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Parametric holomorphy of elliptic eigenvalue problems

Byeong-Ho Bahn

TL;DR

The work presents a general framework to certify parametric holomorphy $(\bm{b},\varepsilon)$-holomorphy for solutions of parametric PDEs, then specializes to elliptic eigenvalue problems to show that ground eigenpairs of both linear and semilinear EVPs are holomorphic with explicit single- and multi-variable derivative bounds. The results leverage a Banach-space implicit-function approach to handle eigenvalue ordering and crossing, avoiding detailed multi-index machinery while still obtaining sharp factorial-type bounds for derivatives. These bounds underlie dimension-independent approximation strategies, including DeepOnet-based solvers and quasi-M Monte Carlo methods, by enabling robust universal expansions and QMC error guarantees. Practically, the framework provides verifiable criteria to ensure holomorphy and derivative control under affine coefficient dependence, yielding practical tools for high-dimensional uncertainty quantification in EVP contexts.

Abstract

The study of parameter-dependent partial differential equations (parametric PDEs) with countably many parameters has been actively studied for the last few decades. In particular, it has been well known that a certain type of parametric holomorphy of the PDE solutions allows the application of deep neural networks without encountering the curse of dimensionality. This paper aims to propose a general framework for verifying the desired parametric holomorphy by utilizing the bounds on parametric derivatives. The framework is illustrated with examples of parametric elliptic eigenvalue problems (EVPs), encompassing both linear and semilinear cases. As the results, it will be shown that the ground eigenpairs have the desired holomorphy. Furthermore, under the same conditions, the bounds for the mixed derivatives of the ground eigenpairs are derived. These bounds are well known to take a crucial role in the error analysis of quasi-Monte Carlo methods.

Parametric holomorphy of elliptic eigenvalue problems

TL;DR

The work presents a general framework to certify parametric holomorphy -holomorphy for solutions of parametric PDEs, then specializes to elliptic eigenvalue problems to show that ground eigenpairs of both linear and semilinear EVPs are holomorphic with explicit single- and multi-variable derivative bounds. The results leverage a Banach-space implicit-function approach to handle eigenvalue ordering and crossing, avoiding detailed multi-index machinery while still obtaining sharp factorial-type bounds for derivatives. These bounds underlie dimension-independent approximation strategies, including DeepOnet-based solvers and quasi-M Monte Carlo methods, by enabling robust universal expansions and QMC error guarantees. Practically, the framework provides verifiable criteria to ensure holomorphy and derivative control under affine coefficient dependence, yielding practical tools for high-dimensional uncertainty quantification in EVP contexts.

Abstract

The study of parameter-dependent partial differential equations (parametric PDEs) with countably many parameters has been actively studied for the last few decades. In particular, it has been well known that a certain type of parametric holomorphy of the PDE solutions allows the application of deep neural networks without encountering the curse of dimensionality. This paper aims to propose a general framework for verifying the desired parametric holomorphy by utilizing the bounds on parametric derivatives. The framework is illustrated with examples of parametric elliptic eigenvalue problems (EVPs), encompassing both linear and semilinear cases. As the results, it will be shown that the ground eigenpairs have the desired holomorphy. Furthermore, under the same conditions, the bounds for the mixed derivatives of the ground eigenpairs are derived. These bounds are well known to take a crucial role in the error analysis of quasi-Monte Carlo methods.
Paper Structure (18 sections, 17 theorems, 119 equations)

This paper contains 18 sections, 17 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Motivation and the goal
  3. Contribution
  4. Organization
  5. Notation
  6. Preliminary
  7. $({b},\varepsilon)$- holomorphy
  8. Parametric PDEs
  9. Main results
  10. Parametric EVPs
  11. Linear Eigenvalue Problem
  12. Separately holomorphy of the ground eigenpairs
  13. The bound of parametric derivatives of the ground eigenpairs
  14. The parametric continuity of the ground eigenpairs
  15. Semilinear Eigenvalue Problem
  16. ...and 3 more sections

Key Result

Theorem 3

Let $(\mathcal{X},\|\cdot\|_{\mathcal{X}})$ be a given Banach space and let $u:U\rightarrow \mathcal{X}$ be a continuous function. Suppose that $u$ is separately holomorphic extension. In other words, $u_j(z_j):=u(z_j;\bm{y}_{-j})$ is holomorphic(complex analytic) on some open set $V_j\supset [-1,1] where $\bm{\beta}:=(\beta_j)_{j\geq 1}$ is a sequence of positive numbers and $\bm{\alpha}:=(\alpha

Theorems & Definitions (36)

  • Definition 1
  • Definition 2
  • Theorem 3
  • proof
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 7
  • proof
  • Remark 8
  • ...and 26 more