High-Order Collocation Methods for Differential Equations with Random Inputs

By Dongbin Xiu, Jan S. Hesthaven. SIAM Journal on Scientific Computing, 2005

1,690 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/1408c03eab3e7a2e67144400139a950ffb697d28

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Citations in the corpus

Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* (2011, background): “We mention only the book by Ghanem and Spanos (2007) and the series of papers by Xiu and Karniadakis (2002a), Xiu and Hesthaven (2005), Oden, Babuška, Nobile, Feng and Tempone (2005), Babuška, Nobile and Tempone (2005, 2007a, 2007b), Nobile, Tempone and Webster (2008a, 2008b), Nobile and Tempone…”
Adaptive sparse polynomial chaos expansion based on least angle regression (2011, methodology): “Note that a non intrusive scheme based on Lagrange interpolation of the model response , namely the stochastic collocation method , was also proposed in the past few years as an efficient alternative to SSFEM [ 13 , 14 ] .”
Stochastic collocation and mixed finite elements for flow in porous media (2008, methodology): “Choices of collocation points include tensor product of zeros of orthogonal polynomials [2,39], sparse grid approximations [39,27,28], and probabilistic collocation [23].”
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.”
Robust topology optimization accounting for misplacement of material (2012, methodology): “We should note that more recent uncertainty quantification methods such as the stochastic collocation method (Xiu and Hesthaven 2005) and regression methods (Sudret 2008) can outperform the Monte Carlo method even in high-dimensional problems.”
Polynomial chaos expansions for uncertainty propagation and moment independent sensitivity analysis of seawater intrusion simulations (2015, methodology): “…methods may be classified as projection methods (Ghiocel and Ghanem, 2002; Le Maitre et al., 2002), stochastic or probabilistic collocation methods (Xiu and Hesthaven, 2005), gradient-based methods also known as collocation methods coupled with sensitivity derivatives (Perez, 2008), and regression…”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, methodology): “Stochastic collocation methods (Babuška et al. 2007; Xiu and Hesthaven 2005), on the ot er hand, are “non-intrusive”; these require only a finite number of uncoupled deterministic simulations, with no reformulation of the governing equationsof the forward model.”
Computational Aspects of Stochastic Collocation with Multifidelity Models (2014, background): “Although the main idea is classical in nature, as all sampling methods (random or deterministic) are of collocation type, SC methods have become widely popular in practical problems because of their high-order properties [23], especially in conjunction with the gPC”
Efficient posterior exploration of a high‐dimensional groundwater model from two‐stage Markov chain Monte Carlo simulation and polynomial chaos expansion (2013, methodology): “In contrast, sparse grid approximation methods [e.g., Smolyak, 1963; Gerstner and Griebel, 2003; Xiu and Hesthaven, 2005; Babuska et al., 2007; Nobile et al., 2008a, 2008b] can drastically reduce the required number of collocation points while preserving a high level of accuracy for moderately…”
Variance-based global sensitivity analysis for multiple scenarios and models with implementation using sparse grid collocation (2015, methodology): “…integration (Smolyak, 1963), and have been shown to be an efficient and effective tool to overcome the curse of dimensionality for high dimensional numerical integration and interpolation (Barthelmann et al., 1999; Bungartz and Griebel, 2004; Gerstner and Griebel, 1998; Xiu and Hesthaven, 2005).”
A stochastic collocation approach to Bayesian inference in inverse problems (2009, background): “An alternative to the stochastic Galerkin approach to uncertainty propagation is stochastic collocation [25,27].”