Asymptotic convergence of spectral inverse iterations for stochastic eigenvalue problems

Abstract

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.

Multiparametric shell eigenvalue problems

“…For instance, the second smallest eigenvalue of the Laplace operator on a symmetric domain is of multiplicity two. Thus, as in the example considered in [4] for instance, the second and third eigenvalues of a parameter dependent extension of the problem typically cross within the parameter space.…”

“…The asymptotic convergence of Algorithm 3 has been studied in [4]. In particular we have the following result.…”

“…We then consider the spectral inverse iteration and its subspace iteration variant for computing the coefficients of the solution in the constructed basis. We refer to [10,4] for a detailed description of the algorithms.…”

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

“…A direct projection onto the basis functions will result in large coupled nonlinear systems that can be solved by a Newton-type algorithm [6,13]. Alternatives that do not use nonlinear solvers are stochastic versions of the (inverse) power methods and subspace iteration algorithms [16,17,26,34,38]. These methods have been shown to produce accurate solutions compared with the Monte Carlo or collocation methods.…”

Inexact Methods for Symmetric Stochastic Eigenvalue Problems

“…The authors of [23,33] used a quadrature-based normalization of eigenvectors. Normalization based on a solution of a small nonlinear problem was proposed by Hakula et al [12], and Hakula and Laaksonen [13] also provided an asymptotic convergence theory for the stochastic iteration. In an alternative approach, Ghanem and Ghosh [6,9] proposed two numerical schemes-one based on Newton iteration and another based on an optimization problem (see also [8,10]).…”