Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients

Understand

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 $\ell^p$ 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.