The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, rather than by the spatial approximation order. The convergence and complexity estimates are illustrated by numerical experiments.
Keywords: Parameter-dependent elliptic partial differential equations, stochastic Galerkin method, a posteriori error estimation, adaptive methods, complexity analysis
@article{M2AN_2022__56_6_1955_0,
author = {Bachmayr, Markus and Voulis, Igor},
title = {An adaptive stochastic {Galerkin} method based on multilevel expansions of random fields: {Convergence} and optimality},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1955--1992},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {6},
doi = {10.1051/m2an/2022062},
mrnumber = {4481120},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022062/}
}
TY - JOUR AU - Bachmayr, Markus AU - Voulis, Igor TI - An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1955 EP - 1992 VL - 56 IS - 6 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022062/ DO - 10.1051/m2an/2022062 LA - en ID - M2AN_2022__56_6_1955_0 ER -
%0 Journal Article %A Bachmayr, Markus %A Voulis, Igor %T An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1955-1992 %V 56 %N 6 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022062/ %R 10.1051/m2an/2022062 %G en %F M2AN_2022__56_6_1955_0
Bachmayr, Markus; Voulis, Igor. An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 6, pp. 1955-1992. doi: 10.1051/m2an/2022062
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