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🎓Jérémy BRIANT thesis defense

  Tuesday 12 November 2024From 10h00 at 12h00

  ENSEEIHT, Salle des Thèses C002    

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 GrattonInstitut National Polytechnique de ToulouseDirecteur de thèse
M. Anthony T. WeaverCerfacs – CECIDirecteur de thèse
M. Laurent DebreuInria Grenoble – Rhône-AlpesRapporteur
M. Olivier Le MaîtreCNRS Paris-CentreRapporteur
Mme Clémentine PrieurUniversité Grenoble AlpesExaminatrice
M. Reda El AmriIFP Energie NouvellesExaminateur
Mme Selime GürolCerfacs – CECIExaminatrice
M. Paul MycekCerfacs – CECIInvité
M. Ehouarn SimonInstitut National Polytechnique de ToulouseInvité

CALENDAR

Monday

02

December

2024

Artificial intelligence for computational physics

From Monday 2 December 2024 to Friday 6 December 2024

  Training    

Friday

06

December

2024

CERFACS SEMINAR

Friday 6 December 2024From 10h00 at 11h00

  JCA room, CERFACS    

Monday

09

December

2024

Multi-architecture parallelism using Kokkos/C++ library

From Monday 9 December 2024 to Wednesday 11 December 2024

  Training    

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