Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients *

Markus Bachmayr (1) , Albert Cohen (1) , Giovanni Migliorati (1)

1 LJLL - Laboratoire Jacques-Louis Lions

Approximation of elliptic PDEs with random diffusion coefficients typically requires a representation of the diffusion field in terms of a sequence y = (y j) j≥1 of scalar random variables. One may then apply high-dimensional approximation methods to the solution map y → u(y). Although Karhunen-Lò eve 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 wavelet-type 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 Karhunen-Lò eve 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 .

Domaines: Analyse numérique [math.NA]

Markus Bachmayr, Albert Cohen, Giovanni Migliorati. Representations of Gaussian random fields and approximation of elliptic PDEs with lognormal coefficients *. Journal of Fourier Analysis and Applications, 2018, 24, pp.621-649. ⟨hal-01352284⟩