The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations

By Dongbin Xiu, G. Em Karniadakis. SIAM Journal on Scientific Computing, 2002

4,961 citations recorded by the source corpus.

Subfields: Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7692786e62ae71b16a2998c74316a5a8e107eb82

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Citations in the corpus

High-Order Collocation Methods for Differential Equations with Random Inputs (2005, background): “The generalizations, also called generalized polynomial chaos, employ non-Hermite polynomials to improve efficiency for a wider class of problems and include global polynomial expansions based on hypergeometrical polynomials [53, 52, 54], piecewise polynomial expansions [2, 12], and wavelet basis ex”
Arbitrary polynomial chaos expansion method for uncertainty quantification and global sensitivity analysis in structural dynamics (2020, background): “Most recently, PCE is further extended by Wan and Karniadakis 106 [40] and Witteveen and Bijl [41] so that it can be suitable for modeling the arbitrary 107 probability distributed random variables.”
Numerical Methods and Smolyak Quadrature for Nonlinear Stochastic Partial Differential Equations (2003, neutral): “[54].”
Metamodel-based sensitivity analysis: polynomial chaos expansions and Gaussian processes (2016, neutral): “…to the squared norm of In general orthogonal bases may be obtained by applying the Gram-Schmidt orthogonalization procedure, e.g. to the canonical family of monomials 1 , x, x 2 , . . . . (cid:8) For standard distributions, the associated families of orthogonal polynomials are well-known [64].”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, neutral): “We mention, for instance, the so-called Karhunen–Loève [39, 40] or a polynomial chaos (PC) expansion [59, 63].”
Efficient posterior exploration of a high‐dimensional groundwater model from two‐stage Markov chain Monte Carlo simulation and polynomial chaos expansion (2013, methodology): “For instance, Jacobi polynomials are typically associated with a beta distribution, Hermite polynomials with a Gaussian distribution, Laguerre polynomials with a gamma distribution, and Legendre polynomials, that represent a special case of the Jacobi polynomials, with an uniform distribution [Xiu a”
Polynomial chaos expansions for dependent random variables (2019, methodology): “The poor performance we observed here is consistent with results observed in the univariate case [64] and the multivariate results observed in [19].”
A stochastic finite element procedure for moment and reliability analysis (2006, methodology): “Projection method This method was used by (Puiget al., 2002, Xiu and Karniadakis, 2002, Field and Grigoriu, 2004).”
Generalized spectral decomposition method for solving stochastic finite element equations : Invariant subspace problem and dedicated algorithms (2008, methodology): “For the approximation at the stochastic level, we use a generalized polynomial chaos approximation of order p [21,22].”
Simulation-based optimal Bayesian experimental design for nonlinear systems (2011, methodology): “Many options exist, ranging from projection-based model reduction [45, 46] to spectral methods based on polynomial chaos (PC) expansions [47, 22, 23, 48, 49, 50, 51].”
Modeling Multibody Systems with Uncertainties. Part I: Theoretical and Computational Aspects (2006, methodology): “A generalization of the Cameron and Martin theorem [57] was employed in [ 46 ] to conclude that each type of Askey chaos converges to any L2 functional in the L2 sense in the”