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Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces

Carlo Marcati, Christoph Schwab, Jakob Zech

TL;DR

The paper addresses the sparsity of Wiener-Hermite polynomial chaos expansions for holomorphic maps on Gaussian spaces that arise in coefficient-to-solution mappings of linear elliptic PDEs with lognormal coefficients. It develops an abstract framework for countably-parametric holomorphy in Hilbert spaces, proving weighted $\ell^p$-summability and $N$-term approximation rates that are free from the Curse of Dimensionality, and then verifies these results for PDE operators with log-Gaussian inputs. A constructive Hermite approximation scheme is presented with provable convergence rates, and the coefficient-to-solution map for a linear elliptic PDE is shown to satisfy the required holomorphy assumptions, yielding practical, fast-converging surrogate models. The findings extend prior work to unbounded Gaussian parameter spaces and offer improved summability ranges, with implications for uncertainty quantification, operator learning, and neural operators.

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.

Sparsity for Infinite-Parametric Holomorphic Functions on Gaussian Spaces

TL;DR

The paper addresses the sparsity of Wiener-Hermite polynomial chaos expansions for holomorphic maps on Gaussian spaces that arise in coefficient-to-solution mappings of linear elliptic PDEs with lognormal coefficients. It develops an abstract framework for countably-parametric holomorphy in Hilbert spaces, proving weighted -summability and -term approximation rates that are free from the Curse of Dimensionality, and then verifies these results for PDE operators with log-Gaussian inputs. A constructive Hermite approximation scheme is presented with provable convergence rates, and the coefficient-to-solution map for a linear elliptic PDE is shown to satisfy the required holomorphy assumptions, yielding practical, fast-converging surrogate models. The findings extend prior work to unbounded Gaussian parameter spaces and offer improved summability ranges, with implications for uncertainty quantification, operator learning, and neural operators.

Abstract

We investigate the sparsity of Wiener polynomial chaos expansions of holomorphic maps 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 . We establish weighted summability results for the Wiener-Hermite coefficient sequences of images of affine-parametric expansions of the log-Gaussian input under . These results give rise to -term approximation rate bounds for the full range of input summability exponents . We show that these approximation rate bounds apply to parameter-to-solution maps for elliptic diffusion PDEs with lognormal coefficients.
Paper Structure (20 sections, 17 theorems, 149 equations)

This paper contains 20 sections, 17 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Existing Results
  3. Contributions
  4. Layout
  5. Notation
  6. Preliminaries
  7. Gaussian measures and spaces
  8. Infinite parametric holomorphy
  9. Sparsity for $(b, \xi, 2, \delta, X)$-holomorphic functions
  10. Constructive Hermite approximation
  11. Parametric holomorphy for operators between Hilbert spaces
  12. Application: Coefficient-to-solution map for a linear elliptic PDE
  13. Conclusions and future work
  14. Results on $(b,\xi,2,\delta, X)$-holomorphic functions
  15. Proof of Lemma \ref{['lemma:cnu']}
  16. ...and 5 more sections

Key Result

Theorem 3.1

Let $u$ be $({\bm{b}},\xi,2, \delta,X)$-holomorphic for some ${\boldsymbol{0}} \neq {\bm{b}} \in \ell^p({\mathbb{N}})$, with a $p\in (0,2)$. Let $M\in{\mathbb{N}}$ with $M\geq 2/p$, and let ${\boldsymbol{\varrho}} \in (0, \infty)^\infty$ be such that Then, where the weight sequence $\beta_{\boldsymbol{\nu}}(M,{\boldsymbol{\varrho}})$ is defined in eq:beta. Furthermore,

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2: (${\bm{b}},\xi,r, \delta, X$)-Holomorphy
  • Remark 2.1
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • Theorem 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 31 more