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Analytic and Gevrey class regularity for parametric elliptic eigenvalue problems and applications

Alexey Chernov, Tung Le

TL;DR

The paper develops a Gevrey-δ regularity theory for the smallest eigenpair of parametric elliptic eigenvalue problems with countably many parameters, providing sharp derivative bounds that scale like (|ν|!)^δ. A novel alternative-to-factorial technique replaces traditional factorial-based induction, enabling optimal analyticity and Gevrey regularity under nonaffine coefficient parametrizations. The results yield dimension-robust convergence guarantees for Gauss-Legendre quadrature and quasi-Monte Carlo integration, with practical validation through numerical experiments on analytic and Gevrey coefficient models. This work extends parametric EVP analysis to infinite-dimensional parameter spaces and supports efficient uncertainty quantification for PDEs with random coefficients.

Abstract

We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions where the coefficients (and hence the solution $u$) may depend on a parameter $y$. For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyse Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients.

Analytic and Gevrey class regularity for parametric elliptic eigenvalue problems and applications

TL;DR

The paper develops a Gevrey-δ regularity theory for the smallest eigenpair of parametric elliptic eigenvalue problems with countably many parameters, providing sharp derivative bounds that scale like (|ν|!)^δ. A novel alternative-to-factorial technique replaces traditional factorial-based induction, enabling optimal analyticity and Gevrey regularity under nonaffine coefficient parametrizations. The results yield dimension-robust convergence guarantees for Gauss-Legendre quadrature and quasi-Monte Carlo integration, with practical validation through numerical experiments on analytic and Gevrey coefficient models. This work extends parametric EVP analysis to infinite-dimensional parameter spaces and supports efficient uncertainty quantification for PDEs with random coefficients.

Abstract

We investigate a class of parametric elliptic eigenvalue problems with homogeneous essential boundary conditions where the coefficients (and hence the solution ) may depend on a parameter . For the efficient approximate evaluation of parameter sensitivities of the first eigenpairs on the entire parameter space we propose and analyse Gevrey class and analytic regularity of the solution with respect to the parameters. This is made possible by a novel proof technique which we introduce and demonstrate in this paper. Our regularity result has immediate implications for convergence of various numerical schemes for parametric elliptic eigenvalue problems, in particular, for elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients.
Paper Structure (12 sections, 14 theorems, 142 equations, 2 figures)

This paper contains 12 sections, 14 theorems, 142 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Deficiency of the real-variable inductive argument
  4. The falling factorial estimates as a remedy
  5. Gevrey-class and analytic functions
  6. Regularity assumptions and the formulation of the main result
  7. Elliptic eigenvalue problems with countably many parameters
  8. Parametric regularity: Proof of the main result
  9. Applications and numerical experiments
  10. Gauss-Legendre quadrature
  11. Quasi-Monte Carlo method for Gevrey functions
  12. APPENDIX

Key Result

Theorem 2.2

\newlabelthm:ana-bound-1d0 Let $f \in C^\infty(I)$ for some open interval $I$. The function $f$ is real analytic if and only if for each $y_0 \in I$, there is an open interval $J$, with $y_0 \in J \subseteq I$, and constants $R > 0$ and $C > 0$ such that the derivatives of $f$ satisfy The radius of convergence $\rho$ of the power series power-series-1d at some $y_0 \in I$ can be determined as th

Figures (2)

  • Figure 1: The quadrature error $\varepsilon^{(1)}_n$ with respect to the number $n$ of quadrature points (left) and the quadrature error $\varepsilon^{(2)}_n$ with respect to $m = n^{1/3}$ (right).
  • Figure 2: Convergence of the QMC errors $\varepsilon_n^{{\rm QMC},(1)}$ and $\varepsilon_n^{{\rm QMC},(2)}$, and the MC errors $\varepsilon_n^{{\rm MC},(1)}$ and $\varepsilon_n^{{\rm MC},(2)}$ for the analytic and the Gevrey class setting.

Theorems & Definitions (34)

  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Proof 1
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 2.8
  • Theorem 3.2
  • ...and 24 more