Inexact Methods for Symmetric Stochastic Eigenvalue Problems

Abstract

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. Sousedík, H. C. Elman, SIAM/ASA Journal on Uncertainty Quantification 4(1), pp. 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.

Stochastic Galerkin methods for linear stability analysis of systems with parametric uncertainty

“…From this perspective, our study can be viewed as an extension of the setup from [6] to problems with viscosity given in the form of stochastic expansion and their efficient solution using stochastic Galerkin method. However, more importantly, we illustrate that the inexact methods for stochastic eigenvalue problems proposed recently by Lee and Sousedík [19] can be also applied to problems with nonsymmetric matrix operator 1 . This in general allows to perform a linear stability analysis for other types of problems as well.…”

“…For the mean-based preconditioner we used Re = 0.97, but the preconditioner appeared to be far less sensitive to a specific value of Re , and Re = 1 otherwise. We note that this way the algorithms are still formulated for a generalized nonsymmetric eigenvalue problem unlike in [19], where we studied symmetric problems and in implementation we used a Cholesky factorization of the mass matrix in order to formulate a standard eigenvalue problem.…”

“…Our approach is based on inexact Newton iteration: the linear systems with Jacobian matrices are solved using GMRES, for which we also develop several preconditioners. The preconditioners are motivated by our prior work on (truncated) hierarchical preconditioning [32,19], see also [2]. For an overview of literature on solving eigenvalue problems in the context of spectral stochastic finite element methods we refer to [1,19,29] and the references therein.…”

“…The preconditioners are motivated by our prior work on (truncated) hierarchical preconditioning [32,19], see also [2]. For an overview of literature on solving eigenvalue problems in the context of spectral stochastic finite element methods we refer to [1,19,29] and the references therein. Recently, Hakula and Laaksonen [15] studied crossing of eigenmodes in the stochastic parameter space, and Elman and Su [9] developed a low-rank inverse subspace iteration.…”

An efficient reduced‐order method for stochastic eigenvalue analysis

“…Several improvements 7,9 are proposed to reduce computational costs. Other extensions of PC‐based methods are to solve stochastic eigenvalue problems by combing PC expansion and deterministic numerical techniques, for example, power method, 10 inverse power method, 11‐13 subspace iteration 14‐16 . Other methods are also proposed to solve stochastic eigenvalue problems.…”