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Title: Inverse Subspace Iteration for Spectral Stochastic Finite Element Methods

Abstract

We study random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its eigenvalues and eigenvectors represented using polynomial chaos expansions. We formulate a version of stochastic inverse subspace iteration, which is based on the stochastic Galerkin finite element method, and we compare its accuracy with that of Monte Carlo and stochastic collocation methods. The coefficients of the eigenvalue expansions are computed from a stochastic Rayleigh quotient. Our approach allows the computation of interior eigenvalues by deflation methods, and we can also compute the coefficients of multiple eigenvectors using a stochastic variant of the modified Gram-Schmidt process. The effectiveness of the methods is illustrated by numerical experiments on benchmark problems arising from vibration analysis.

Authors:
 [1];  [2]
  1. Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States)
  2. Univ. of Maryland, College Park, MD (United States)
Publication Date:
Research Org.:
Univ. of Maryland Baltimore County (UMBC), Baltimore, MD (United States)
Sponsoring Org.:
National Science Foundation (NSF); USDOE
OSTI Identifier:
1418636
Grant/Contract Number:  
SC0009301; DMS1418754; DMS1521563.
Resource Type:
Accepted Manuscript
Journal Name:
SIAM/ASA Journal on Uncertainty Quantification
Additional Journal Information:
Journal Volume: 4; Journal Issue: 1; Journal ID: ISSN 2166-2525
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING

Citation Formats

Sousedík, Bedřich, and Elman, Howard C. Inverse Subspace Iteration for Spectral Stochastic Finite Element Methods. United States: N. p., 2016. Web. doi:10.1137/140999359.
Sousedík, Bedřich, & Elman, Howard C. Inverse Subspace Iteration for Spectral Stochastic Finite Element Methods. United States. doi:10.1137/140999359.
Sousedík, Bedřich, and Elman, Howard C. Thu . "Inverse Subspace Iteration for Spectral Stochastic Finite Element Methods". United States. doi:10.1137/140999359. https://www.osti.gov/servlets/purl/1418636.
@article{osti_1418636,
title = {Inverse Subspace Iteration for Spectral Stochastic Finite Element Methods},
author = {Sousedík, Bedřich and Elman, Howard C.},
abstractNote = {We study random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric positive-definite matrix operator, we explore the performance of algorithms for computing its eigenvalues and eigenvectors represented using polynomial chaos expansions. We formulate a version of stochastic inverse subspace iteration, which is based on the stochastic Galerkin finite element method, and we compare its accuracy with that of Monte Carlo and stochastic collocation methods. The coefficients of the eigenvalue expansions are computed from a stochastic Rayleigh quotient. Our approach allows the computation of interior eigenvalues by deflation methods, and we can also compute the coefficients of multiple eigenvectors using a stochastic variant of the modified Gram-Schmidt process. The effectiveness of the methods is illustrated by numerical experiments on benchmark problems arising from vibration analysis.},
doi = {10.1137/140999359},
journal = {SIAM/ASA Journal on Uncertainty Quantification},
number = 1,
volume = 4,
place = {United States},
year = {2016},
month = {2}
}

Journal Article:
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