Mean field-games were introduced by Lasry and Lions to capture the nature of Nash-equilibria in stochastic differential games as the number of (indistinguishable) players tends to infinity. In certain situations they lead to a forward/backward system coupling an Hamilton-Jacobi-Bellman equation with a Fokker-Planck equation which is the optimality system for the optimal control of a Fokker-Planck equation. In this talk, I will explain how these problems, when the Hamiltonian is quadratic, may be viewed as an entropy minimization problem which has its roots in the Schrödinger bridge problem considered by Schrödinger himself. I will review some aspects of the Schrödinger bridge theory as well as its connections with optimal transport. One interest of the entropic viewpoint is that it opens the way to fast algorithms building upon the Sinkhorn scaling algorithm. Joint work with J.-D. Benamou, S. Di Marino and L. Nenna
We consider a phase field model of Cahn-Hilliard type for incompressible two-phase flows. The phase fluxes are proportional to the phase chemical potentials rather than to the generalised chemical potential as in the classical degenerate Cahn-Hilliard model. Our model can be interpreted as a constrained Wasserstein gradient flow. The existence of solution is established thanks to the convergence of the minimising movement scheme.
Blood flow simulation can provide an efficient tool to clinicians in the phase of diagnosis. The distribution of the blood flow and pressure into the arteries help to identify the location of ischemia caused by stenosis and to quantify the severity of a lesion through the fractional flow reserve (FFR) measure. In this work, we will give a 2D reproduction of the flow in a coronary tree, based on a Non-Newtonian flow model. The domain of simulation corresponds to a patient-specific stenotic left coronary artery, extracted from 2D angiography using image segmentation techniques. We try also to give some simulations in 3D. The values of the coronary pressure will be used to estimate the fractional flow reserve (FFR) in order to contribute to the clinical decision making.
This talk aims at illustrating some methods for the asymptotical analysis of optimal control problems. We use examples in the context of groundwater pollution. The spatio-temporal objective takes into account the economic trade off between the pollutant use –for instance fertilizer– and the cleaning costs. It is constrained by a hydrogeological PDEs model for the spread of the pollution in the aquifer. We rigorously derive, by asymptotic analysis, the effective optimal control problem for contaminant species that are slightly concentrated in the aquifer. On the other hand, the mathematical analysis of the optimal control problems is performed and we prove in particular that the latter effective problem is well-posed. Furthermore, a stability property of the optimal control process is provided: any optimal solution of the properly scaled problem tends to the optimal solution of the effective problem as the characteristic pollutant concentration decreases. Finally we give some results in game theory.
Nonlinear parabolic equations, which degenerate into elliptic, hyperbolic or ordinary differential equations, arise in many problems of hydrodynamics. This talk is focused on one specific example resulting from the modeling of two-phase flow in porous media. The system of equations describing the two-phase filtration consist of an elliptic equation for the pressure field and a degenerate nonlinear parabolic equation for the saturation of a given phases. In addition, since natural porous media are highly heterogeneous, the coefficients and the parameters describing the nonlinearities are defined as the functions discontinuous with respect to space. The combination of heterogeneities and nonlinearities in the equations leads to the discontinuous solutions, which have to be accurately captured by the numerical method. We will look at some of the numerical discretization schemes allowing for an accurate representation of a discontinuous solution. In addition we will discuss some methods for solving the resulting systems of nonlinear algebraic equations.
Recently, there is high interest in adapting and solving PDEs on data which is given by arbitrary graphs and networks. The demand for such methods is motivated by existing and potential future applications, such as in machine learning and mathematical image processing. Indeed, any kind of data can be represented by a graph in an abstract form in which the vertices are associated to the data and the edges correspond to relationships within the data. In this talk we will explore some of theses PDEs, their connections with non local continuous counterpart and their applications in image and point cloud processing.
The non-local p-Laplacian evolution equation and variational regularization, governed by a given kernel, have applications in various areas of science and engineering. In particular, there are modern tools for massive data processing (including signals, images, geometry), and machine learning tasks such as classification. In practice, however, these models are implemented in discrete form (in space and time, or in space for variational regularization) as a numerical approximation to a continuous problem, where the kernel is replaced by an adjacency matrix of graph. Yet few results on the consistency of these discretizations are available. In particular it is largely open to determine when do the solutions of either the evolution equation or the variational problem of graph-based tasks converge (in an appropriate topology), as the number of vertices increases, to a well-defined object in the continuum setting, and if yes, at which rate. In this work, we lay the foundations to address these questions. Combining tools from graph theory, convex analysis, non-linear semigroup theory and evolution equations, we give a rigorous interpretation to the continuous limit of the discrete non-local p-Laplacian evolution and variational problems on graphs. More specifically, we consider a sequence of (deterministic) graphs converging to a so-called limit object known as the graphon. If the continuous p-Laplacian evolution and variational problems are properly discretized on this graph sequence, we prove that the solutions of the sequence of discrete problems converge to the solution of the continuous problem governed by the graphon, when the number of graph vertices grows to infinity. Along the way, we provide a consistency/error estimate which is of interest in its own. In turn, this allows to establish the convergence rates for different graph models. In particular, we point out the role of the graphon geometry/regularity. For random graph sequences, using sharp deviation inequalities, we deliver non-asymptotic convergence rates in probability and exhibit the different regimes depending on p and the regularity of the graphon and the initial condition.
