A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data

By Ivo Babuska, Fabio Nobile, Raúl Tempone. SIAM Review, 2007

1,653 citations recorded by the source corpus.

Subfields: Environmental Engineering, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/951e0597e2a24045fd9a51bb8497dce4d93f4cb2

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Citations in the corpus

Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “4 Stochastic collocation and multifidelity Stochastic collocation methods [14, 92] compute statistics such as the expectation (24) and the variance (25) by using a deterministic quadrature rule instead of the Monte Carlo method.”
Solving elliptic problems with non-Gaussian spatially-dependent random coefficients (2009, methodology): “Regarding the stochastic collocation finite element method for the model problem (8), the L2ðC; H(1)0ðDÞÞ approximation was studied in [16,28], and the L1ðC; H(1)0ðDÞÞ approximation was analyzed in [29], where sparse collocation points were generated by a full tensor product with anisotropic Chebysh”
Complexity Analysis of stochastic gradient methods for PDE-constrained optimal Control Problems with uncertain parameters (2021, background): “…(see, e.g., [8,27] and the references therein), as well as on the numerical approximation of (uncontrolled) PDEs with random coeﬃcients ( e.g., [4, 23, 35]), the analysis of PDE constrained control problems under uncertainty is much more recent and incomplete, although the topic has received…”
Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs (2010, methodology): “D b(y, x) ∂φi ∂x ∂φj ∂x dx, y ∈ Γ, D = [0, 1].”
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs (2011, background): “This is the point of view adopted for example in [1,2,13,15,19,22,23].”
Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs (2010, methodology): “This is the point of view adopted for example in [14, 16, 4, 13, 3, 18, 19].”
Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison (2011, background): “In comparison, the SG-ATD solution has 837 sdof with weights α = [1, 2, 3, 4], 434 sdof with the experimental weights α = [1, 2.”
Stochastic collocation and mixed finite elements for flow in porous media (2008, methodology): “6) satisfies the smoothness assumption in [2].”
An efficient SFE method using Lagrange polynomials: Application to nonlinear mechanical problems with uncertain parameters (2007, background): “[4] to provide rigorous convergence analysis and demonstrate exponential convergence in the probability space for an elliptic partial differential equation with random input.”