Together with Anna Shalova and Mark Peletier, we study the limiting dynamics of a large class of noisy gradient descent systems in the overparameterized regime. In this regime the set of global minimizers of the loss is large, and when initialized in a neighbourhood of this zero-loss set a noisy gradient descent algorithm slowly evolves along this set.… Read the rest

# Author Archives: André

# Preprint: Variational convergence of exchange-driven stochastic particle systems in the thermodynamic limit

Together with Chun Yin Lam we consider the thermodynamic limit of mean-field stochastic particle systems on a complete graph. The evolution of occupation number at each vertex is driven by particle exchange with its rate depending on the population of the starting vertex and the destination vertex, including zero-range and misanthrope process.… Read the rest

# Preprint: Modelling and numerical analysis for a durotaxis model in “Piezo1-induced durotaxis of pancreatic stellate cells depends on TRPC1 and TRPV4 channels”

In this interdisciplinary project with a group from the Institute of Physiology II in Münster and the Department of Medicine at Duke, together with Angela Stevens we contributed a mathematical model and numerical analysis for a phenomenological model of durotaxis with mechanosensitive ion channels.… Read the rest

# Preprint: A structure preserving discretization for the Derrida-Lebowitz-Speer-Spohn equation based on diffusive transport:

Together with Daniel Matthes, Eva-Maria Rott and Giuseppe Savaré we propose a spatial discretization of the fourth-order nonlinear DLSS equation on the circle. Our choice of discretization is motivated by a novel gradient flow formulation with respect to a metric that generalizes martingale transport.… Read the rest

# Conference announcement: PDE & Probability in interaction: functional inequalities, optimal transport and particle systems

Together with Pierre Monmarché (Sorbonne Université), Julien Reygner (École des Ponts ParisTech), and Marielle Simon (Université de Lille), we are delighted to announce the upcoming workshop “PDE & Probability in interaction: functional inequalities, optimal transport and particle systems”.

The event will be held from January 22 to 26, 2024, at CIRM in Marseille.… Read the rest

# Preprint: Graph-to-local limit for the nonlocal interaction equation

Together with Antonio Esposito and Georg Heinze, we study a class of nonlocal partial differential equations presenting a tensor-mobility, in space, obtained asymptotically from nonlocal dynamics on localising infinite graphs. Our strategy relies on the variational structure of both equations, being a Riemannian and Finslerian gradient flow, respectively.… Read the rest

# Preprint: Variational convergence of the Scharfetter-Gummel scheme

Together with Anastasiia Hraivoronska and Oliver Tse, we explore the convergence of the Scharfetter-Gummel scheme for the aggregation-diffusion equation using a variational approach. Our investigation involves obtaining a novel gradient structure for the finite volume scheme that works consistently for any nonnegative diffusion constant, which allows us to study the discrete-to-continuum and zero-diffusion limits simultaneously.… Read the rest

# Published: On a class of nonlocal continuity equations on graphs:

The article with Antonio Esposito and Francesco Saverio Patacchini on *a class of nonlocal continuity equations on graphs* got published in the European Journal of Applied Mathematics. This is a follow-up work on our previous work also with Dejan Slepcev, where we introduced evolutions on graphs based on Upwind interpolation.… Read the rest

# Published: Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients

The paper with Víctor Navarro-Fernández on Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients got published at ESAIM: Mathematical Modelling and Numerical Analysis (M2AN). In the revision (also on arXiv:2201.10411), we arrived at uniform errors estimate in the diffusion constant also in the limit of vanishing diffusion.… Read the rest

# Preprint: Covariance-modulated optimal transport and gradient flows

Together with Martin Burger, Franca Hoffmann, Daniel Matthes and Matthias Erbar, we investigate a new dynamical optimal transport distance in which the energy to be minimised is modulated by the covariance matrix of the distribution. Such transport metrics arise naturally in mean-field limits of certain ensemble Kalman methods for solving inverse problems.… Read the rest