Discretizing the Fokker-Planck equation with second-order accuracy: a dissipation driven approach

• Clément Cancès (Inria)

Room: room 112 We propose a fully discrete finite volume scheme for the standard Fokker-Planck equation. The space discretization relies on the well-known square-root approximation, which falls into the framework of two-point flux approximations. Our time discretization is novel and relies on a tailored nonlinear mid-point rule, designed to accurately capture the dissipative structure of the model. We establish well-posedness for the scheme, positivity of the solutions, as well as a fully discrete energy-dissipation inequality mimicking the continuous one. We then prove the rigorous convergence of the scheme under mildly restrictive conditions on the unstructured grids, which can be easily satisfied in practice. Numerical simulations show that our scheme is second order accurate both in time and space, and that one can solve the discrete nonlinear systems arising at each time step using Newton's method with low computational cost.

Lagrangian scheme for nonlinear Fokker-Planck-type equations

• Bertram Düring

Room: room 112 Many nonlinear diffusion equations can be interpreted as gradient flows whose dynamics are driven by internal energies and given external potentials, examples include the heat equation and the porous medium equation. When solving these equations numerically, schemes that respect the equations’ special structure are of particular interest. In this talk we present a Lagrangian scheme for nonlinear diffusion equations. For discretisation of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. We present numerical experiments for the porous medium equation in two space dimensions.

Vector valued optimal transport: from dynamic to Kantorovich formulations

• Katy Craig

Room: room 112 Motivated by applications in multispecies PDE and classification of vector valued measures, we develop a unified theory that connects four existing notions of vector valued optimal transport. We prove a sharp inequality relating the four notions, showing they are bi-Holder equivalent, and compare and contrast the properties of each metric from the perspective of gradients flows and linearization.

The Riemannian geometry of Sinkhorn divergences

• Bernhard Schmitzer

Room: room 112 Optimal transport provides an intuitive and robust way to compare probability measures with applications in many areas of mathematics. This holds in particular for the Wasserstein-2 distance with its formal Riemannian structure. While entropic regularization of optimal transport has several favourable effects, such as improved statistical sample complexity, it destroys this metric structure. The de-biased Sinkhorn divergence is a partial remedy, as it is positive, definite, and its sublevel sets induce the weak* topology. However, it does not satisfy the triangle inequality. We resolve this issue by considering the Hessian of the Sinkhorn divergence as a Riemannian tensor and study the induced distance. In this talk we outline the key steps of this construction, the corresponding induced notion of tangent space, some early results on the distance, and open directions for future work.

Remarks on JKO steps for the Fisher information

• Guillaume Carlier

Room: room 112 It is by now well-known that the quantum drift diffusion equation which is a highly nonlinear fourth-order evolution equation is the Wasserstein gradient flow of the Fisher information. It is therefore natural to investigate whether JKO steps are tractable both from an analytical and computational viewpoint. In this talk, based on a joint work with Daniel Matthes and Jean-David Benamou, I will discuss some aspects of this problem.

Particle discretization of Wasserstein gradient flows

• Quentin Merigot

Room: room 112 In this talk, I will present an approach to particle-based discretizations of Wasserstein gradient flows based on the Moreau-Yosida regularization of the underlying energies. This approach allows to approximate some evolution PDEs (such as Fokker-Planck, porous media or crowd motion models) with interacting particle systems, the interaction being through a mesh which is canonically associated to each point cloud by the regularized energy. These schemes are numerically appealing, but their numerical analysis seems difficult. One of the reason is that the driving ODEs have spurious stationary points, which do not correspond to Wasserstein critical points of the energy of the continuous gradient flow. I will nonetheless mention some convergence results, and explain proof techniques.

