An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations

By Xiang Ma, Nicholas Zabaras. Journal of Computational Physics, 2009

504 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d2d343f07f044005b3c764b217e4e52ce7ae6c83

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Citations in the corpus

A non-adapted sparse approximation of PDEs with stochastic inputs (2010, methodology): “Among them are the multi-scale model reduction of [33] and the sparse decomposition of [60,8,6,7] for the stochastic Galerkin technique, adaptive sparse regression approaches of [10,11] for PC expansions, anisotropic and adaptive sparse grids of [50,44] for the stochastic collocation scheme, and the”
Surrogate and reduced-order modeling: a comparison of approaches for large-scale statistical inverse problems [Chapter 7] (2010, methodology): “Collocation methods using sparse grids (Bieri et al. 2009; Ganapathysubramanian and Zabaras 2007; Ma and Zabaras 2009; Nobile et al. 2008; Smolyak 1963; Xiu and Hesthaven 2005), offer great efficiency and ease of impleentation for higherdimensional problems.”
Variance-based global sensitivity analysis for multiple scenarios and models with implementation using sparse grid collocation (2015, background): “, 2014 showed a highly non-smoothed likelihood function for a groundwater reactive transport model), the sparse grid methods may not perform well until a large number of sparse grid points are used (Ganapathysubramanian and Zabaras, 2007; Ma and Zabaras, 2009).”
Surrogate accelerated sampling of reservoir models with complex structures using sparse polynomial chaos expansion (2015, methodology): “Among them are the multi-scale model reduction of [57] and the sparse decomposition of [60, 59, 58] for the stochastic Galerkin technique, anisotropic and adaptive sparse grids of [51, 62, 61] for the stochastic collocation scheme, and low-rank solution approximations of [63, 64, 65, 78].”
A reduced spectral function approach for the stochastic finite element analysis (2011, methodology): “More recently efficient collocation methods have been proposed [26,27].”
Bayesian Model and Dimension Reduction for Uncertainty Propagation: Applications in Random Media (2017, methodology): “Amongst existing methods for uncertainty propagation, those based on (generalized) polynomial chaos expansions (gPC, [81]) have grown into prominence in recent years with the development of non-intrusive, stochastic collocation approaches [85, 43].”
ASSESSMENT OF COLLOCATION AND GALERKIN APPROACHES TO LINEAR DIFFUSION EQUATIONS WITH RANDOM DATA (2011, background): “Additional modifications to the stochastic collocation for adaptively dealing with very high dimensional problems are considered in [18, 19].”
Multi-output separable Gaussian process: Towards an efficient, fully Bayesian paradigm for uncertainty quantification (2013, methodology): “A partial remedy of the situation can be found by using sparse grids (SG) based on the Smolyak algorithm [3], which have a weaker dependence on the dimensionality of the problem (see [4–6] and the adaptive version developed by our group [7]).”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, background): “Approximation of low regularity functions in high dimensions is particularly difficult; see, for instance, [38, 37, 41, 16, 17].”
A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regime (2019, methodology): “Instead of viewing the encoding step (dimension reduction) separately from the surrogate modeling task Ma and Zabaras (2011); Xing et al. (2016), both are integrated and trained simultaneously, ensuring that the encoded, latent representation of the microstructure is maximally predictive of the soug”
Computational Aspects of Stochastic Collocation with Multifidelity Models (2014, unknown): “, [1, 2, 3, 4, 5, 7, 8, 12, 15, 17, 19, 21].”
Deep Convolutional Encoder‐Decoder Networks for Uncertainty Quantification of Dynamic Multiphase Flow in Heterogeneous Media (2018, methodology): “, 2017] or local response nonlinearity [Ma and Zabaras, 2009; Mo et al., 2017; Zhang et al., 2013].”