Model approximation and reduction for partial differential equations with probabilistic interpretation.

Summary In this thesis, we are interested in the numerical solution of models governed by partial differential equations that admit a probabilistic interpretation. In a first step, we consider partial differential equations in high dimension. Based on a probabilistic interpretation of the solution which allows to obtain point estimates of the solution via Monte-Carlo methods, we propose an algorithm combining an adaptive interpolation method and a variance reduction method to approximate the solution on its whole definition domain. In a second step, we focus on reduced basis methods for parameterized partial differential equations. We propose two gluttonous algorithms based on a probabilistic interpretation of the error. We also propose a discrete optimization algorithm that is probably approximately correct in relative accuracy and that allows us, for these two gluttonous algorithms, to judiciously select a snapshot to add to the reduced basis based on the probabilistic representation of the approximation error.