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

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 O(h 2 + N −1+δ ), where δ > 0 is arbitrary and the hidden constant is independent of the truncation dimension, which needs to grow as h → 0 and N → ∞. As for the analogous PDE source problem, the conditions under which our error bounds hold depend on a parameter p ∈ (0, 1) representing the summability of the terms in the series expansions of the coefficients. Although the eigenvalue problem is nonlinear, which means it is generally considered harder than the source problem, in almost all cases (p = 1) 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.

Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems II: Efficient algorithms and numerical results

“…In this section we briefly summarise the relevant material on variational EVPs, two-grid FE methods and QMC methods. For further details we refer the reader to the references indicated throughout or to [19].…”

“…The Krein-Rutmann Theorem ensures that the smallest eigenvalue is simple, and then in [19,Prop. 2.4] it was shown that the spectral gap can be bounded away from 0 independently of y.…”

“…The latter two classes perform poorly for high-dimensional problems, so in order to handle the high-dimensionality of the parameter space, sparse and low-rank versions of those methods have been developed, see e.g., [1,24,17]. Furthermore, to improve upon classical Monte Carlo, while still performing well in high dimensions, the present authors with their colleagues have analysed the use of quasi-Monte Carlo methods [19,20].…”

Subspace Reduction for Stochastic Planar Elasticity

“…This global normalization constraint in fact makes the problem nonlinear as opposed to its deterministic counterpart. Moreover, current mathematical analysis (see, e.g., [4,7]) is based on second-order model problems where it is guaranteed that the smallest mode is well-isolated and hence the question of subspace convergence is still not resolved and is the subject of current research.…”

Bounding the Spectral Gap for an Elliptic Eigenvalue Problem with Uniformly Bounded Stochastic Coefficients

“…As an example of the important role that the spectral gap plays in error analysis, consider the random elliptic eigenvalue problem from [8]. There, an algorithm using dimension truncation, Quasi-Monte Carlo (QMC) quadrature and finite element (FE) methods was used to approximate the expectation with respect to the stochastic parameters of the smallest eigenvalue.…”