Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations

By Hermann G. Matthies, Andreas Keese. 2005

617 citations recorded by the source corpus.

Subfields: Environmental Engineering, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/368c6f59b75924cfa7f17f64350650b2ba2ba986

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In Environmental Engineering

In Statistical and Nonlinear Physics

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Citations in the corpus

Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients (2007, background): “Tensor-product discretization or tensor-product collocation grids in the parametric variable result, however, in superalgebraic complexity rates (see, e.g. Babuška et al., 2004; Frauenfelder et al., 2005; Matthies & Keese, 2005).”
Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison (2011, background): “[11, 27, 13] and references therein), or in anisotropic tensor product polynomial spaces [2, 8, 14].”
Uncertainty propagation in CFD using polynomial chaos decomposition (2006, methodology): “In the following section, we discuss one of these approaches, which has been applied in the context of UQ in [21, 35 ].”
Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems (2008, methodology): “The deterministic forward model, originally specified by (a system of) partial differential equations, is thus replaced by stochastic PDEs; numerical approaches to such systems, in which random fields appear as boundary conditions or coefficients, have seen extensive development [9, 16, 32–35].”
The stochastic finite element method: Past, present and future (2009, neutral): “[1–3,19,22,28,29,41,66,67, 94,108,113,187,189].”
Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions (2017, methodology): “In the intrusivemethod, aweak form of the mathematical model is developed based on the minimization of the residual in the balance equation in the Galerkin scheme where an alteration in the original finite element code is imposed (Ghanem and Spanos 1991; Matthies and Keese 2005).”
Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats (2014, background): “It will be shown that under additional assumptions the approximation error depends only on smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials.”
Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs (2010, neutral): “AMS Subject Classification: 65F30, 65F50, 65N35, 65F10 Key words: elliptic equations, stochastic PDEs, the Karhunen-Loève expansion, separable approximation, quantics-TT tensors, preconditioners, tensor-truncated iteration.”
An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis (2010, neutral): “[3, 4, 5].”
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].”
Solving parametric PDE problems with artificial neural networks (2017, methodology): “For example the stochastic Galerkin method [12, 15] expands the random solution using certain prefixed basis functions (i.”
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures (2006, methodology): “To solve differential equations with stochastic inputs following the procedure established by Ghanem and Spanos, the random solution is expanded spectrally by PC, and a Galerkin projection scheme is subsequently used to transform the original stochastic problem into a deterministic one with a larger”