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Stochastic PDE eigenvalue problems often arise in the field of uncertaintyquantification, whereby one seeks to quantify the uncertainty in an eigenvalue,or its eigenfunction. In this paper we present an efficient multilevelquasi-Monte Carlo (MLQMC) algorithm for computing the expectation of thesmallest eigenvalue of an elliptic eigenvalue problem with stochasticcoefficients. Each sample evaluation requires the solution of a PDE eigenvalueproblem, and so tackling this problem in practice is notoriouslycomputationally difficult. We speed up the approximation of this expectation infour ways: we use a multilevel variance reduction scheme to spread the workover a hierarchy of FE meshes and truncation dimensions; we use QMC methods toefficiently compute the expectations on each level; we exploit the smoothnessin parameter space and reuse the eigenvector from a nearby QMC point to reducethe number of iterations of the eigensolver; and we utilise a two-griddiscretisation scheme to obtain the eigenvalue on the fine mesh with a singlelinear solve. The full error analysis of a basic MLQMC algorithm is given inthe companion paper [Gilbert and Scheichl, 2022], and so in this paper we focuson how to further improve the efficiency and provide theoretical justificationfor using nearby QMC points and two-grid methods. Numerical results arepresented that show the efficiency of our algorithm, and also show that thefour strategies we employ are complementary.

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