Abstracts

The purpose of this elementary work is to study the behavior of the Sinkhorn algorithm - the most powerful tool to compute solutions of the Schrödinger problem - in the case when the latter has no solution. Putting aside the question of integrability of the data, this can typically happen when the support of the reference measure is far from being a Cartesian product. Our main result is that in this context, the algorithm admits exactly two limit points, each of them being the solution of a Schrödinger problem with modified data, themselves characterized as solutions of entropic optimization problems. In addition, the geometric mean of these two limit points is the solution to a relaxed problem, where the marginal constraints are replaced by marginal penalizations. We were motivated by an application of the Schrödinger problem to the analysis of single cell data, where the underlying reference process is a jump process whose jumps correspond to RNA bursts, and hence are nonnegative on each coordinate. This is a joint work with Elias Ventre.

I will present the Semi-Geostrophic equations, a model designed to understanding mid-latitude atmospheric front formations. I will recall how Optimal Transport theory provide a framework for its mathematical analysis and numerical simulations. I will then describe a recent work investigating the use of Entropic Optimal Transport in this framework. Finally I will raise questions on the possibility to develop a diffusive Semi-Geostrophic model.

We consider a class of control problems where we study the minimization of cost in the pathwise sense. We derive the associated Bellman's dynamical programming principle in the pathwise sense and show that the corresponding Hamilton-Jacobi-Bellman equation is well-posed in the class of stochastic viscosity solutions. Moreover, we give a characterization of the drift of the control problem.

As well known, the Schroedinger problem does not exactly correspond to the Schroedinger equation of Quantum Mechanics. I will explain how the Schroedinger equation nevertheless admits a variational principle in terms of a complex version of the Schroedinger problem. Connections with the theory of random matrices will also be addressed.

Coupling methods provide with a powerful toolbox to quantify the long time behaviour of diffusion processes. In particular, it has been recently shown that coupling by reflection allows to prove exponential convergence results in Wasserstein distance without having to impose pointwise convexity assumptions on the drift field. The purpose of this talk is to detail the construction of a variant of coupling by reflection that applies to controlled diffusion processes and yields uniform in time gradient (and Hessian) estimates for the solution of the corresponding Hamilton-Jacobi-Bellman equations. In turn, these estimates are shown to be key to prove various kind of exponential turnpike properties for the optimal processes and optimal processes.

First we recall the link between Incompressible Euler Equation and Optimal transport (throughout a riemannien submersion) and some of the results that can be deduced using this geometrical link, for instance Brenier generalized geodesics, Polar projection, Lagrangian numerical scheme. We them aim to explain why the couple [Camassa-Holm equation//H^{\div}/ Unbalanced Optimal transport] share the same geometrical structure and discuss which of the previous results can be extended in this case. In particular a definition of a pression for the Camassa-Holm equation naturally arises in this framework.

Here, we introduce a price-formation model where a large number of small players can store and trade a commodity such as electricity. Our model is a constrained mean-field game (MFG) where the price is a Lagrange multiplier for the supply vs. demand balance condition. This model can also be regarded as a constrained optimal transport problem, where the center of mass of the probability distribution is given. We establish the existence of a unique solution using a fixed-point argument. In particular, we show that the price is well-defined, and it is a Lipschitz function of time. Then, we examine the case where the supply is stochastic and we study linear-quadratic models. Finally, we discuss a recurrent neural network method to numerically approximate these problems.

Stochastic parametrisations of the interactions among disparate scales of motion in fluid convection are often used for estimating prediction uncertainty, which can arise due to inadequate model resolution, or incomplete observations, especially in dealing with atmosphere and ocean dynamics, where viscous and diffusive dissipation effects are negligible. This talk discusses a family of three different types of stochastic parameterisations for the classical Euler-Boussinesq (EBC) equations for buoyant ideal incompressible fluid flow under gravity in a vertical plane.
Based on a joint work with Wei Pan, preprint is available at arXiv:2205.09461

In this talk, we follow the framework of stochastic geometric mechanics inspired by Schrödinger's problem, to investigate the stochastic integrability. We first introduce stochastic flows generated by second-order elliptic operators and their stochastic characteristics. Then we use second-order symplectic and contact geometry to solve nonlinear second-order elliptic and parabolic PDEs, by interpreting stochastic Hamiltonian systems and stochastic contact Hamiltonian systems as characteristic equations. We prove a stochastic Kelvin's circulation theorem, which leads to the second-order Poincaré-Cartan integral invariant and relative integral invariant. Finally, we also study complete solutions of the Hamilton-Jacobi-Bellman equation via stochastic conserved quantities. This is a joint work with J.-C. Zambrini.

