  We consider and analyze applying a spectral inverse iteration algorithm and
its subspace iteration variant for computing eigenpairs of an elliptic operator
with random coefficients. With these iterative algorithms the solution is
sought from a finite dimensional space formed as the tensor product of the
approximation space for the underlying stochastic function space, and the
approximation space for the underlying spatial function space. Sparse
polynomial approximation is employed to obtain the first one, while classical
finite elements are employed to obtain the latter. An error analysis is
presented for the asymptotic convergence of the spectral inverse iteration to
the smallest eigenvalue and the associated eigenvector of the problem. A series
of detailed numerical experiments supports the conclusions of this analysis.
Numerical experiments are also presented for the spectral subspace iteration,
and convergence of the algorithm is observed in an example case, where the
eigenvalues cross within the parameter space. The outputs of both algorithms
are verified by comparing to solutions obtained by a sparse stochastic
collocation method.

Numerical Analysis;Numerical Analysis