604 RWTH Publication No: 854284        2022       
TITLE An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality
AUTHORS Markus Bachmayr, Igor Voulis
ABSTRACT 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
DOI 10.1051/m2an/2022062
PUBLICATION ESAIM: M2AN, Volume 56, Number 6, November-December 2022, pp 1955 - 1992