In 1966 V.Arnold suggested a group-theoretic approach to ideal hydrodynamics via the geodesic flow of the right-invariant energy metric on the group of volume-preserving diffeomorphisms of the flow domain. In the talk we discuss geodesic and Hamiltonian frameworks to include fluid flows with vortex sheets. It turns out that the corresponding dynamics is related to a certain groupoid of pairs of volume-preserving diffeomorphisms with common interface. We also discuss some ramifications of this groupoid framework to the Euler-Arnold equations. This is a joint work with A.Izosimov.

I will present a non-random and random models for wave turbulence in their relation with the random and non-random models for the water turbulence. Then I’ll discuss the recent progress in their study.

In a seminal article published in 1986, extending Schrödinger's results, Jean-Claude Zambrini obtained the equations of motion of the entropic interpolation between two marginal measures in a Brownian heat bath with a scalar potential. These equations were expressed in terms of a stochastic Newton equation. Recently, Giovanni Conforti recasted them in terms of a Newton equation on the Otto-Wasserstein space. These results were obtained under strong regularity conditions. We extend them relaxing these regularity assumptions as much as possible. This is a joint work with Giovanni Conforti.

I will focus on recent developments concerning the structure of conjugate points along geodesics in the group of volume-preserving diffeomorphisms (incompressible geometry). As in the classical riemannian case these are singular values of the associated exponential map. In this case the riemannian structure on the diffeomorphism group (viewed as a configuration space of an ideal fluid) comes from the kinetic energy. Time permitting, I will comment on the compressible case and some relations to optimal transport.

In this talk I discuss how quantization theory and complexification give rise to a finite-dimensional model of optimal transport on compact Kähler manifolds, particularly the sphere. This is joint work with Erik Jansson.

We present a stochastic branching algorithm for the numerical solution of fully nonlinear parabolic PDEs with arbitrary gradient nonlinearities. This algorithm extends the classical Feynman-Kac formula by propagating information on nonlinearities on the branches of random trees. The algorithm is implemented using neural networks and applied to the numerical solution of the Navier-Stokes equation. It also provides numerical solutions in fully nonlinear examples that are not treated by backward stochastic differential equations or the classical Galerkin method.
Joint work with Guillaume Penent and Jiang Yu Nguwi

G. Penent and N. Privault. A fully nonlinear Feynman-Kac formula with derivatives of arbitrary orders. ArXiv:2201.03882, 2022.
J.Y. Nguwi, G. Penent and N. Privault. A deep branching solver for fully nonlinear partial differential equations. ArXiv:2203.03234, 2022.

The Euler-Poincaré approach to the description of continuum mechanics will be presented in a general framework, useful for both complex fluids and elastic materials. Emphasis will be put on complex fluids. Then, topological conservation laws will be introduced based on the theory of differential character valued momentum maps.

We revisit unbalanced optimal transport in the conic formulation from Liero/Mielke/Savaré and state an associated Schrödinger problem on the cone over the state space. The latter can be cast in the general framework of entropic regularization of linear programming. In particular, the classical Daroch-Ratcliff iterative scaling algorithm applies, which generalizes the Sinkhorn algorithm in the balanced case. The talk is based on joint work with Simon Telen, Bernd Sturmfels and François-Xavier Vialard

We will discuss solvability of some equations of motion of inextensible networks. These problems can be expressed as systems of PDE that involve unknown Lagrange multipliers and non-standard boundary conditions related to the freely moving junctions. They can also be formally interpreted as a gradient flows on a certain submanifold of the Otto-Wasserstein space of probability measures. A link to fluid dynamics will be also explained. This is a joint work with Ayk Telciyan.