Dimension–Adaptive Tensor–Product Quadrature

By Thomas Gerstner, Michael Griebel. Computing, 2003

590 citations recorded by the source corpus.

Subfields: Applied Mathematics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/88806bcc479d32663bdb004bf0fb59191dcf5a8f

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Citations in the corpus

Likelihood approximation by numerical integration on sparse grids (2008, background): “A c Gerstner and Griebel (2003) suggest a ‘dimension adaptive’ algorithm based on sparse grids integration which uses a similar approach but searches for a suﬃcient accuracy level in each dimension separately. c When increasing the accuracy, g ( · ) has to evaluated at the additional nodes only and…”
Sparse Gauss-Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation (2011, methodology): “The dimensional adaptive quadraturemethod based on sparse grids was proposed in [35] to solve the numerical integration of multivariate functions.”
Approximation of high-dimensional parametric PDEs * (2015, methodology): “2 A very similar greedy algorithm was proposed in [39] in the slightly different context of adaptive quadrature, that is, when we want to approximate the integral of u over the domain U rather than u itself.”
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…”
ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS (2012, methodology): “We also compare our results with the dimension adaptive algorithm [14], in the implementation proposed in [17] and available at”
Sparse-grid, reduced-basis Bayesian inversion (2015, methodology): “, J , may have very different influence on the output of interest, we relax the total degree structure |i| ≤ q of the Smolyak SG construction introduced in the last section and adopt a dimension-adaptive construction for a generalized sparse grid [26].”
A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids - Part II: Transient simulation using space-time separated representations (2007, background): “Moreover, some recent works [9] on sparse grids prove its ability for integrating functions defined in really high-dimensional spaces (an example in dimension 256 was reported in that paper), enlarging the perspectives towards the solution of multidimensional partial differential equations.”
Sparse grid collocation schemes for stochastic natural convection problems (2007, methodology): “We closely follow the dimension adaptive quadrature approach developed by Gerstner and Griebel [36] and further developed by Klimke [32].”
Adaptive sparse polynomial chaos expansion based on least angle regression (2011, methodology): “On the one hand, the dimension-adaptive tensor product quadrature technique outlined in [15] was applied to the stochastic collocation scheme by [16].”
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, unknown): “For instance, see [8, 19] where anisotropic Smolyak formulas have been proposed.”