07069938X. URL: https://epubs.siam.org/doi/10.1137/07069938X (visited on
12/01/2022).

[87]

S. S. Chen, D. L. Donoho and M. A. Saunders. ‘Atomic decomposition by basis pursuit’. In: SIAM journal
on scientific computing 20.1 (1999), pp. 33–61.

[88] L. Chizat, P. Roussillon, F. Léger, F.-X. Vialard and G. Peyré. ‘Faster Wasserstein Distance Estimation with
the Sinkhorn Divergence’. In: Neural Information Processing Systems. Advances in Neural Information
Processing Systems. Vancouver, Canada, Dec. 2020. URL: https://hal.archives-ouvertes.fr/hal
-02867271.

[89] L. Condat. ‘Discrete total variation: new definition and minimization’. In: SIAM J. Imaging Sci. 10.3

(2017), pp. 1258–1290. URL: https://doi.org/10.1137/16M1075247.

[90] C. Cotar, G. Friesecke and C. Kluppelberg. ‘Density Functional Theory and Optimal Transportation with
Coulomb Cost’. In: Communications on Pure and Applied Mathematics 66.4 (2013), pp. 548–599. DOI:
10.1002/cpa.21437. URL: http://dx.doi.org/10.1002/cpa.21437.

[91] M. J. P. Cullen. A Mathematical Theory of Large-Scale Atmosphere/Ocean Flow. Imperial College Press,

2006. URL: https://books.google.fr/books?id=JxBqDQAAQBAJ.

[92] M. J. P. Cullen, W. Gangbo and G. Pisante. ‘The semigeostrophic equations discretized in reference and
dual variables’. In: Arch. Ration. Mech. Anal. 185.2 (2007), pp. 341–363. DOI: 10.1007/s00205-006-004
0-6. URL: http://dx.doi.org/10.1007/s00205-006-0040-6.

[93] M. J. P. Cullen, J. Norbury and R. J. Purser. ‘Generalised Lagrangian solutions for atmospheric and

oceanic flows’. In: SIAM J. Appl. Math. 51.1 (1991), pp. 20–31.

[94] M. Cuturi. ‘Sinkhorn Distances: Lightspeed Computation of Optimal Transport’. In: Proc. NIPS. Ed. by

C. J. C. Burges, L. Bottou, Z. Ghahramani and K. Q. Weinberger. 2013, pp. 2292–2300.

[95] L. De Pascale and J. Louet. ‘A Study of the Dual Problem of the One-Dimensional L

-Optimal Transport
Problem with Applications’. In: Journal of Functional Analysis 276.11 (1st June 2019), pp. 3304–3324. DOI:
10.1016/j.jfa.2019.02.014. URL: https://www.sciencedirect.com/science/article/pii
/S0022123619300643 (visited on 12/01/2022).

∞

[96] E. J. Dean and R. Glowinski. ‘Numerical methods for fully nonlinear elliptic equations of the Monge-

Ampère type’. In: Comput. Methods Appl. Mech. Engrg. 195.13-16 (2006), pp. 1344–1386.

[97] V. Duval and G. Peyré. ‘Exact Support Recovery for Sparse Spikes Deconvolution’. English. In: Foundations
of Computational Mathematics (2014), pp. 1–41. DOI: 10.1007/s10208-014-9228-6. URL: http://dx
.doi.org/10.1007/s10208-014-9228-6.

[98] C. Fernandez-Granda. ‘Support detection in super-resolution’. In: Proc. Proceedings of the 10th Interna-

tional Conference on Sampling Theory and Applications (2013), pp. 145–148.

26

[99]

J. Feydy, T. Séjourné, F.-X. Vialard, S.-I. Amari, A. Trouvé and G. Peyré. ‘Interpolating between Optimal
Transport and MMD using Sinkhorn Divergences’. working paper or preprint. Oct. 2018. URL: https:
//hal.science/hal-01898858.

[100] G. Friesecke and D. Vögler. ‘Breaking the Curse of Dimension in Multi-Marginal Kantorovich Optimal
Transport on Finite State Spaces’. In: SIAM Journal on Mathematical Analysis 50.4 (2018), pp. 3996–4019.
DOI: 10.1137/17M1150025. eprint: https://doi.org/10.1137/17M1150025. URL: https://doi.o
rg/10.1137/17M1150025.