Comparing probability distributions is a fundamental problem in data sciences. Simple norms and divergences, such as the total variation and the relative entropy, only compare densities in a point-wise manner and fail to capture the geometric nature of the problem. Maximum Mean Discrepancies (MMD, Euclidean norms defined through a kernel) and Optimal Transport costs (OT) are the two main classes of distances between measures that provide some geometrical understanding, metrizing the convergence in law. In this talk, I will present the Sinkhorn divergences, which is a new family of geometric divergences that interpolates between MMD and OT. These divergences rely on a new notion of metric entropy, which ensures its positivity and metrization of the convergence in law. On the practical side, these divergences can be computed on large scale problem thanks to Sinkhorn's algorithm, and finds numerous application in imaging and machine learning. This is a joint work with Jean Feydy , Thibault S ejourn é, Francois-Xavier Vialard, Shun-ichi Amari and Alain Trouvé.
The talk concerns new results on the metric character of the Hamilton-Jacobi equation with obstacle (HJO). We will introduce an explicit formula of game theory type for the intrinsic distance which shows how the Hamilton-Jacobi equation sorts out some kind of least worst strategy to handle the obstacle. Some applications concerning the shapes of an overflowing sandpile and a brimful lake over an arbitrary landscape of varying height will be given to motivate the results. The talk aims also to address some open questions and new directions for the study of the corresponding dynamic using gradient flow in a metric space and optimal mass transportation.
We will discuss properties of solutions to aggregation-diffusion models appearing in many biological models such as cell adhesion, organogenesis and pattern formation. We will concentrate on typical behaviours encountered in systems of these equations assuming different interactions between species under a global volume constraint. In the case of reaction-diffusion systems, a splitting schemes based on optimal transportation ideas can be used to show global existence of solutions with seggregation behavior.
The classical Lojasiewicz inequality and its extensions by Simon and Kurdyka have been a considerable impact on the analysis of the large time behaviour of gradient flow in Hilbert spaces. Our aim is to adapt the classical Kurdyka- Lojasiewicz and Lojasiewicz-Simon inequalities to the general framework gradi- ent flow in metric spaces. We show that the validity of a Kurdyka- Lojasiewicz inequality imply trend to equilibrium in the metric sense, and the Kurdyka- Lojasiewicz inequality has the advantage to derive decay estimates of the trend to equilibrium and finite time of extinction. Also we study the relation be- tween Kurdyka- Lojasiewicz inequality and the existence of talweg. The entropy method have proved to be very useful to study the large time behaviour of solutions to many EDP’s. This method is based in the entropy-entropy pro- duction/disipation (EEP) inequality, which correspond to Kurdyka- Lojasiewicz inequality, and also in the entropy transportation (ET) inequality. We show that for geodesically convex functionals Kurdyka- Lojasiewicz inequality and entropy transportation (ET) inequality are equivalent. We apply our general results to gradient flow in Banach spaces and in spaces of probability measures with Wasserstein distances. For the energy functional associated with a doubly- nonlinear equations on R N we obtain the equivalence between Lojasiewicz- Simon inequality, generalized log-Sobolev inequality and p-Talagrand inequality; also we get decay estimates for its solutions. Finally we apply our results to metric spaces with Ricci curvature bounds from below, getting that, in this con- text, a p-Talagrand inequality is equivalent to a Lojasiewicz-Simon inequality. Joint work with Daniel Hauer (Sydney University)
We study the heat flow on metric random walk spaces and give different characterizations of egodicity. Under positivity of Ollivier-Ricci curvature or Bakry-Emery curvature condition we show that a Poincarés inequality and a transport-information inequality hold true.
We approximate the regular solutions of the incompressible Euler equation by flows of ODEs taking values in finite-dimentional spaces. This approach à la Brenier relies on the one hand on Arnold's interpretation of the Euler equation as geodesics in the space of measure-preserving diffeomorphisms, and on the other hand on the semi-discrete Optimal Transport. This approach is naturally associated with a numerical scheme, which will be shown to converge towards regular solutions of the incompressible Euler equation. This approach also works with fluid structure interactions. Collaborations with Quentin Mérigot, Bruno Lévy and Erica Schwindt.