From inverse optimal transport to global trade

• Marie-Therese Wolfram

Room: room Fokko-du-Cloux In this talk I will focus on two challenging problems in applied optimal transport: inferring unknown cost functions in noisy optimal transport plans and leveraging deep learning to infer trading barriers in international commodity trade. We start by discussing the classic optimal transportation problems studied by Gaspard Monge and Leonid Kantorovich, before focusing on the respective inverse problem, so-called inverse optimal transport. Hereby we wish to infer the underlying transportation cost from solutions that are corrupted by noise. Then we generalize this approach to identify transport costs in global food and agricultural trade. Our analysis reveals that he global South suffered disproportionately from the war in Ukraine's impact on wheat markets. Additionally, it examines the effects of free-trade agreements, trade disputes with China, and Brexit's impact on British-European trade, uncovering hidden patterns not evident from trade volumes alone.

Diffusion: not just discretized, but also quantized

• Daniel Matthes

Room: room Fokko-du-Cloux Around 2005, Degond, Mehats and Ringhofer proposed a novel approach for the derivation of quantum fluid models (known as QHD, QET etc) from first principles: they apply a moment method to the quantum Boltzmann equation, using a BGK collision operator for moment closure. In the simplest case, the resulting fluid model is the non-local quantum drift diffusion equation nlQDD. It turns out that this equation is formally the Wasserstein gradient flow of the relative von Neumann entropy. In our (so far unsuccessful) attempt to understand the appearance of nlQDD's gradient flow structure, we have re-done the derivation by moment closure consistently on the level of spatial discretization, and we are able to replicate (albeit still not understand) the variational form in the discrete setting, along with essentially all the relevant estimates. We can further pass to the continuous limit, thereby generalizing en passant the so far only result on existence of weak solutions from close-to-equilibrium to large data. Finally, I will discuss a discretized DLSS equation that arises in the semi-classical expansion of nlQDD, and even bears two gradient flow structures: the Wasserstein one and another, second order one. This second gradient flow structure is more deeply analyzed in the presentation of Andre Schlichting. This is mainly joint work with Eva-Maria Rott; the DLSS part is a joint work with Andre Schlichting and Giuseppe Savare.

Derivation of the fourth order DLSS equation with nonlinear mobility via chemical reactions

• André Schlichting

Room: room Fokko-du-Cloux We provide a derivation of the fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider on the discretized circle the rate equation for the process where pairs of particles sitting on the same side jump simultaneously to the two neighboring sites, and the reverse jump where a pair of particles sitting on a common site jump simultaneously to the side in the middle. Depending on the rates, in the vanishing mesh size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. We identify the limiting gradient structure to be driven by entropy with respect to a generalization of the diffusive transport type with nonlinear mobility via EDP convergence. Furthermore, the DLSS equation with nonlinear mobility of the power type shares qualitative similarities with the fast diffusion and porous medium equations, since we find traveling wave solutions with algebraic tails and polynomial compact support, respectively. Joint work with Alexander Mielke and Artur Stephan. The pure DLSS part is with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré.

Finding Wasserstein saddle points "without" optimal transport

• Rafael Bailo

Room: room Fokko-du-Cloux In this talk, we will present a numerical scheme to approximate the saddle points of a Wasserstein gradient flow. Our approach is based on known techniques for Hilbert spaces and is derived from a formal JKO scheme. Unlike geodesic approaches, ours does not require solving any optimal transport problems. We will showcase the performance of the method and validate it over several examples. This is work in collaboration with Jeroen Wapstra (TU Eindhoven).

Hilbertian degenerations and inhibition models

• Bertrand Maury

Room: room Fokko-du-Cloux Gradient flows in Hilbert spaces are conditioned by the choice of an underlying metric so that a given functional may give rise to infinitely many gradient flows. We shall investigate in this talk the possible limit behaviors of the associated evolution problems when the metric degenerates, by considering sequences of metrics which are « less and less equivalent » to the canonical one. In particular, we shall be interested in the possibility to incorporate inhibition behaviors in crowd motion models, by attributing a (infinitely) larger mass to individuals considered as prioritary according to some criteria, like simply being the closest to some common objective. We will show how this asymptotic approach transforms a crowd model on the gradient flow type into a cascade of differential inclusions. In a more exploratory part, we shall presents some attempts to apply this strategy to the setting of Wasserstein gradient flows, by having the L2 like norm on the « tangent space » degenerate.