[101] U. Frisch, S. Matarrese, R. Mohayaee and A. Sobolevski. ‘Monge-Ampère-Kantorovitch (MAK) recon-

struction of the eary universe’. In: Nature 417.260 (2002).

[102] A. Galichon, P. Henry-Labordère and N. Touzi. ‘A stochastic control approach to No-Arbitrage bounds
given marginals, with an application to Loopback options’. In: submitted to Annals of Applied Probability
(2011).

[103] W. Gangbo and R. McCann. ‘The geometry of optimal transportation’. In: Acta Math. 177.2 (1996),
pp. 113–161. DOI: 10.1007/BF02392620. URL: http://dx.doi.org/10.1007/BF02392620.

[104] E. Ghys. ‘Gaspard Monge, Le mémoire sur les déblais et les remblais’. In: Image des mathématiques,

CNRS (2012). URL: http://images.math.cnrs.fr/Gaspard-Monge,1094.html.

[105]

[106]

I. Guo and G. Loeper. ‘Path Dependent Optimal Transport and Model Calibration on Exotic Derivatives’.
In: SSRN Electron. J. (Jan. 2018). Available at doi:10.2139/ssrn.3302384. DOI: 10.2139/ssrn.3302384.

J. Haskovec, P. Markowich, B. Perthame and M. Schlottbom. ‘Notes on a PDE System for Biological
Network Formation’. In: Nonlinear Analysis. Nonlinear Partial Differential Equations, in Honor of Juan
Luis Vázquez for His 70th Birthday 138 (1st June 2016), pp. 127–155. DOI: 10.1016/j.na.2015.12.018.
URL: https://www.sciencedirect.com/science/article/pii/S0362546X15004344 (visited on
10/06/2021).

[107] D. D. Holm, J. T. Ratnanather, A. Trouvé and L. Younes. ‘Soliton dynamics in computational anatomy’.

In: NeuroImage 23 (2004), S170–S178.

[108] B. J. Hoskins. ‘The mathematical theory of frontogenesis’. In: Annual review of fluid mechanics, Vol. 14.

Palo Alto, CA: Annual Reviews, 1982, pp. 131–151.

[109] M. Jacobs, W. Lee and F. Léger. ‘The back-and-forth method for Wasserstein gradient flows’. In: ESAIM:

Control, Optimisation and Calculus of Variations 27 (2021), p. 28.

[110] M. Jacobs and F. Léger. ‘A fast approach to optimal transport: The back-and-forth method’. In: Numer.
Math. 146 (2020), pp. 513–544. DOI: 10.1007/s00211-020-01154-8. URL: https://doi.org/10.10
07/s00211-020-01154-8.

[111] W. Jäger and S. Luckhaus. ‘On explosions of solutions to a system of partial differential equations
modelling chemotaxis’. In: Trans. Amer. Math. Soc. 329.2 (1992), pp. 819–824. DOI: 10.2307/2153966.
URL: http://dx.doi.org/10.2307/2153966.

[112] R. Jordan, D. Kinderlehrer and F. Otto. ‘The variational formulation of the Fokker-Planck equation’. In:

SIAM J. Math. Anal. 29.1 (1998), pp. 1–17.

[113] L. Kantorovitch. ‘On the translocation of masses’. In: C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942),

pp. 199–201.

[114] T. Lachand-Robert and E. Oudet. ‘Minimizing within convex bodies using a convex hull method’. In:
SIAM Journal on Optimization 16.2 (Jan. 2005), pp. 368–379. URL: https://hal.archives-ouvertes
.fr/hal-00385109.

[115]

J.-M. Lasry and P.-L. Lions. ‘Mean field games’. In: Jpn. J. Math. 2.1 (2007), pp. 229–260. DOI: 10.1007/s
11537-007-0657-8. URL: http://dx.doi.org/10.1007/s11537-007-0657-8.

[116] L. Lebrat, F. de Gournay, J. Kahn and P. Weiss. ‘Optimal Transport Approximation of 2-Dimensional
Measures’. en. In: SIAM Journal on Imaging Sciences 12.2 (Jan. 2019), pp. 762–787. DOI: 10.1137/18M11
93736. URL: https://epubs.siam.org/doi/10.1137/18M1193736 (visited on 25/03/2021).

