592 RWTH Publication No: 956772        2015       
TITLE Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients
AUTHORS Markus Bachmayr, Albert Cohen, Ronald DeVore, Giovanni Migliorati
ABSTRACT Elliptic partial differential equations with diffusion coefficients of lognormal form, that is a=exp(b), where b is a Gaussian random field, are considered. We study the ℓp summability properties of the Hermite polynomial expansion of the solution in terms of the countably many scalar parameters appearing in a given representation of b. These summability results have direct consequences on the approximation rates of best n-term truncated Hermite expansions. Our results significantly improve on the state of the art estimates available for this problem. In particular, they take into account the support properties of the basis functions involved in the representation of b, in addition to the size of these functions. One interesting conclusion from our analysis is that in certain relevant cases, the Karhunen-Loève representation of b may not be the best choice concerning the resulting sparsity and approximability of the Hermite expansion.
KEYWORDS stochastic PDEs, lognormal coefficients, n-term approximation, Hermite polynomials
DOI 10.1051/m2an/2016051
PUBLICATION ESAIM: M2AN, 51, (2017), 341–363