Direct and Inverse Problems for PDEs with Random Coefficients - Abstract
We consider elliptic partial differential equations with diffusion coefficients of lognormal form, that is $a=exp(b)$ where $b$ is a Gaussian random field. For such problems, 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 established for this problem by Hoang and Schwab. In particular, the new estimates take into account the support properties of the basis functions involved in the representation of $b$, in addition to the size of these functions. Furthermore, we also outline new results in the same spirit for affinely parametrized diffusion coefficients, which in a similar manner improve on known estimates by Cohen, DeVore, and Schwab. One interesting conclusion from our analysis in the lognormal case is that in certain relevant examples, the Karhunen-Loève representation of $b$ is not the best choice in terms of the sparsity and approximability of the resulting Hermite expansion. This is joint work with Albert Cohen, Ronald DeVore, and Giovanni Migliorati.