Approximation and model reduction for partial differential equations with probabilistic interpretation

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 part, we consider partial differential equations in high dimension. Based on a probabilistic interpretation of the solution which allows to obtain pointwise evaluations of the solution using Monte-Carlo methods, we propose an algorithm combining an adaptive interpolation method and a variance reduction method to approximate the global solution. In a second part, we focus on reduced basis methods for parametric partial differential equations. We propose two greedy algorithms based on a probabilistic interpretation of the error. We also propose a discrete optimization algorithm probably approximately correct in relative precision which allows us, for these two greedy algorithms, to judiciously select a snapshot to add to the reduced basis based on the probabilistic representation of the approximation error.

Directors:

• Professeur Anthony Nouy (Centrale Nantes )
• Professeur Clémentine Prieur (Université Grenoble Alpes )
• MCF Marie Billaud-Friess (Centrale Nantes )

Raporteurs:

• DR Mireille Bossy (INRIA Sophia Antipolis )
• Professeur Benjamin Jourdain (ENPC )

Examinators:

• Professeur Tony Lelièvre (ENPC )
• MCF HDR Pierre Etoré (Grenoble INP )