A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations

By Anthony Nouy. 2007

301 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Mechanics of Materials, Statistics and Probability

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “Convergence results in this direction can be found in [3,39,40,84–86, A number of improvements to the simple scheme outlined above have been proposed to increase its convergence speed, see, e.g., [4, 95, 200, 202].”
Results and Questions on a Nonlinear Approximation Approach for Solving High-dimensional Partial Differential Equations (2008, methodology): “This method has also been used in [8, 9] under the name Power type Generalized Spectral Decomposition for solving high-dimensional problems in the context of uncertainty quantification.”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, methodology): “3 Generalized spectral decomposition method In this section, we recall the basics of the Generalized Spectral Decomposition method (GSD) [36,37,42], which is a method for the a priori construction of a separated representation of the solution u of (4):”
A non-adapted sparse approximation of PDEs with stochastic inputs (2010, methodology): “Among them are the multi-scale model reduction of [33] and the sparse decomposition of [60,8,6,7] for the stochastic Galerkin technique, adaptive sparse regression approaches of [10,11] for PC expansions, anisotropic and adaptive sparse grids of [50,44] for the stochastic collocation scheme, and the”
A reduced spectral function approach for the stochastic finite element analysis (2011, background): “Recently Nouy [33,34] discussed the possibility of an optimal spectral decomposition.”
The LATIN multiscale computational method and the Proper Generalized Decomposition (2010, methodology): “This technique has already been used in the context of stochastic problems [19] and multidimensional problems [1, 2, 6].”
Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, unknown): “Basic PGDs are based on a progressive construction of the sequence um, where at each step, an additional elementary tensor ⊗k=1w (k) m is added to the previously computed decomposition um−1 [2,26,30].”
A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations (2010, background): “…in other contexts: separation of physical variables and parameters (or random variables) in the context of parametrized (or stochastic ) PDEs [27] [28] [32] [29] 3 , with a possible additional separation of parameters [11] [30] 4 , separation of coordinate variables in multi-dimensional…”
F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig on Minimal Subspaces in Tensor Representations on Minimal Subspaces in Tensor Representations (unknown): “…solution of problems defined in high-dimensional tensor spaces, such as PDEs arising in stochastic calculus [1] [4] [9] (e.g., Fokker-Planck equation), stochastic parametric PDEs arising in uncertainty quantification with spectral approaches [18] [8] [19], and quantum chemistry (cf., e.g., [25]).”
Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics (2012, unknown): “In [21, 22, 25], diﬀerent deﬁnitions and algorithms were proposed for generating the approximate representations (13) of the solution.”