Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation

By Ivo Babuska, Raúl Tempone, Georgios E. Zouraris. 2005

378 citations recorded by the source corpus.

Subfields: Computational Mechanics, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e53a7d75415efa48b00de5459c07bcd1d4406bcb

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Citations in the corpus

Surrogate accelerated sampling of reservoir models with complex structures using sparse polynomial chaos expansion (2015, methodology): “The choice of the polynomial chaos expansions as a proxy model was pioneered by Ghanem and Spanos [27] and has been applied to various engineering problems [28, 29, 30, 31, 57, 33, 77].”
An extended stochastic finite element method for solving stochastic partial differential equations on random domains (2008, background): “In the case of a deterministic domain, the reader can refer to [2,27] for a detailed and comprehensive study of the regularity requirements on the data (C, f and F ) leading to the desired properties.”
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, methodology): “We focus our interest next on comparing the anisotropic sparse collocation method with its isotropic counterpart, as well as with the anisotropic adaptive full tensor product method, proposed in [3, 12], and the standard Monte Carlo method.”
Efficient Analysis of High Dimensional Data in Tensor Formats (2012, neutral): “(14) [1–3, 14, 17, 22, 28, 36, 37, 43, 44, 48, 49].”
Karhunen-Loève approximation of random fields by generalized fast multipole methods (2006, background): “Due to the high cost of the stochastic Galerkin FEM for large M ([5, 8]), considerable computational work could (and should) be spent on finding ‘optimal’ (with respect to an error measure for a− aM ) separated approximations (1.”
A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, positive): “Our final interest then, is to compare our isotropic sparse tensor product method with the Monte Carlo approach and also, the anisotropic full tensor product method, proposed in [4].”
eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces (2010, methodology): “In [20, 21], an eXtended Stochastic Finite Element Method (X-SFEM) has been proposed, which is based on the mariage between the deterministic X-FEMmethod [22, 23] and Galerkin spectral stochastic methods [6, 5].”
Generalized spectral decomposition method for solving stochastic finite element equations : Invariant subspace problem and dedicated algorithms (2008, background): “Several choices have been proposed for the construction of a stochastic approximation basis in L(2)(Θ, dP ): polynomial chaos [1], generalized polynomial chaos [21,22], finite elements [6,4], or multi-wavelets [23,24].”
A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations (2007, neutral): “[13]) can then be written as follows: find u ∈ V ⊗ S such that A(u, v) = B(v) ∀v ∈ V ⊗ S, (2) where”
Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations (2009, background): “Well-posedness of problem (35) still results from a similar analysis of forms A and B [5, 9 , 16, 39, 83, 119].”
The stochastic finite element method: Past, present and future (2009, neutral): “[10–12,30,41,58,113,124].”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, methodology): “To study the convergence of the tensor product collocation method, we increase the order p for the approximating polynomial spaces, Pp(Γ), following the adaptive algorithm described on page 1287 of [5].”