Averaging, Diffusion Approximation in Infinite Dimension - Theory and Numerics – ADA

Our project is to treat multiscale models which are both infinite-dimensional and stochastic with a theoretic and computational approach. Multiscale analysis and multiscale numerical approximation for infinite-dimensional problems (partial differential equations) is an extensive part of contemporary mathematics, with such wide topics as hydrodynamic limits, homogenization, design of asymptotic-preserving schemes. Multiscale models in a random or stochastic context have been analysed and computed essentially in finite dimension (ordinary/stochastic differential equations), or in very specific domains, mainly the propagation of waves, of partial differential equations. The technical difficulties of our project are due to the stochastic aspect of the problems (this brings singular terms in the equations, which are difficult to understand with a pure PDE's analysis approach) and to their infinite-dimensional character. These two aspects, combined, typically raise compactness and computational issues. Our aim is to create the new tools, analytical, probabilistic and numerical ones, which are required to understand a large class of stochastic multiscale partial differential equations, that includes some kinetic and dispersive equations. Our aim is to derive reduced equations. In the different regimes we are interested in, and, particularly, in the diffusive regime (diffusion-approximation), that leads to limit equations with white noise. We will investigate the derivation of reduced equations in different context and for various models:
- collisional kinetic equations with a Vlasov forcing term induced by (resp. a collisional kernel perturbed by) an external, or coupled, Markov process (kinetic equations for plasmas or fluids, resp. modelling of motion by run-and-tumble),
- limit Boltzmann to Navier-Stokes under stochastic forces,
- collisional or non-collisional kinetic equations with a stochastic drag force term also induced by an external, or coupled, Markov process (models of sprays in turbulent flows, stochastic Cucker-Smale models, stochastic Landau damping),
- dispersive models for the propagation of waves (e.g. Zakharov system, Klein-Gordon-Zakharov system, stochastic NLS equations).

The numerical approximation of these models raises the following issues, which we will investigate. First, the construction and analysis of asymptotic-preserving schemes for equations in which the small parameter affects equally the deterministic ans stochastic terms. This concerns the design of schemes unconstrained by the small parameters for the original equations. Numerical schemes for the approximation of the reduced equations furnished by the theoretical analysis are another approach that we will develop. This requires the computation of the coefficients of the reduced equations. We will build accurate Heterogeneous Multiscale Methods (HMM) to that purpose. HMM in our context (kinetic, dispersive stochastic equations) have never been developed. Several questions of numerical analysis are related. Regarding these questions, we will analyse the efficiency of the the numerical schemes in the approximation of invariant measures or auto-correlations, their orders of convergence, and develop strategies to reduce the overall cost, like high-order integrators for invariant distributions, variance reduction strategies in Monte-Carlo methods.