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Semilinear elliptic eigenvalue problem: Parametric analyticity and the uncertainty quantification

Byeong-Ho Bahn

TL;DR

This work extends uncertainty quantification for elliptic eigenvalue problems to nonlinear, parametric settings with affine parameter dependence. By proving ground-state analyticity via an implicit-function approach and establishing a uniform spectral gap, the authors derive factorial-type bounds for mixed derivatives of the ground state and energy. These bounds enable dimension-truncation and quasi-Monte Carlo error analyses that yield dimension-independent convergence rates for estimating expectations of eigenvalues and functionals of the eigenfunction. The results unify the nonlinear case with prior linear theory, and the total error bounds provide practical guidance for QMC-based UQ in nonlinear PDEs, including connections to physical models like the Gross–Pitaevskii equation.

Abstract

In this paper, to the best of our knowledge, we make the first attempt at studying the parametric semilinear elliptic eigenvalue problems with the parametric coefficient and some power-type nonlinearities. The parametric coefficient is assumed to have an affine dependence on the countably many parameters with an appropriate class of sequences of functions. In this paper, we obtain the upper bound estimation for the mixed derivatives of the ground eigenpairs that has the same form obtained recently for the linear eigenvalue problem. The three most essential ingredients for this estimation are the parametric analyticity of the ground eigenpairs, the uniform boundedness of the ground eigenpairs, and the uniform positive differences between ground eigenvalues of linear operators. All these three ingredients need new techniques and a careful investigation of the nonlinear eigenvalue problem that will be presented in this paper. As an application, considering each parameter as a uniformly distributed random variable, we estimate the expectation of the eigenpairs using a randomly shifted quasi-Monte Carlo lattice rule and show the dimension-independent error bound.

Semilinear elliptic eigenvalue problem: Parametric analyticity and the uncertainty quantification

TL;DR

This work extends uncertainty quantification for elliptic eigenvalue problems to nonlinear, parametric settings with affine parameter dependence. By proving ground-state analyticity via an implicit-function approach and establishing a uniform spectral gap, the authors derive factorial-type bounds for mixed derivatives of the ground state and energy. These bounds enable dimension-truncation and quasi-Monte Carlo error analyses that yield dimension-independent convergence rates for estimating expectations of eigenvalues and functionals of the eigenfunction. The results unify the nonlinear case with prior linear theory, and the total error bounds provide practical guidance for QMC-based UQ in nonlinear PDEs, including connections to physical models like the Gross–Pitaevskii equation.

Abstract

In this paper, to the best of our knowledge, we make the first attempt at studying the parametric semilinear elliptic eigenvalue problems with the parametric coefficient and some power-type nonlinearities. The parametric coefficient is assumed to have an affine dependence on the countably many parameters with an appropriate class of sequences of functions. In this paper, we obtain the upper bound estimation for the mixed derivatives of the ground eigenpairs that has the same form obtained recently for the linear eigenvalue problem. The three most essential ingredients for this estimation are the parametric analyticity of the ground eigenpairs, the uniform boundedness of the ground eigenpairs, and the uniform positive differences between ground eigenvalues of linear operators. All these three ingredients need new techniques and a careful investigation of the nonlinear eigenvalue problem that will be presented in this paper. As an application, considering each parameter as a uniformly distributed random variable, we estimate the expectation of the eigenpairs using a randomly shifted quasi-Monte Carlo lattice rule and show the dimension-independent error bound.
Paper Structure (19 sections, 22 theorems, 173 equations)

This paper contains 19 sections, 22 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Main problem.
  3. Motivation and the goal.
  4. The summary of our contribution.
  5. Organization.
  6. Notation.
  7. Preliminaries
  8. Parametric semilinear eigenvalue problem
  9. Parametric analyticity of the ground state.
  10. Uniform bounds of smallest eigenvalues.
  11. Bounding the mixed derivatives of the ground eigenpair.
  12. Application to Uncertainty Quantification
  13. Quasi-Monte Carlo methods.
  14. Dimension truncation error.
  15. Quasi-Monte Carlo error.
  16. ...and 4 more sections

Key Result

theorem 1

There exists a ground eigenpair $(u,\lambda)$ of semi1 where $u$ is unique up to the modulus $|\cdot|$. Moreover, if $(d,p)\in \mathcal{A}$, the unique non-negative real-valued solution $|u|$ is indeed strictly positive in $\Omega$.

Theorems & Definitions (42)

  • theorem 1
  • lemma 1
  • proof
  • definition 1
  • theorem 2
  • proof
  • lemma 2
  • proof
  • lemma 3
  • proof
  • ...and 32 more