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Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions
Author(s) -
Ghosh Debraj,
Ghanem Roger
Publication year - 2007
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2066
Subject(s) - polynomial chaos , eigenvalues and eigenvectors , mathematics , stochastic process , polynomial , classification of discontinuities , basis (linear algebra) , projection (relational algebra) , convergence (economics) , mathematical optimization , mathematical analysis , monte carlo method , algorithm , geometry , statistics , physics , quantum mechanics , economics , economic growth
Abstract Given their mathematical structure, methods for computational stochastic analysis based on orthogonal approximations and projection schemes are well positioned to draw on developments from deterministic approximation theory. This is demonstrated in the present paper by extending basis enrichment from deterministic analysis to stochastic procedures involving the polynomial chaos decomposition. This enrichment is observed to have a significant effect on the efficiency and performance of these stochastic approximations in the presence of non‐continuous dependence of the solution on the stochastic parameters. In particular, given the polynomial structure of these approximations, the severe degradation in performance observed in the neighbourhood of such discontinuities is effectively mitigated. An enrichment of the polynomial chaos decomposition is proposed in this paper that can capture the behaviour of such non‐smooth functions by integrating a priori knowledge about their behaviour. The proposed enrichment scheme is applied to a random eigenvalue problem where the smoothness of the functional dependence between the random eigenvalues and the random system parameters is controlled by the spacing between the eigenvalues. It is observed that through judicious selection of enrichment functions, the spectrum of such a random system can be more efficiently characterized, even for systems with closely spaced eigenvalues. Copyright © 2007 John Wiley & Sons, Ltd.