Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces

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Abstract
We investigate the sparsity of Wiener polynomial chaos expansions of holomorphic maps [Formula: see text] on Gaussian Hilbert spaces. 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 [Formula: see text]. We establish weighted summability results for the Wiener–Hermite coefficient sequences of images of affine-parametric expansions of the log-Gaussian input under [Formula: see text]. These results give rise to [Formula: see text]-term approximation rate bounds for the full range of input summability exponents [Formula: see text]. We show that these approximation rate bounds apply to parameter-to-solution maps for elliptic diffusion PDEs with lognormal coefficients.
Funding Information
  • Deutsche Forschungsgemeinschaft (543965776)
  • Italian Ministry of University and Research (P2022NC97R)
  • Italian Ministry of University and Research (202292JW3F)
  • GNCS-INdAM

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