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Sparsity for infinite-parametric holomorphic functions on Gaussian Spaces

by C. Marcati and Ch. Schwab and J. Zech

(Report number 2025-14)

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.

Keywords: Holomorphy, Elliptic PDEs, Wiener-Hermite coefficients, Gaussian measure spaces, $N$-term approximation

@Techreport{MSZ25_1135,
  author = {C. Marcati and Ch. Schwab and J. Zech},
  title = {Sparsity for infinite-parametric holomorphic functions on Gaussian Spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-14},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-14.pdf },
  year = {2025}
}

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