Sparsity for infinite-parametric holomorphic functions on Gaussian Spaces
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2025-05
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Report
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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.
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published
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2025-14
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Seminar for Applied Mathematics, ETH Zurich
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Subject
Holomorphy; Ellptic PDEs; Wiener-Hermite coefficients; Gaussian measure spaces; $N$-term approximation
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03435 - Schwab, Christoph / Schwab, Christoph