Approximate methods for stochastic eigenvalue problems

摘要

We consider the discretization and solution of eigenvalue problems of elliptic operators with random coefficients. For solving the resulting systems of equations we present a new and efficient spectral inverse iteration based on the stochastic Galerkin approach with respect to a polynomial chaos basis. The curse of dimensionality inherent in normalization over parameter spaces is avoided by a solution of a non-linear system of equations defining the Galerkin coefficients. For reference we also present an algorithm for adaptive stochastic collocation. Functionality of the algorithms is demonstrated by applying them on four examples of a given model problem. Convergence of the Galerkin-based method is analyzed and the results are tested against the collocated reference solutions and theoretical predictions.