Sparse grid collocation schemes for stochastic natural convection problems

By Baskar Ganapathysubramanian, Nicholas Zabaras. Journal of Computational Physics, 2007

422 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/5793e3269352707db38c0f0815f7c824967fd308

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Citations in the corpus

Stochastic Collocation Algorithms Using l1-Minimization for Bayesian Solution of Inverse Problems (2015, methodology): “The idea of a stochastic collocation method (SCM) for Bayesian inference was proposed by Marzouk and Xiu in [28], who presented a sparse grid collocation method [2, 18, 40, 42] using the Smolyak algorithm to construct a solution of the stochastic surrogate model.”
Convergence Analysis for Stochastic Collocation Methods to Scalar Hyperbolic Equations with a Random Wave Speed (2010, background): “Collocation methods have been studied and used in different disciplines for uncertainty quantification (see, e.g., Tatang [11], Xiu and Hesthaven [15], Keese and Matthies [8] Ganapathysubramanian and Zabaras [5]).”
A reduced spectral function approach for the stochastic finite element analysis (2011, methodology): “More recently efficient collocation methods have been proposed [26,27].”
eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces (2010, methodology): “A rst way to solve a PDE de ned on a random domain consists in using direct spectral stochastic approaches (interpolation [8, 9, 10, 11, 12], projection [13, 14, 15], regression [16]), or any other direct stochastic method such as Monte-Carlo or quasi Monte-Carlo simulations [17], coupled with a det”
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, methodology): “In this respect, the proposed method differs from the dimension-adaptive method developed in [15] and applied in the context of PDEs in [13], where the sparse approximation structure has to be uncovered only by successive enrichment, and does not exploit any a priori information of the solution.”
An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations (2009, methodology): “In recent papers [34] [35], the authors have applied this method to various stochastic problems.”
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients (2009, background): “Stochastic Collocation has gained much attention recently from the computational community, see for instance the works [3, 10, 23, 20, 32].”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, methodology): “Several methods have been proposed for the definition and computation of the approximate solution: L2 projection [19, 21], interpolation [3, 14, 49, 51, 54], regression [6] or Galerkin projections [2, 16, 18, 35].”
An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis (2010, methodology): “On the one hand, the dimension-adaptive tensor product quadrature technique outlined in [14] was applied to the stochastic collocation scheme by [15].”
Stability in the almost everywhere sense: A linear transfer operator approach (2010, background): “Recent developments in sparse collocation-based numerical methods for the solution of partial differential equation [15,16] provide a promising future direction for the computation of Lyapunov density in higher-dimensional dynamical systems.”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, methodology): “In the framework of PDEs with random input data, the sparse grid stochastic collocation method has been proposed in [61] and analyzed in [46] (see also [29, 26]).”
Variance-based global sensitivity analysis for multiple scenarios and models with implementation using sparse grid collocation (2015, background): “…is highly non-smooth (Shi et al., 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).”
ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS (2012, neutral): “[2, 12, 29]).”
Deep Convolutional Encoder‐Decoder Networks for Uncertainty Quantification of Dynamic Multiphase Flow in Heterogeneous Media (2018, background): “…computational cost, in which the training samples for surrogate construction are adaptively selected, e.g., based on the importance of dimensions [ Ganapathysubramanian and Zabaras , 2007; Liao et al. , 2017] or local response nonlinearity [ Ma and Zabaras , 2009; Mo et al. , 2017; Zhang et al. ,…”