Ana Djurdjevac: Nonlinear SPDE models of particle systems

Speaker
Ana Djurdjevac, Freie Univ. Berlin

Abstract
Interacting particle systems provide flexible and powerful models that are useful in many application areas such as sociology (agents), molecular dynamics (proteins) etc. However, particle systems with large numbers of particles are very complex and difficult to handle, both analytically and computationally. Therefore, a common strategy is to derive effective equations that describe the time evolution of the empirical particle density. Our aim is to derive and study continuum models for the mesoscopic behavior of particle systems. Contrary to recent work in the field, we are interested in finite size effects and will not consider the infinite particle limit. We will introduce nonlinear and non-Gaussian models that provide a more faithful representation of the evolution of the empirical density of a given particle system than the usual linear Gaussian perturbations around the hydrodynamic limit models. In particular, we want to study the well-posedness of these nonlinear SPDE models and to control the weak error of the SPDE approximation. A prototypical example that we will consider is the formal identification of a finite system of diffusions with the singular Dean-Kawasaki SPDE. This is the joint work with H. Kremp, N. Perkowski and X. Ji. Additionally, we aim to study the numerical approximation of these types of nonlinear SPDEs. However, the standard finite volume approximation implicitly requires a sufficiently large number of particles to ensure both the positivity of the solution and an accurate approximation of the stochastic flux. To address this challenge, we extend hybrid algorithms for particle systems to cases where the density is low. We develop criteria for determining the threshold by comparing higher-order statistics from the finite volume method with particle simulations. We then demonstrate the use of these criteria for dynamic adaptation in both two- and three-dimensional spatial settings. This is a joint work with J. Bell and A. Almgren.