Quantifying Uncertainty
Ghosh, D. and Ghanem, R.G.. "Stochastic convergence acceleration through basis enrichment of polynomial chaos expansions" International Journal for Numerical Methods is Engineering. 73
(2).
JAN 2008.
pp. 162--184.
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 (c) 2007 John Wiley & Sons, Ltd.
Ghanem, R.G. and Ghosh, D.. "Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition" International Journal for Numerical Methods is Engineering. 72
(4).
OCT 22 2007.
pp. 486--504.
A new procedure for characterizing the solution of the eigenvalue problem in the presence of uncertainty is presented. The eigenvalues and eigenvectors are described through their projections on the polynomial chaos basis. An efficient method for estimating the coefficients with respect to this basis is proposed. The method uses a Galerkin-based approach by orthogonalizing the residual in the eigenvalue-eigenvector equation to the subspace spanned by the basis functions used for approximation. In this way, the stochastic problem is framed as a system of deterministic non-linear algebraic equations. This system of equations is solved using a Newton-Raphson algorithm. Although the proposed approach is not based on statistical sampling, the efficiency of the proposed method can be significantly enhanced by initializing the non-linear iterative process with a small statistical sample synthesized through a Monte Carlo sampling scheme. The proposed method offers a number of advantages over existing methods based on statistical sampling. First, it provides an approximation to the complete probabilistic description of the eigensolution. Second, it reduces the computational overhead associated with solving the statistical eigenvalue problem. Finally, it circumvents the dependence of the statistical solution on the quality of the underlying random number generator. Copyright (C) 2007 John Wiley & Sons, Ltd.