Analysis of quasi-Monte Carlo methods for elliptic eigenvalue problems with stochastic coefficients

TL;DR

This paper analyzes the use of quasi-Monte Carlo methods for efficiently estimating the expected fundamental eigenvalue in stochastic elliptic eigenvalue problems, providing error bounds that depend on discretization and sampling parameters.

Contribution

It introduces a rigorous error analysis for quasi-Monte Carlo methods applied to stochastic eigenvalue problems, including uniform spectral gap bounds and convergence rates.

Findings

Error bounds of order $oldsymbol{h^2 + N^{-1+oldsymbol{ ext{delta}}}}$ for the eigenvalue expectation

Spectral gap remains uniformly positive across all random realizations

Error convergence rates match those of source problems despite nonlinearity

Abstract

We consider the forward problem of uncertainty quantification for the generalised Dirichlet eigenvalue problem for a coercive second order partial differential operator with random coefficients, motivated by problems in structural mechanics, photonic crystals and neutron diffusion. The PDE coefficients are assumed to be uniformly bounded random fields, represented as infinite series parametrised by uniformly distributed i.i.d. random variables. The expectation of the fundamental eigenvalue of this problem is computed by (a) truncating the infinite series which define the coefficients; (b) approximating the resulting truncated problem using lowest order conforming finite elements and a sparse matrix eigenvalue solver; and (c) approximating the resulting finite (but high dimensional) integral by a randomly shifted quasi-Monte Carlo lattice rule, with specially chosen generating vector. We prove error estimates for the combined error, which depend on the truncation dimension $s$, the finite element mesh diameter $h$, and the number of quasi-Monte Carlo samples $N$. Under suitable regularity assumptions, our bounds are of the particular form $\mathcal{O}(h^2+N^{-1+\delta})$, where $\delta>0$ is arbitrary and the hidden constant is independent of the truncation dimension, which needs to grow as $h\to 0$ and $N\to\infty$. Although the eigenvalue problem is nonlinear, which means it is generally considered harder than the analogous source problem, in almost all cases we obtain error bounds that converge at the same rate as the corresponding rate for the source problem. The proof involves a detailed study of the regularity of the fundamental eigenvalue as a function of the random parameters. As a key intermediate result in the analysis, we prove that the spectral gap (between the fundamental and the second eigenvalues) is uniformly positive over all realisations of the random problem.