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Bounding the spectral gap for an elliptic eigenvalue problem with uniformly bounded stochastic coefficients

Alexander D. Gilbert, Ivan G. Graham, Robert Scheichl, Ian H. Sloan

TL;DR

The work addresses bounding the spectral gap for a random elliptic eigenproblem with infinitely many stochastic parameters. It uses a variational formulation, shows Lipschitz dependence of eigenvalues on the parameter, and constructs a compact reparameterization to apply compactness arguments to obtain a uniform positive gap $\lambda_2(y)-\lambda_1(y) \ge \delta$. The main contribution is a rigorous uniform gap bound under a decay condition on the coefficient terms, supported by 1D numerical experiments that indicate a nonzero gap for affine coefficients and highlight open questions for lognormal coefficients. This result underpins robust error analyses for high-dimensional stochastic eigenproblems and improves reliability of uncertainty quantification methods that rely on spectral gaps.

Abstract

A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenvalue. When we are interested in the smallest or fundamental eigenvalue, we call this the spectral or fundamental gap. In a recent manuscript [Gilbert et al., arXiv:1808.02639], the current authors, together with Frances Kuo, studied an elliptic eigenvalue problem with homogeneous Dirichlet boundary conditions, and with coefficients that depend on an infinite number of uniformly distributed stochastic parameters. In this setting, the eigenvalues, and in turn the eigenvalue gap, also depend on the stochastic parameters. Hence, for a robust error analysis one needs to be able to bound the gap over all possible realisations of the parameters, and because the gap depends on infinitely-many random parameters, this is not trivial. This short note presents, in a simplified setting, an important result that was shown in the paper above. Namely, that, under certain decay assumptions on the coefficient, the spectral gap of such a random elliptic eigenvalue problem can be bounded away from 0, uniformly over the entire infinite-dimensional parameter space.

Bounding the spectral gap for an elliptic eigenvalue problem with uniformly bounded stochastic coefficients

TL;DR

The work addresses bounding the spectral gap for a random elliptic eigenproblem with infinitely many stochastic parameters. It uses a variational formulation, shows Lipschitz dependence of eigenvalues on the parameter, and constructs a compact reparameterization to apply compactness arguments to obtain a uniform positive gap . The main contribution is a rigorous uniform gap bound under a decay condition on the coefficient terms, supported by 1D numerical experiments that indicate a nonzero gap for affine coefficients and highlight open questions for lognormal coefficients. This result underpins robust error analyses for high-dimensional stochastic eigenproblems and improves reliability of uncertainty quantification methods that rely on spectral gaps.

Abstract

A key quantity that occurs in the error analysis of several numerical methods for eigenvalue problems is the distance between the eigenvalue of interest and the next nearest eigenvalue. When we are interested in the smallest or fundamental eigenvalue, we call this the spectral or fundamental gap. In a recent manuscript [Gilbert et al., arXiv:1808.02639], the current authors, together with Frances Kuo, studied an elliptic eigenvalue problem with homogeneous Dirichlet boundary conditions, and with coefficients that depend on an infinite number of uniformly distributed stochastic parameters. In this setting, the eigenvalues, and in turn the eigenvalue gap, also depend on the stochastic parameters. Hence, for a robust error analysis one needs to be able to bound the gap over all possible realisations of the parameters, and because the gap depends on infinitely-many random parameters, this is not trivial. This short note presents, in a simplified setting, an important result that was shown in the paper above. Namely, that, under certain decay assumptions on the coefficient, the spectral gap of such a random elliptic eigenvalue problem can be bounded away from 0, uniformly over the entire infinite-dimensional parameter space.

Paper Structure

This paper contains 5 sections, 3 theorems, 47 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Variational eigenvalue problems
  3. Bounding the spectral gap
  4. Numerical results
  5. Conclusion

Key Result

Lemma 1

Let ${\boldsymbol{\alpha}} \in \ell^q$ for some $1 < q < \infty$. The set $U({\boldsymbol{\alpha}}) \subset \ell^\infty$ given by is a compact subset of $\ell^\infty$.

Figures (5)

  • Figure 1: Estimate of the minimum of the spectral gap and estimate of the distance to the true minimum: affine coefficient \ref{['eq:a_general']} with $a_0 = c_0 = 1$, $a_{\min} = 0.18$, $a_{\max} = 1.82$.
  • Figure 2: Estimate of the minimum of the spectral gap and estimate of the distance to the true minimum: affine coefficient \ref{['eq:a_general']} with $a_0 = 1$, $c_0 = 1.223$, $a_{\min} = 2\times10^{-4}$, $a_{\max} = 2$.
  • Figure 3: Estimate of the minimum of the spectral gap and estimate of the distance to the true minimum: affine coefficient \ref{['eq:a_general']} with $a_0 = 1$, $c_0 = 0.5$, $a_{\min} = 0.59$, $a_{\max} = 1.41$.
  • Figure 4: Estimate of the minimum of the spectral gap and estimate of the distance to the true minimum: log-normal coefficient \ref{['eq:lognormal']} with $a_{\min} = a_* = 0$ and $c_0 = 1$.
  • Figure 5: Estimate of the minimum of the spectral gap and estimate of the distance to the true minimum: log-normal coefficient \ref{['eq:lognormal']} with $a_{\min} = a_* = 0.18$ and $c_0 = 1$.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Theorem 3
  • proof