Numerical Methods and Smolyak Quadrature for Nonlinear Stochastic Partial Differential Equations

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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]).”
Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations (2009, methodology): “Therefore, we generally opt for standard high order quadrature techniques (sparse quadrature for high stochastic dimensions [59, 83]) in order to reduce the number of integration points.”
A non-intrusive stochastic Galerkin approach for modeling uncertainty propagation in deformation processes (2007, methodology): “Methods such as h and p adaptive discretization of the support space, sparse grid and dimension adaptive quadrature techniques [21] can significantly reduce the curse of dimensionality.”