Institut de recherche mathématique avancée

Résumé : In this presentation, I will introduce a hybrid finite volume method on general polygonal and polyhedral meshes for the modeling of an anisotropic cross-diffusion system arising from a mesoscopic stochastic process describing diffusion in solids, under a size-exclusion constraint.
This system possesses an entropy structure, which is exploited to define the numerical scheme in terms of (discrete) entropy variables, and is thus preserved at the discrete level.
This structure makes it possible to prove the existence of nonnegative discrete solutions satisfying the size-exclusion constraint, as well as mass conservation, and to establish the convergence of the scheme under mesh refinement.
To the best of our knowledge, this is the first work proposing and analyzing a structure-preserving hybrid finite volume scheme for anisotropic cross-diffusion systems on general polygonal and polyhedral meshes.

The preprint associated with this presentation is:
V. Ehrlacher, A. Massimini, J. Moatti. Structure-preserving hybrid finite volume scheme for an anisotropic cross-diffusion system, 2026. Preprint, HAL : hal-05589824

Résumé : In nuclear fusion, simulations are essential for understanding and controlling tokamak instabilities, phenomena that can severely damage reactors. Neural approaches for solving partial differential equations (PDEs) are gaining interest due to their mesh-free nature, flexibility, and scalability. These methods rely on neural networks as approximation spaces instead of classical polynomial bases, and this work investigates the efficiency of several neural techniques applied to plasma simulations. We first study stationary elliptic equations, with particular attention to the Grad–Shafranov equation, solved using Physics-Informed Neural Networks (PINNs). We then address time-dependent problems such as anisotropic diffusion, relying on adapted neural schemes, including Discrete PINNs and Neural Galerkin methods. In both cases, the Natural Gradient method is employed to significantly accelerate and stabilize the optimization process during training.

Résumé : TBA

Résumé : Proof-of-Work blockchains rely on miners who continuously spend resources in exchange for uncertain rewards. To reduce this uncertainty, many miners join Pay-Per-Share mining pools where they receive regular payments while the pool manager absorbs the risk. This creates a natural question: how should miners choose between competing pools, and how should pools set their fees and payout policies? This talk discusses a stochastic model for this problem in which the surplus of a mining pool is described by a two-sided jump process. The first part presents a mean-variance approach to miners' allocation across pools and explains how this leads to a Nash equilibrium between competing pool managers. The second part turns to the pool manager's dividend problem, formulated in an expected discounted dividend framework. Using ideas from actuarial risk theory, it is shown how barrier strategies arise naturally, under which conditions they are optimal, and how the optimal barrier can be computed numerically. The aim is to understand the economic incentives behind mining pools, and their possible impact on the decentralization of Proof-of-Work blockchains.

Résumé : In this talk we discuss the convergence of adaptive FEM for strongly monotone nonlinear partial differential equations arising from the minimization of a convex energy functional. Rather than focusing on specific algorithmic implementations, we establish a generalized theoretical framework that identifies sufficient conditions for an adaptive scheme to contract the energy difference at an optimal rate relative to the number of degrees of freedom. The presented abstract point of view is subsequently shown to include many existing error estimation approaches (local residuals, flux equilibration, linear residual liftings, energy descent), thereby providing a unified proof for their optimal convergence. A key contribution of our recent work is the introduction of computable local equivalence factors yielding a computable a posteriori bound on the contraction rate which can then be used for an improved local refinement procedure. Finally, we will discuss technical challenges posed by higher-order discretizations and show how the arising oscillations can be managed through local higher-order flux-equilibration.