  We study two inexact methods for solutions of random eigenvalue problems in
the context of spectral stochastic finite elements. In particular, given a
parameter-dependent, symmetric matrix operator, the methods solve for
eigenvalues and eigenvectors represented using polynomial chaos expansions.
Both methods are based on the stochastic Galerkin formulation of the eigenvalue
problem and they exploit its Kronecker-product structure. The first method is
an inexact variant of the stochastic inverse subspace iteration [B.
Soused\'{\i}k, H. C. Elman, SIAM/ASA Journal on Uncertainty Quantification
4(1), pp. 163--189, 2016]. The second method is based on an inexact variant of
Newton iteration. In both cases, the problems are formulated so that the
associated stochastic Galerkin matrices are symmetric, and the corresponding
linear problems are solved using preconditioned Krylov subspace methods with
several novel hierarchical preconditioners. The accuracy of the methods is
compared with that of Monte Carlo and stochastic collocation, and the
effectiveness of the methods is illustrated by numerical experiments.
