Cerfacs Enter the world of high performance ...

Multilevel Monte Carlo methods for the estimation of discretized field statistics in geoscience

Ecole Doctorale 475: Mathématiques, Informatique et Télécommunications de Toulouse (EDMITT)

Lien Zoom :
https://inp-toulouse-fr.zoom.us/j/99813718403?pwd=88o5Fqkn7IhHyhop4MbxXFgZidQhbi.1
Meeting ID : 998 1371 8403
Mot de passe : 576513

Variance reduction multifidelity methods for the estimation of statistics are increasingly used in more domains as an improvement over the classical Monte Carlo method. These methods were originally designed to estimate scalar statistics. However, in some applications, in geosciences for example, the quantities of interest which statistics we want to estimate can be random vectors or random fields. In the case of a random vector which represent a discretized field, using a multifidelity estimator is not straightforward, especially if the inputs and outputs of the different models do not have the same dimension among the different fidelity levels. Transfer operators between the levels need to be introduced. This thesis is about the adaptation of the multilevel Monte Carlo (MLMC) estimator for the estimation of discretized fields in geosciences and the analysis of the transfer operators, allowing to choose operators that reduce the variance of the MLMC. A spectral analysis of the MLMC estimator is performed to better understand the effects of the transfer operators. Numerical experiments on simplified problems show that the MLMC deteriorates the estimation of high frequency components of a discretized field compared to a Monte Carlo estimator. A theoretical analysis on a specific class of problems, similar to analysis developed for multigrid methods, allows for a better understanding of the disparities in the MLMC estimation of the different components of a discretized field. Following those results, the transfer operators are modified to include a filtering step inspired from multigrid methods. Adding filters leads to a better estimation of low and high frequency components, and to a decrease in the total variance of the estimator. The use of methods similar to the multilevel best linear unbiased estimator (MLBLUE) can help in choosing the transfer operators that reduce the variance of the estimator. Those improvements were applied to the estimation of the discretized variance field of a diffusion-based covariance operator. Numerical experiments support the conclusions of the theoretical analysis on a more complex problem and demonstrates the improvements brought by the proposed estimator compared to a crude MLMC. This research work contributes to expand the range of MLMC applications to other domains, expecially geosciences, and offer a better understanding of the method by establishing links with multigrid methods. It also allows for improvements of the MLMC estimator by highlighting the importance of the transfer operator choice.

Jury

M. Serge Gratton	Institut National Polytechnique de Toulouse	Directeur de thèse
M. Anthony T. Weaver	Cerfacs – CECI	Directeur de thèse
M. Laurent Debreu	Inria Grenoble – Rhône-Alpes	Rapporteur
M. Olivier Le Maître	CNRS Paris-Centre	Rapporteur
Mme Clémentine Prieur	Université Grenoble Alpes	Examinatrice
M. Reda El Amri	IFP Energie Nouvelles	Examinateur
Mme Selime Gürol	Cerfacs – CECI	Examinatrice
M. Paul Mycek	Cerfacs – CECI	Invité
M. Ehouarn Simon	Institut National Polytechnique de Toulouse	Invité