  Stochastic PDE eigenvalue problems often arise in the field of uncertainty
quantification, whereby one seeks to quantify the uncertainty in an eigenvalue,
or its eigenfunction. In this paper we present an efficient multilevel
quasi-Monte Carlo (MLQMC) algorithm for computing the expectation of the
smallest eigenvalue of an elliptic eigenvalue problem with stochastic
coefficients. Each sample evaluation requires the solution of a PDE eigenvalue
problem, and so tackling this problem in practice is notoriously
computationally difficult. We speed up the approximation of this expectation in
four ways: 1) we use a multilevel variance reduction scheme to spread the work
over a hierarchy of FE meshes and truncation dimensions; 2) we use QMC methods
to efficiently compute the expectations on each level; 3) we exploit the
smoothness in parameter space and reuse the eigenvector from a nearby QMC point
to reduce the number of iterations of the eigensolver; and 4) we utilise a
two-grid discretisation scheme to obtain the eigenvalue on the fine mesh with a
single linear solve. The full error analysis of a basic MLQMC algorithm is
given in the companion paper [Gilbert and Scheichl, 2021], and so in this paper
we focus on how to further improve the efficiency and provide theoretical
justification of the enhancement strategies 3) and 4). Numerical results are
presented that show the efficiency of our algorithm, and also show that the
four strategies we employ are complementary.
