Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures

By Xiaoliang Wan, G. Em Karniadakis. SIAM Journal on Scientific Computing, 2006

531 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d11a7eac2fefd40ab4e0a03a19a99189fc92a522

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In Civil and Structural Engineering

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Citations in the corpus

An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations (2009, methodology): “The multi-element generalized polynomial chaos method (MEgPC ) was introduced to address discontinuities in the random space while preserving the convergence rate of the gPC method [18] [19] [20].”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, background): “where the Hα(ξ ) form a basis of classical stochastic approximation spaces, such as polynomial or piecewise polynomial spaces [10, 31, 43, 47 , 52].”
Multi-fidelity Gaussian process regression for prediction of random fields (2017, methodology): “, multivariate orthogonal polynomials selected from the Askey scheme [35,33,7,32], or generalized bi-orthogonal series expansion [5,30].”
Polynomial chaos for the approximation of uncertainties: Chances and limits (2008, methodology): “In [53, 54] Wan and Karniadakis introduced an adaptive method, similar to spectral element methods in deterministic problems, which we will discuss below.”
Polynomial chaos expansions for uncertainty propagation and moment independent sensitivity analysis of seawater intrusion simulations (2015, background): “Fourth, with the introduction of generalized PCEs (gPCE) (Xiu and Karniadakis, 2002, 2003a), multi-element gPCEs (Wan and Karniadakis, 2006; Prempraneerach et al., 2010) and arbitrary PCEs (aPCEs) (Oladyshkin and Nowak, 2012), PCEs can handle many probability distribution types (e.g., normal,…”
A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems (2010, unknown): “Finally, let us mention that long-time integration errors are a major problem with the polynomial chaos theory, which has been addressed by Wan and Karniadakis [61] who developed a multi-element generalized polynomial chaos (ME-gPC) method.”
Data-driven uncertainty quantification using the arbitrary polynomial chaos expansion (2012, methodology): “The ME-gPC is the first adaptive piecewise approach helping to deal with discontinuity of distributions or of model responses, and provides the desired adaptation to a wide spectrum of distributions.”
Solving elliptic problems with non-Gaussian spatially-dependent random coefficients (2009, methodology): “One option to reduce the errors 1 and 2 is the multielement extension of polynomial chaos methods [14,15], where the parametric space is decomposed for extra h-type convergence.”
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients (2009, methodology): “Clearly, other forms of approximation other than global polynomials are possible and have been explored in recent years, such as wavelet approximations [17, 19] or piecewise polynomials [1, 30].”
Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems (2018, background): “They are the deterministic version of Equation 26 and 28.”
Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison (2011, background): “Other global polynomial spaces has been considered recently, see for instance [5, 24], as well as different approximation spaces such as piecewise polynomials [2, 12, 25 ].”
Bayesian updating via bootstrap filtering combined with data-driven polynomial chaos expansions: methodology and application to history matching for carbon dioxide storage in geological formations (2013, background): “Because the reduced model is merely a polynomial and has exploitable properties due to the orthogonal basis, it is vastly faster than the original one and offers a large playground for stochastic analysis [24, 57, 85], risk assessment [58, 84], and global sensitivity analysis [10, 57, 79].”
Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations (2009, methodology): “This kind of approach then makes sense in the context of adaptive approximation procedures [60, 74, 81, 127, 128].”