Solution of stochastic partial differential equations using Galerkin finite element techniques

By M. Deb, Ivo Babuska, J. Tinsley Oden. 2001

485 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/67f5c2be8da9ab71c91515f7d96e1c08fb71d976

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In Civil and Structural Engineering

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

A compressed sensing approach for partial differential equations with random input data (2012, background): “In particular, both a priori and a posteriori approximation error results are available [14, 22, 32].”
Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations (2004, methodology): “The approximation error in the approach [14, 15] can then be bounded in terms of deterministic quantities, as described in this work.”
Numerical Methods and Smolyak Quadrature for Nonlinear Stochastic Partial Differential Equations (2003, unknown): “Assuming for a moment that there is no stochastic dependence, then we choose for the possible solutions the space (9) W = {u∈W1 p(R)|usatisfies essential boundary conditions },”
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.”
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, background): “) and polynomial approximation in the probability domain, either by full polynomial spaces [30, 22, 16], tensor product polynomial spaces [2, 12, 25], or piecewise polynomial spaces [8, 2, 19, 28, 21].”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, result): “This approach has been considered by several authors [4, 20, 27, 62, 32, 43].”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, methodology): “For the case where ξi is composed bymi independent random variables, classical choices consist in introducing orthogonal complete polynomial basis [19,53] (classical polynomial chaos basis), or piecewise polynomial basis [11,46,30].”
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].”