4 research outputs found
Multilevel quasi-Monte Carlo for random elliptic eigenvalue problems I: Regularity and error analysis
Random eigenvalue problems are useful models for quantifying the uncertainty
in several applications from the physical sciences and engineering, e.g.,
structural vibration analysis, the criticality of a nuclear reactor or photonic
crystal structures. In this paper we present a simple multilevel quasi-Monte
Carlo (MLQMC) method for approximating the expectation of the minimal
eigenvalue of an elliptic eigenvalue problem with coefficients that are given
as a series expansion of countably-many stochastic parameters. The MLQMC
algorithm is based on a hierarchy of discretisations of the spatial domain and
truncations of the dimension of the stochastic parameter domain. To approximate
the expectations, randomly shifted lattice rules are employed. This paper is
primarily dedicated to giving a rigorous analysis of the error of this
algorithm. A key step in the error analysis requires bounds on the mixed
derivatives of the eigenfunction with respect to both the stochastic and
spatial variables simultaneously. An accompanying paper [Gilbert and Scheichl,
2021], focusses on practical extensions of the MLQMC algorithm to improve
efficiency, and presents numerical results
Asymptotic convergence of spectral inverse iterations for stochastic eigenvalue problems
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.Peer reviewe