A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It relies on a domain decomposition method which introduces several subdomains of interest (called patches) containing the different sources of uncertainties and non-linearities. An iterative algorithm is then introduced, which requires the solution of a sequence of linear global problems (with deterministic operators and uncertain right-hand sides), and non-linear local problems (with uncertain operators and/or right-hand sides) over the patches. Non-linear local problems are solved using an adaptive sampling-based least-squares method for the construction of sparse polynomial approximations of local solutions as functions of the random parameters. Consistency, convergence and robustness of the algorithm are proved under general assumptions on the semi-linear elliptic operator. A convergence acceleration technique (Aitken’s dynamic relaxation) is also introduced to speed up the convergence of the algorithm. The performances of the proposed method are illustrated through numerical experiments carried out on a stationary non-linear diffusion-reaction problem.
Accepté le :
DOI : 10.1051/m2an/2018025
Mots-clés : Uncertainty quantification, non-linear elliptic stochastic partial differential equation, multiscale, domain decomposition, sparse approximation
@article{M2AN_2018__52_5_1763_0,
author = {Nouy, Anthony and Pled, Florent},
title = {A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1763--1802},
publisher = {EDP-Sciences},
volume = {52},
number = {5},
year = {2018},
doi = {10.1051/m2an/2018025},
zbl = {1479.35971},
mrnumber = {3878605},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2018025/}
}
TY - JOUR AU - Nouy, Anthony AU - Pled, Florent TI - A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1763 EP - 1802 VL - 52 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2018025/ DO - 10.1051/m2an/2018025 LA - en ID - M2AN_2018__52_5_1763_0 ER -
%0 Journal Article %A Nouy, Anthony %A Pled, Florent %T A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1763-1802 %V 52 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2018025/ %R 10.1051/m2an/2018025 %G en %F M2AN_2018__52_5_1763_0
Nouy, Anthony; Pled, Florent. A multiscale method for semi-linear elliptic equations with localized uncertainties and non-linearities. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1763-1802. doi : 10.1051/m2an/2018025. http://www.numdam.org/articles/10.1051/m2an/2018025/
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