An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data

By Fabio Nobile, Raúl Tempone, Clayton G. Webster. SIAM Journal on Numerical Analysis, 2008

578 citations recorded by the source corpus.

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

View Paper: https://www.semanticscholar.org/paper/da8c82e5e8feba3529bb633cf4e66f11bdbad983

How researchers see this paper

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

In Computational Theory and Mathematics

In Environmental Engineering

In Statistics and Probability

Citations in the corpus

Polynomial chaos expansions for dependent random variables (2019, methodology): “Specifically we use the following index sets presented in [45]:”
N-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs (2014, background): “We mention the classic reference [12] and the more recent publications [17, 25, 24, 2, 21, 20, 4, 18, 10] and the references there.”
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients (2009, methodology): “Remark 4 (Anisotropic Smolyak) The work [22] proposed and analyzed a novel anisotropic sparse grid stochastic collocation method that is based on a weighted version of the Smolyak formula (32), namely one penalizes the use of approximation levels on input random variables which have little influence”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, methodology): “Their systematic choice can be based either on a priori or a posteriori information and is motivated by the regularity of the solution and the error estimates derived in [45].”
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems (2011, methodology): “In applications with smooth parameter dependence, one can often avoid the use of a parameter sample size that grows exponentially with p. Sparse grid techniques have been successfully used in collocation methods for stochastic and parameter-dependent PDEs, see, e.g., [35] [36] [5] [6] [9].”
A least-squares approximation of partial differential equations with high-dimensional random inputs (2009, background): “More recently, there has been a great amount of attention to collocation-based techniques that take advantage of existing deterministic solvers for moderate Oð10Þ number of uncertain variables [24,18,27,1,20,19].”
Uncertainty quantification in chemical systems (2009, methodology): “Other local constructions have also been developed, employing Lagrange interpolants [34–41].”
A non-adapted sparse approximation of PDEs with stochastic inputs (2010, methodology): “Among them are the multi-scale model reduction of [32] and the sparse decomposition of [55, 8, 6, 7, 10] for the stochastic Galerkin technique, anisotropic and adaptive sparse grids of [46, 41] for the stochastic collocation scheme, and low-rank solution approximations of [48, 49, 33].”