Approximation and model reduction for partial differential equations with probabilistic interpretation
Speciality : Mathématiques Appliquées
28/06/2021 - 14:00 Arthur Macherey (Centrale Nantes ) En visioconférence: https://www.youtube.com/watch?v=iJydWzBvYo4
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 )