593

RWTH Publication No: 956768 2016 
TITLE 
Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients 
AUTHORS 
Markus Bachmayr, Albert Cohen, Giovanni Migliorati 
ABSTRACT 
Approximation of elliptic PDEs with random diffusion coefficients typically requires a representation of the diffusion field in terms of a sequence y=(yj)j≥1 of scalar random variables. One may then apply highdimensional approximation methods to the solution map y↦u(y). Although KarhunenLoève representations are commonly used, it was recently shown, in the relevant case of lognormal diffusion fields, that they do not generally yield optimal approximation rates. Motivated by these results, we construct wavelettype representations of stationary Gaussian random fields defined on bounded domains. The size and localization properties of these wavelets are studied, and used to obtain polynomial approximation results for the related elliptic PDE which outperform those achievable when using KarhunenLoève representations. Our construction is based on a periodic extension of the random field, and the expansion on the domain is then obtained by simple restriction. This makes the approach easily applicable even when the computational domain of the PDE has a complicated geometry. In particular, we apply this construction to the class of Gaussian processes defined by the family of Matérn covariances. 
KEYWORDS 
Gaussian random fields, Lognormal diffusion problems, Periodic continuation, Karhunen–Loève expansions,
Wavelets 
DOI 
10.1007/s0004101795395 
PUBLICATION 
Journal of Fourier Analysis and Applications, volume 24, pages621–649, (2018) 