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.

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.

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.

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

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

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.

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

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.

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.

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

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

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

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

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

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

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

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

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.

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

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.

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

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.

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.

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