[117] C. Léonard. ‘A survey of the Schrödinger problem and some of its connections with optimal transport’.
In: Discrete Contin. Dyn. Syst. 34.4 (2014), pp. 1533–1574. DOI: 10.3934/dcds.2014.34.1533. URL:
http://dx.doi.org/10.3934/dcds.2014.34.1533.

27

[118] A. S. Lewis. ‘Active sets, nonsmoothness, and sensitivity’. In: SIAM Journal on Optimization 13.3 (2003),

pp. 702–725.

[119] B. Li, F. Habbal and M. Ortiz. ‘Optimal transportation meshfree approximation schemes for Fluid and

plastic Flows’. In: Int. J. Numer. Meth. Engng 83:1541–579 83 (2010), pp. 1541–1579.

[120] G. Loeper. ‘A fully nonlinear version of the incompressible Euler equations: the semigeostrophic system’.

In: SIAM J. Math. Anal. 38.3 (2006), 795–823 (electronic).

[121] G. Loeper and F. Rapetti. ‘Numerical solution of the Monge-Ampére equation by a Newton’s algorithm’.

In: C. R. Math. Acad. Sci. Paris 340.4 (2005), pp. 319–324.

[122]

S. G. Mallat. A wavelet tour of signal processing. Third. Elsevier/Academic Press, Amsterdam, 2009.

[123] B. Maury, A. Roudneff-Chupin and F. Santambrogio. ‘A macroscopic crowd motion model of gradient
flow type’. In: Math. Models Methods Appl. Sci. 20.10 (2010), pp. 1787–1821. DOI: 10.1142/S021820251
0004799. URL: http://dx.doi.org/10.1142/S0218202510004799.

[124] Q. Mérigot. ‘A multiscale approach to optimal transport’. In: Computer Graphics Forum 30.5 (2011),

pp. 1583–1592.

[125] M. I. Miller, A. Trouvé and L. Younes. ‘Geodesic Shooting for Computational Anatomy’. In: Journal of
Mathematical Imaging and Vision 24.2 (Mar. 2006), pp. 209–228. URL: http://dx.doi.org/10.1007
/s10851-005-3624-0.

[126]

J.-M. Mirebeau. ‘Adaptive, Anisotropic and Hierarchical cones of Discrete Convex functions’. In: Nu-
merische Mathematik 132.4 (2016). 35 pages, 11 figures. (Second version fixes a small bug in Lemma 3.2.
Modifications are anecdotic.), pp. 807–853. URL: https://hal.archives-ouvertes.fr/hal-00943
096.

[127] E. Oudet and F. Santambrogio. ‘A Modica-Mortola Approximation for Branched Transport and Ap-
plications’. en. In: Archive for Rational Mechanics and Analysis 201.1 (July 2011), pp. 115–142. DOI:
10.1007/s00205-011-0402-6. URL: http://link.springer.com/10.1007/s00205-011-0402-6
(visited on 06/01/2022).

[128] B. Pass. ‘Uniqueness and Monge Solutions in the Multimarginal Optimal Transportation Problem’. In:

SIAM Journal on Mathematical Analysis 43.6 (2011), pp. 2758–2775.

[129] B. Pass and N. Ghoussoub. ‘Optimal transport: From moving soil to same-sex marriage’. In: CMS Notes

45 (2013), pp. 14–15.

[130] F.-P. Paty and M. Cuturi. Regularized Optimal Transport is Ground Cost Adversarial. 2020. arXiv: 2002.0

3967 [stat.ML].

[131] H. Raguet, J. Fadili and G. Peyré. ‘A Generalized Forward-Backward Splitting’. In: SIAM Journal on
Imaging Sciences 6.3 (2013), pp. 1199–1226. DOI: 10.1137/120872802. URL: http://hal.archives-o
uvertes.fr/hal-00613637/.

[132] L. Rudin, S. Osher and E. Fatemi. ‘Nonlinear total variation based noise removal algorithms’. In: Physica
D: Nonlinear Phenomena 60.1 (1992), pp. 259–268. URL: http://dx.doi.org/10.1016/0167-2789(9
2)90242-F.

