Approximation creuses et de faible rang pour les fonctions multivariées: Applications en quantification d'incertitudes

摘要

Uncertainty quantification has been a topic of significant research in computational engineering since early worksin stochastic finite element method. Over past decades, several methods based on classical results in analysis and approximation theory have been proposed. However for problems involving high stochastic dimension, these methods are limited by the so called "curse of dimensionality" as the underlying approximation space increases exponentially with dimension. Resolution of these high dimensional problems "non intrusively" (where we cannot access or modify model source code), is indeed often difficult with only a partial information in the form of a few model evaluations. Given computation and time resource constraints, methods addressing these issues are needed. The present thesis exploits recent developments in low rank and sparse approximations to propose methods that take into account both low rank and sparsity structures of high dimensional functions and can thus provide sufficiently accurate approximation with few sample evaluations. The number of parameters to estimate in these sparse low rank tensor formats is linear in stochastic dimension with few non zero parameters that can be estimated efficiently by sparse regularization techniques. The proposed methods are integrated with clustering and classification approach for approximation of discontinuous and irregular functions.