  We develop new multilevel Monte Carlo (MLMC) methods to estimate the
expectation of the smallest eigenvalue of a stochastic convection-diffusion
operator with random coefficients. The MLMC method is based on a sequence of
finite element (FE) discretizations of the eigenvalue problem on a hierarchy of
increasingly finer meshes. For the discretized, algebraic eigenproblems we use
both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi
(IRA), providing an analysis of the cost in each case. By studying the variance
on each level and adapting classical FE error bounds to the stochastic setting,
we are able to bound the total error of our MLMC estimator and provide a
complexity analysis. As expected, the complexity bound for our MLMC estimator
is superior to plain Monte Carlo. To improve the efficiency of the MLMC
further, we exploit the hierarchy of meshes and use coarser approximations as
starting values for the eigensolvers on finer ones. To improve the stability of
the MLMC method for convection-dominated problems, we employ two additional
strategies. First, we consider the streamline upwind Petrov--Galerkin
formulation of the discrete eigenvalue problem, which allows us to start the
MLMC method on coarser meshes than is possible with standard FEs. Second, we
apply a homotopy method to add stability to the eigensolver for each sample.
Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte
Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster
convergence of QMC, this improves the overall complexity. We provide detailed
numerical results comparing our different strategies to demonstrate the
practical feasibility of the MLMC method in different use cases. The results
support our complexity analysis and further demonstrate the superiority over
plain Monte Carlo in all cases.
