The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations

Xiu, Dongbin

doi:10.21236/ada460654

Cited by 396 publications

(758 citation statements)

References 14 publications

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“…For a stochastic system with Gaussian random variables, the polynomials are Hermite polynomials in terms of multidimensional Gaussian random variables (Ghanem and Spanos 1991). The generalized polynomial chaos provides a more efficient way to represent non-Gaussian problems (Xiu and Karniadakis 2002). We highlight that polynomial chaos expansion is often used to represent a random process/field in the literature while in this work polynomial chaos expansion is employed to represent a smooth function about random variables n i p n o .…”

Section: Probabilistic Collocation Methodsmentioning

confidence: 99%

A comparative study of numerical approaches to risk assessment of contaminant transport

Zhang

Shi

Chang

et al. 2010

Stoch Environ Res Risk Assess

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In risk analysis, a complete characterization of the concentration distribution is necessary to determine the probability of exceeding a threshold value. The most popular method for predicting concentration distribution is Monte Carlo simulation, which samples the cumulative distribution function with a large number of repeated operations. In this paper, we first review three most commonly used Monte Carlo (MC) techniques: the standard Monte Carlo, Latin Hypercube sampling, and Quasi Monte Carlo. The performance of these three MC approaches is investigated. We then apply stochastic collocation method (SCM) to risk assessment. Unlike the MC simulations, the SCM does not require a large number of simulations of flow and solute equations. In particular, the sparse grid collocation method and probabilistic collocation method are employed to represent the concentration in terms of polynomials and unknown coefficients. The sparse grid collocation method takes advantage of Lagrange interpolation polynomials while the probabilistic collocation method relies on polynomials chaos expansions. In both methods, the stochastic equations are reduced to a system of decoupled equations, which can be solved with existing solvers and whose results are used to obtain the expansion coefficients. Then the cumulative distribution function is obtained by sampling the approximate polynomials. Our synthetic examples show that among the MC methods, the Quasi Monte Carlo gives the smallest variance for the predicted threshold probability due to its superior convergence property and that the stochastic collocation method is an accurate and efficient alternative to MC simulations.

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Section: Probabilistic Collocation Methodsmentioning

confidence: 99%

A comparative study of numerical approaches to risk assessment of contaminant transport

Zhang

Shi

Chang

et al. 2010

Stoch Environ Res Risk Assess

View full text Add to dashboard Cite

show abstract

“…This section is devoted to recalling some basic facts about polynomial chaos (see, e.g., [10,25,33]), as well as to setting the notation.…”

Section: (Wiener) Polynomial Chaosmentioning

confidence: 99%

“…The system of orthonormal polynomials which gives rise to a generalized polynomial chaos, similar to the Wiener chaos, is determined by the density function; for instance, the uniform density obviously leads to the Legendre polynomials. We refer to [33] for more details.…”

Section: (Wiener) Polynomial Chaosmentioning

confidence: 99%

“…The most popular examples of such expansions are the KarhunenLoève expansions (see, e.g., [22]), which require the knowledge of the eigenfunctions of the covariance kernel of the random input, and the Polynomial Chaos expansions (also termed (generalized) Wiener Chaos expansions [32,33]), which exploit the classical technology of weighted orthogonal polynomials. The stochastic problem is then transformed into a deterministic one in higher dimension.…”

Section: Introductionmentioning

confidence: 99%

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A fictitious domain approach to the numerical solution of PDEs in stochastic domains

Canuto

Kozubek

2007

Numer. Math.

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We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is "spectral" in the polynomial chaos order, in any subdomain which does not contain the random boundaries. Mathematics Subject Classification (1991)

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“…The difficulty is largely computational-additional dimensions introduced into the problem by the random treatment of microstructural material and morphological parameters lead to computational intractability when the material response includes damage and nonlinearity. As Fish and Wu [13] recently demonstrated, model order reduction approaches for the multiscale solvers e.g., [8,24,41], as well as efficient uncertainty quantification algorithms [6,15,39] are critical to the development of computationally tractable stochastic multiscale modeling of composite materials.…”

Section: Introductionmentioning

confidence: 99%

Multiscale modeling of failure in composites under model parameter uncertainty

2015

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This manuscript presents a multiscale stochastic failure modeling approach for fiber reinforced composites. A homogenization based reduced-order multiscale computational model is employed to predict the progressive damage accumulation and failure in the composite. Uncertainty in the composite response is modeled at the scale of the microstructure by considering the constituent material (i.e., matrix and fiber) parameters governing the evolution of damage as random variables. Through the use of the multiscale model, randomness at the constituent scale is propagated to the scale of the composite laminate. The probability distributions of the underlying material parameters are calibrated from unidirectional composite experiments using a Bayesian statistical approach. The calibrated multiscale model is exercised to predict the ultimate tensile strength of quasi-isotropic openhole composite specimens at various loading rates. The effect of random spatial distribution of constituent material properties on the composite response is investigated.

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The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations

Cited by 396 publications

References 14 publications

A comparative study of numerical approaches to risk assessment of contaminant transport

A comparative study of numerical approaches to risk assessment of contaminant transport

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

Multiscale modeling of failure in composites under model parameter uncertainty

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