[Paper Review] Semilinear elliptic eigenvalue problem: Parametric analyticity and the uncertainty quantification

Byeong-Ho Bahn|arXiv (Cornell University)|Aug 6, 2023

Probabilistic and Robust Engineering DesignDecision Sciences3 citations

TL;DR

This paper establishes parametric analyticity and mixed derivative bounds for ground eigenpairs in semilinear elliptic eigenvalue problems with affine parametric coefficients and power-type nonlinearities. It derives dimension-independent convergence rates for quasi-Monte Carlo methods in uncertainty quantification by proving that the eigenpair's Taylor coefficients decay sufficiently fast under $\bm{\beta} \in \ell^q$ for $q \in (0,1]$, enabling robust expectation estimation with error bounds independent of dimension.

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.

Motivation & Objective

• To analyze the parametric analyticity and regularity of ground eigenpairs in semilinear elliptic eigenvalue problems with affine parametric coefficients.
• To derive upper bounds for mixed derivatives of eigenpairs that mirror those in linear eigenvalue problems.
• To establish uniform boundedness and positive spectral gaps for the ground eigenvalues of linearized operators under parametric variation.
• To apply these results to uncertainty quantification using randomly shifted quasi-Monte Carlo methods.
• To prove dimension-independent error bounds for the expectation of eigenpairs when parameters are uniformly distributed.

Proposed method

• Utilizes parametric analyticity of eigenpairs in the parameter space $\bm{y} \in [-1/2, 1/2]^\mathbb{N}$, derived from the structure of the semilinear operator.
• Establishes uniform boundedness of the ground eigenpair and positive lower bounds on spectral gaps between the first and second eigenvalues of linearized operators.
• Applies a falling factorial technique to remove $\varepsilon > 0$ from derivative bounds, achieving sharp decay estimates of the form $C(|\bm{\nu}|!)^{1+\varepsilon}\bm{\beta}^{\bm{\nu}}$.
• Employs unanchored $\ell^2$-type norms and weighted function spaces to control mixed derivative growth in the parametric domain.
• Derives error bounds for randomly shifted quasi-Monte Carlo rules using the $\ell^r$-summability of the coefficient sequence $\bm{\beta}$ with $r \in (0,1]$.
• Constructs a weight sequence $\gamma_{\bm{u}}$ to ensure the $C_{\theta,s}$ constant in the error bound remains independent of dimension $s$.

Experimental results

Research questions

• RQ1 Can the parametric analyticity of ground eigenpairs be established for semilinear elliptic eigenvalue problems with affine parametric coefficients and power-type nonlinearities?
• RQ2 What is the sharp decay rate of mixed derivatives of the eigenpair components in the parametric domain?
• RQ3 Can dimension-independent convergence rates be achieved for quasi-Monte Carlo methods in the uncertainty quantification of such eigenvalue problems?
• RQ4 How does the choice of $\bm{\beta} \in \ell^q$ influence the convergence rate of the QMC approximation?
• RQ5 What conditions on the parametric coefficients ensure uniform boundedness and spectral gaps for the linearized operators?

Key findings

• The mixed derivatives of the ground eigenpair satisfy an upper bound of the form $C(|\bm{\nu}|!)^{1+\varepsilon}\bm{\beta}^{\bm{\nu}}$, with $\varepsilon > 0$ removed via a falling factorial technique, yielding sharp decay.
• The eigenpair remains uniformly bounded and the spectral gap between the first and second eigenvalues is uniformly bounded away from zero across the parametric domain.
• For $\bm{\beta} \in \ell^q$ with $q \in (0,1]$, the error bound for the randomly shifted quasi-Monte Carlo rule is $C \varphi(N)^{-(1/2\theta)}$, with $\theta \in [q/(2-q), 1)$, achieving dimension-independent convergence.
• When $q=1$, setting $\theta=1$ yields a bounded $C_{1,s}$ under the condition $\sum_{j \geq 1} \beta_j < \sqrt{6}$, ensuring convergence with rate $C \varphi(N)^{-1}$.
• The error rate achieves $C \varphi(N)^{-(1/2\theta)}$ with $\theta$ chosen such that $\theta \geq q/(2-q)$, allowing for optimal rates up to $C \varphi(N)^{-(1/q - 1/2)}$ when $q \in (2/3, 1)$.
• The construction of weights $\gamma_{\bm{u}}$ ensures the $C_{\theta,s}$ term in the error bound is independent of the number of parameters $s$, enabling dimension-independent analysis.