Random simulation in infinite dimension – SIMALIN

Our project aims at going beyond existing results concerning the analysis and the implementation of numerical algorithms for two classes of stochastic partial differential equations: semilinear parabolic like Allen-Cahn, Cahn-Hilliard, Burgers, Navier-Stokes and FitzHugh-Nagumo equations and hyperbolic scalar conservation laws like inviscid Burgers equations. These equations are perturbed by a noise term, formally represented by the time derivative of a Wiener process, defined as a series of Brownian motions developed on a Hilbert basis. Until now, in many applications, only low-dimensional noise perturbations, or even only noise which is white in time but does not depend on space, have been considered.

One of the main theoretical and computational difficulties is that for such equations, both the solution and the stochastic forcing depend on time and space, and are thus infinite dimensional and random objects. This is a recent and very active research area, with many remaining open questions compared with the finite dimensional situation where comparison of the strong and weak orders is well-understood. However with infinite dimensional noise the analysis and simulation are much more challenging, since infinite dimensional noise yields low regularity properties for solutions, and in turn low orders of convergence for temporal and spatial numerical approximations.

Building upon recent theoretical advances in the analysis of SPDEs and the effectiveness of some existing sampling methods in finite dimension, we aim to go beyond the state of the art in the following directions: extending results concerning weak rates of convergence for stochastic semi-linear parabolic equations with non-globally Lipschitz continuous coefficients, proving error estimates and rates of convergence for finite volume methods applied to stochastic scalar conservation laws, and designing efficient variance reduction techniques like multilevel Monte Carlo methods, or in the context of rare events. Encouraging results have been obtained about strong and weak orders for a large dimensional noise opening the way to designing higher order schemes.

In parallel with mathematical analysis, it is crucial to validate developments of numerical methods and theoretical error analysis by extensive numerical experiments. The combination of advanced Monte-Carlo methods and of numerical algorithms for PDEs is a non-trivial procedure in practice, which is quite often ignored in the literature. This project aims at considering scientific computing aspects of the numerical methods for SPDEs, and in creating adapted code to be able to efficiently deal with issues arising from both the Monte-Carlo and the PDE components of the models. Efficient softwares and libraries with PDE solvers have been developed, but it seems that for SPDEs no tools of this type is available. We aim at providing such a mathematical software.

In conclusion, the project aims at contributing through theoretical and computational breakthroughs by considering all the aspects related to the simulation of SPDEs, from the analysis of properties of the model (regularity, long-time behavior, etc.) to the numerical experiments (implementation, calibration of parameters, reduction of the statistical error). It will be precursor results, with a view to spreading in various applications in mathematics and in the industry.