Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations

By Ivo Babuska, Raúl Tempone, Georgios E. Zouraris. SIAM Journal on Numerical Analysis, 2004

916 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/48582cdc155b3226f609f669be9694503e38f086

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Citations in the corpus

Sparse tensor approximation of parametric eigenvalue problems (2012, background): “, for specific choices of Λ in the literature, see [6, 3, 7, 4, 24, 23] and references therein.”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, methodology): “See [4, 5] for further details on the double orthogonal basis.”
ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS (2012, background): “[1, 15, 19, 27, 30]), while Collocation is a sum of Lagrangian interpolants over the probability space (see e.”
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients (2009, methodology): “We mention, for instance, the works [1, 3, 28, 23, 22], which deal with an elliptic equation with a stochastic diffusion coefficient and either Stochastic Galerkin or Collocation approximations, and the work [5], which deals with an elliptic equation defined in a random domain.”
Long-Term Behavior of Polynomial Chaos in Stochastic Flow Simulations (2006, methodology): “The k-type convergence was shown theoretically in [ 14 ,13] to be”
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, background): “In such cases, a full anisotropic tensor product approximation [2, 1] may still be more effective for a small or modest number of random variables.”
A reduced spectral function approach for the stochastic finite element analysis (2011, background): “Excellent references, for example [5,14–16] are available on this topic.”
Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats (2014, background): “Keywords: tensor approximation, stochastic PDEs, stochastic Galerkin matrix, lowrank tensor formats, uncertainty quantification AMS Subject Classifications: 65N30, 35R60, 15A69”
Discrete least squares polynomial approximation with random evaluations − application to parametric and stochastic elliptic PDEs (2015, methodology): “Projection methods [2,3,9,14] produce quasi-optimal approximations in XΛ for the metric L(2)(Γ,X, ρ) where ρ is a chosen measure in the parameter space.”
A stochastic collocation approach to Bayesian inference in inverse problems (2009, background): “Many studies have been devoted to the convergence properties of gPC, numerically or analytically, and the computational efficiency of gPC methods. See, for example, [ 2 ,8,15,28,29].”
Efficient Analysis of High Dimensional Data in Tensor Formats (2012, neutral): “we require that both κ(x, ω) and 1/κ(x, ω) are essentially bounded (have finite L∞ norm) [2, 37, 34, 44]: κ(x, ω) > 0 a.”
Complexity Analysis of stochastic gradient methods for PDE-constrained optimal Control Problems with uncertain parameters (2021, methodology): “Here, γ ∈ [1, 3] is a parameter representing the efficiency of the linear solver used (e.”
Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients (2011, neutral): “[1,2,6,7,11,12,29,30]).”
A non-adapted sparse approximation of PDEs with stochastic inputs (2010, methodology): “Most notably, the stochastic Galerkin schemes using Polynomial Chaos (PC) bases [38,28,67,2,63] have been successfully applied to a variety of engineering problems and are extremely useful when the number of uncertain parameters is not large.”