Numerical integration using sparse grids

By Thomas Gerstner, Michael Griebel. Numerical Algorithms, 2004

1,005 citations recorded by the source corpus.

Subfields: Applied Mathematics, Atomic and Molecular Physics, and Optics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/fdb25ab36dd1b5e981fcf71eb09fc2c4bdca0403

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Ivan V. Oseledets
2. Nicholas Zabaras
3. E. E. Tyrtyshnikov
4. Fabio Nobile
5. Raúl Tempone
6. Anthony Nouy
7. Youssef M. Marzouk
8. Baskar Ganapathysubramanian

In Atomic and Molecular Physics, and Optics

1. Ivan V. Oseledets
2. Boris N. Khoromskij
3. E. E. Tyrtyshnikov
4. Wolfgang Hackbusch
5. Anthony Nouy
6. Youssef M. Marzouk
7. Nicholas Zabaras
8. M. Espig

In Numerical Analysis

1. Ivan V. Oseledets
2. Anthony Nouy
3. Nicholas Zabaras
4. Fabio Nobile
5. Raúl Tempone
6. Clayton G. Webster
7. Youssef M. Marzouk
8. Baskar Ganapathysubramanian

Citations in the corpus

• Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations (2009, methodology): “⋄ Smolyak tensorization Smolyak tensorization formula can be used in order to drastically decrease the number of integration points when dealing with high stochastic dimension m [118,42,96,100, 84].”
• An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, methodology): “A variant of the Clenshaw-Curtis formulas are the Filippi formulas in which the abscissas at the boundary of the interval are omitted [15].”
• Explicit Constructions of Quasi-Monte Carlo Rules for the Numerical Integration of High-Dimensional Periodic Functions (2007, background): “Another construction of quadrature rules is due to Smolyak [31] and is nowadays called sparse grid, see also [8].”
• A tractable approximation of non-convex chance constrained optimization with non-Gaussian uncertainties (2015, methodology): “Integration nodes of the sparse grid integration technique (Bungartz and Griebel 2004; Gerstner and Griebel 1998) are constructed through a skillful combination of one-dimensional quadrature nodes.”
• A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications (2017, background): “More details regarding the construction of Smolyak sparse grid can be found in [50,49].”
• Adversarial Uncertainty Quantification in Physics-Informed Neural Networks (2018, methodology): “…the most popular methods for scientific computing under uncertainty, such as polynomial chaos expansions [17, 18, 19], sparse grid quadratures [20, 21], multi-level/multi-fidelity Monte Carlo sampling [22, 23], proper orthogonal decomposition [24, 25], and Gaussian process regression models…”
• TT-cross approximation for multidimensional arrays (2010, methodology): “For large values of the dimensionality d such integrals are usually computed by Monte Carlo or Quasi Monte Carlo methods [33], or by sparse grid approaches [34], or sometimes by using analytical decompositions [35].”
• Sparse Gauss-Hermite Quadrature Filter with Application to Spacecraft Attitude Estimation (2011, methodology): “In [33], the sparse-grid methods based on trapezoidal, Clenshaw–Curtis, and Gauss–Patterson rules were compared for numerical integration.”