Generalized spectral decomposition method for solving stochastic finite element equations : Invariant subspace problem and dedicated algorithms

By Anthony Nouy. 2008

171 citations recorded by the source corpus.

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

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “A connection between best rank-1 or, more generally, best rank-m approximations to a nonlinear eigenvalue problem is explained in [202].”
A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach (2011, background): “Other definitions of PGD have been proposed, which have better convergence properties [19, 20].”
Surrogate accelerated sampling of reservoir models with complex structures using sparse polynomial chaos expansion (2015, methodology): “Among them are the multi-scale model reduction of [57] and the sparse decomposition of [60, 59, 58] for the stochastic Galerkin technique, anisotropic and adaptive sparse grids of [51, 62, 61] for the stochastic collocation scheme, and low-rank solution approximations of [63, 64, 65, 78].”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, methodology): “For a detailed description and in depth study of these algorithms, see [36, 37 ].”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, background): “In the context of stochastic problems, an expansion based on the separation of spatial and random variables has been proposed in [58, 59].”
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 PDEs [6]…”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, background): “On the other hand the works [22, 47] (see also the references therein) have pursued generalized spectral decompositions for approximation, seeking reduced bases that are tailored to the solution and which seem to be rather promising.”
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.”
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”
Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos (2013, unknown): “see [22,36–38].”
Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces (2011, unknown): “For many applications, these updating strategies allow recovering good convergence properties of separated representations [31,33,34].”
eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces (2010, methodology): “Finally, system (16) is a system of P ×N which can be solved by using classical Krylov-type iterative solvers [1, 34] or alternative solution techniques such as the Generalized Spectral Decomposition method [35, 36, 37].”
Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations (2009, methodology): “quires the resolution of a simple stochastic algebraic equation (92) (application of operator f for a fixed w) and of a simple deterministic problem (91) (application of mapping F for a given λ). In fact, operator T( ·; ur ) can be interpreted as a deflated version of the initial operator T ≡ T( ·; ”
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.”
The stochastic finite element method: Past, present and future (2009, methodology): “Recently, a generalization of the classical spectral decomposition (truncated K–L expansion) for the solution of the problem interpreted as an ‘‘extended” eigenproblem has been proposed together with ad hoc iterative solution techniques inspired by classical techniques for solving the eigenproblem [”