Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces

Abstract

We investigate the sparsity of Wiener polynomial chaos expansions of holomorphic maps $\mathcal{G}$ on Gaussian Hilbert spaces, as arise in the coefficient-to-solution maps of linear, second order, divergence-form elliptic PDEs with log-Gaussian diffusion coefficient. Representing the Gaussian random field input as an affine-parametric expansion, the nonlinear map becomes a countably-parametric, deterministic holomorphic map of the coordinate sequence $\boldsymbol{y} = (y_j)_{j\in\mathbb{N}} \in \mathbb{R}^\infty$. We establish weighted summability results for the Wiener-Hermite coefficient sequences of images of affine-parametric expansions of the log-Gaussian input under $\mathcal{G}$. These results give rise to $N$-term approximation rate bounds for the full range of input summability exponents $p\in (0,2)$. We show that these approximation rate bounds apply to parameter-to-solution maps for elliptic diffusion PDEs with lognormal coefficients.