Inexact Methods for Symmetric Stochastic Eigenvalue Problems
This work addresses the computational bottleneck of solving eigenvalue problems in spectral stochastic finite elements for uncertainty quantification, but the methods are incremental improvements over existing approaches.
The paper develops two inexact methods for solving symmetric stochastic eigenvalue problems, achieving comparable accuracy to Monte Carlo and stochastic collocation with reduced computational cost, as demonstrated by numerical experiments.
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\'ık, 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.