[133]

J. Solomon, F. de Goes, G. Peyré, M. Cuturi, A. Butscher, A. Nguyen, T. Du and L. Guibas. ‘Convolutional
Wasserstein Distances: Efficient Optimal Transportation on Geometric Domains’. In: ACM Transaction
on Graphics, Proc. SIGGRAPH’15 (2015). to appear.

[134] R. Tibshirani. ‘Regression shrinkage and selection via the Lasso’. In: Journal of the Royal Statistical

Society. Series B. Methodological 58.1 (1996), pp. 267–288.

[135] R. Tovey and V. Duval. ‘Dynamical Programming for off-the-grid dynamic Inverse Problems’. working

paper or preprint. Dec. 2022. URL: https://inria.hal.science/hal-03500048.

[136] S. Vaiter, M. Golbabaee, J. Fadili and G. Peyré. ‘Model Selection with Piecewise Regular Gauges’. In:
Information and Inference (2015). to appear. URL: http://hal.archives-ouvertes.fr/hal-00842
603/.

[137] C. Villani. Optimal transport. Vol. 338. Grundlehren der Mathematischen Wissenschaften [Fundamental
Principles of Mathematical Sciences]. Old and new. Berlin: Springer-Verlag, 2009, pp. xxii+973. DOI:
10.1007/978-3-540-71050-9. URL: http://dx.doi.org/10.1007/978-3-540-71050-9.

28

[138] C. Villani. Topics in optimal transportation. Vol. 58. Graduate Studies in Mathematics. American Math-

ematical Society, Providence, RI, 2003, pp. xvi+370.

[139] X.-J. Wang. ‘On the design of a reflector antenna. II’. In: Calc. Var. Partial Differential Equations 20.3
(2004), pp. 329–341. DOI: 10.1007/s00526-003-0239-4. URL: http://dx.doi.org/10.1007/s0052
6-003-0239-4.

[140] B. Wirth, L. Bar, M. Rumpf and G. Sapiro. ‘A continuum mechanical approach to geodesics in shape

space’. In: International Journal of Computer Vision 93.3 (2011), pp. 293–318.

[141]

J. Wright, Y. Ma, J. Mairal, G. Sapiro, T. S. Huang and S. Yan. ‘Sparse representation for computer vision
and pattern recognition’. In: Proceedings of the IEEE 98.6 (2010), pp. 1031–1044.

[142] M. Yu. ‘Entropic Unbalanced Optimal Transport: Application to Full-Waveform Inversion and Numerical

Illustration’. Theses. Université de Paris, Dec. 2021. URL: https://hal.inria.fr/tel-03512143.

29
<scratchpad>
- Main topic: Research Centre MOKAPLAN (Optimal Transport and Variational Problems)
- Key points: 
  - Development of novel models and numerical solvers for optimal transport and variational problems
  - Applications in various domains, including fluid mechanics, quantum chemistry, and machine learning
  - Focus on non-convex and non-smooth optimization problems
  - Research on numerical methods, including the use of Sinkhorn divergence and entropic regularization
- Structure: The document is divided into several sections, including activity report, research program, application domains, and dissemination.
- Potential artifacts to ignore: 
  - Web-related elements, such as links and HTML tags
  - Unnecessary information, such as dates and funding information
</scratchpad>

<final_summary>
The MOKAPLAN research centre focuses on the development of novel models and numerical solvers for optimal transport and variational problems. The centre has a broad range of applications, including fluid mechanics, quantum chemistry, and machine learning. Researchers are particularly interested in non-convex and non-smooth optimization problems, and are developing new numerical methods to address these challenges. The centre's work has the potential to impact a wide range of fields, from physics and chemistry to computer science and engineering.
The MOKAPLAN research centre focuses on the development of novel models and numerical solvers for optimal transport and variational problems. The centre has a broad range of applications, including fluid mechanics, quantum chemistry, and machine learning. Researchers are particularly interested in non-convex and non-smooth optimization problems, and are developing new numerical methods to address these challenges. The centre's work has the potential to impact a wide range of fields, from physics and chemistry to computer science and engineering.