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

TL;DR

This paper investigates the sparsity and approximation rates of Hermite polynomial expansions for solutions to elliptic PDEs with lognormal coefficients, improving existing estimates by considering basis support properties.

Contribution

It provides new summability results for Hermite expansions that enhance understanding of approximation rates, surpassing previous estimates by incorporating basis support considerations.

Findings

Improved $oldsymbol{ extit{ ext{n}}}$-term approximation rates for Hermite expansions.

Identification of conditions where Karhunen-Loève basis is suboptimal for sparsity.

Enhanced theoretical understanding of polynomial approximation in stochastic PDEs.

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 $\ell^p$ 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. 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-Lo\`eve representation of $b$ may not be the best choice concerning the resulting sparsity and approximability of the Hermite expansion.