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...ere λ(ω) is a random vector and α = (α1, α2, . . . ) is a vector of non-negative indices. We denote the probability density function of the random vector λ by ρ(λ). Generalized polynomial chaos (gPC) =-=[25]-=-, provides a framework for representing second-order stochastic processes r ∈ L2(Γ,X) for arbitrary distributions of λ by the following expansion: r(λ) = ∞∑ |α|=0 aαHα(λ), (2) where |α| =∑i αi is the ...

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...nem and Spanos, 1991), random eigenfunction expansion: (Adhikari and Manohar, 1999), stochastic reduced basis method : (Nair and Keane, 2002, Sachdeva et al., 2006a,b), Wiener−Askey chaos expansion: (=-=Wan and Karniadakis, 2006-=-, Xiu and Karniadakis, 2002, 2003), domain decomposition method : (Sarkar et al., 2006a,b). 3. Monte carlo simulation Simulation methods: and other methods (Hurtado and Barbat, 1998, Papadrakakis and ...

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...n new interest for this method, but application tasks demanded further adaptation of the method to general distributions. For this reason, several PCE extensions have been developed in the last years =-=[23, 35, 37, 38, 39]-=-. A theoretical framework for arbitrary multivariate probability measures was laid out in [30]. These polynomial chaos expansion methods have been shown to be effective in a large amount of applicatio...

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...kis developed MEgPC [25,26]. This method discretizes the probability space into nonoverlapping partitions. Within each partition, the traditional single element gPC is performed. Summing element integrations provide a complete integration of the full probability space. The algorithm presented adaptively partitioned the space based on estimates of error convergence. When an error estimate deteriorated to a specified point the element was split. The initial work was developed for the GPM methodology using uniform distributions. MEgPC was subsequently extended to arbitrary distributions in Refs. [27] and [28]. Foo developed a collocation-based MEgPC in Ref. [29] and further extended the method to support higher dimensions using ANOVA methods in Ref. [30]. As an alternative to MEgPC, Witteveen and Iaccarino developed a similar multi-element method based on gPC called the simplex elements stochastic collocation method. This method adaptively partitions the probability space using simplex elements coupled with Newton–Cotes quadrature. Their method has shown an O(n) convergence as long as the approximating polynomial order is increased with the number of uncertainties. Another approach to add...

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...65]). The intrusive solution of general stochastic PDEs via spectral methods is relatively recent and it was made possible by the introduction of the generalized Polynomial Chaos (gPC) spectral basis =-=[70, 7, 73, 66, 72]-=-. 34 3.2 Product Structure of the Ambient Space We assume that the solution u of the weak form 3.1.3 lives in a suitable functional space S that we call ambient space. We further assume that S is a se...

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...lti-element gPC (MEgPC) [29, 30]. This method discretizes the probability space into non-overlapping partitions. Within each partition the traditional single element gPC is performed. Summing element integrations provides a complete integration of the full probability space. The algorithm presented adaptively partitioned the space based on estimates of error convergence. When an error estimate deteriorated to a specified point the element was split. The initial work was developed for the GPM methodology using uniform distributions. MEgPC was subsequently extended to arbitrary distributions in [31, 32]. Foo developed a collocation-based MEgPC in [33] and further extended the method to support higher dimensions using ANOVA methods in [34]. As an alternative to MEgPC, Witteveen and Iaccarino developed a similar multi-element method based on gPC called the simplex elements stochastic collocation (SESC) method. This method adaptively partitions the probability space using simplex elements coupled with Newton-Cotes quadrature. Their method has shown an O(n) convergence as long as the approximating polynomial order is increased with the number of uncertainties. Another approach to addressing the ...