Modeling Uncertainty in Steady State Diffusion Problems via Generalized Polynomial Chaos

By Dongbin Xiu, G. Em Karniadakis. 2002

615 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Environmental Engineering, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/52f963c18853392662df9f601ac5107af20f260d

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In Computational Theory and Mathematics

In Environmental Engineering

In Statistics and Probability

Citations in the corpus

Stochastic spectral methods for efficient Bayesian solution of inverse problems (2005, methodology): “[33] used generalized polynomial chaos [34] for uncertainty quantification in fluid-structure interactions and in diffusion problems [35], while Debusschere et al.”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, background): “4) (see also [32, 4, 27, 62] and the references therein).”
Assessment of spectral representation and Karhunen–Loève expansion methods for the simulation of Gaussian stochastic fields (2007, methodology): “[1,2,5,7,8,15,33,45].”
An efficient, high-order perturbation approach for flow in random porous media via Karhunen-Loève and polynomial expansions (2004, methodology): “Unlike in the polynomial chaos expansion approach of Ghanem and Spanos [8] or the generalized polynomial chaos expansion of Xiu and Karni- adakis [28] where all the equations governing the coefficients are coupled, the equations from the present approach are recursive in that the high-order equation”
A new stochastic approach to transient heat conduction modeling with uncertainty (2003, methodology): “(8), following a similar procedure as in [35].”
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, neutral): “We mention, among others, the Spectral Galerkin method [3, 4, 16, 20, 22, 25, 27, 36], Stochastic Collocation [5, 24, 29, 35], perturbation methods or Neumann expansions [1, 17,30,34].”
N-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs (2014, background): “We mention the classic reference [12] and the more recent publications [17, 25, 24, 2, 21, 20, 4, 18, 10] and the references there.”
An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations (2009, methodology): “The gPC was successfully applied to model uncertainty propagation in various applications [8] [9] [10].”
Uncertainty quantification in chemical systems (2009, methodology): “GPC constructions have been used successfully in various physical systems [25, 28, 29].”
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients (2009, background): “[11, 21, 33, 2, 28]) which consists in projecting the original equation onto a polynomial subspace which could be constructed as the span of tensor product polynomials or polynomials having total degree smaller that a given integer p.”
A Polynomial Chaos-Based Kalman Filter Approach for Parameter Estimation of Mechanical Systems (2010, methodology): “Xiu extended the approach to general formulations based on Wiener-Askey polynomials family [43], and applied it to fluid mechanics [44-46].”
Uncertainty propagation using Wiener-Haar expansions (2004, background): “Similar to the WHe [1,8] and other spectral representations [ 20 ], the wavelet expansion (Eq. (21)) is an orthonormal approximation of the stochastic process.”