  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.
