Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations

By Anthony Nouy. 2009

202 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/35e30f61da4061ed7667fef5c2dfeb290594368c

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Citations in the corpus

Model Order Reduction for Linear and Nonlinear Systems: A System-Theoretic Perspective (2014, background): “Reduced basis approaches for stochastic problems are reviewed in [53] whereas a review of computational stochastic methods (also model reduction techniques) with emphasis on spectral stochastic approaches can be found in [147].”
Greedy algorithms for high-dimensional non-symmetric linear problems (2012, methodology): “to solve high-dimensional Fokker-Planck equations in the context of kinetic models for polymers [1] and Nouy in the context of uncertainty quantification [15], under the name of Progressive Generalized Decomposition (PGD) methods.”
Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, methodology): “The proposed GSD de nition can then be interpreted as an a priori Galerkin model reduction technique, where none of the reduced approximation spaces VM and SM are selected a priori (see [38] for the connection with other model reduction techniques).”
Routes for Efficient Computational Homogenization of Nonlinear Materials Using the Proper Generalized Decompositions (2010, background): “This is the basic idea of spectral approaches in stochastic analysis, which are receiving a growing interest in computational science [ 34 ].”
A reduced spectral function approach for the stochastic finite element analysis (2011, methodology): “Another class of methods which have been used widely in the literature is known as the spectral methods (see [1] for a recent review).”
A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach (2011, unknown): “) and (random) parameters [19–21,25].”
Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics (2012, methodology): “This parametric vision, combined with approximation theory and numerical analysis, brought to the so-called spectral stochastic methods [10, 33, 18, 23, 36, 16].”
A PGD-based homogenization technique for the resolution of nonlinear multiscale problems (2013, background): “Subsequently, a number of extensions were proposed, such as the separation of coordinate variables in [30], the separation of spatial and stochastic parameters in [31] or the separation of the time axis in a multidimensional time space [32].”
Doubly Spectral Stochastic Finite-Element Method for Linear Structural Dynamics (2011, methodology): “Problems of structural dynamics in which the uncertainty in specifying the stiffness and mass of the structure are modeled within the framework of random fields and can be treated by using the stochastic finite-element method (Ghanem and Spanos 1991; Sudret and Der-Kiureghian 2000; Stefanou 2009; No”
A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations (2010, background): “…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] [1] [2]…”
A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties (2013, background): “Numerous developments have been realized for the propagation using functional approaches (see reviews [1, 2, 3, 4]).”
eXtended Stochastic Finite Element Method for the numerical simulation of heterogeneous materials with random material interfaces (2010, background): “In the last two decades, a growing interest has been devoted to spectral stochastic methods [1, 2, 3, 4] for the propagation of uncertainties through physical models governed by stochastic partial di erential equations (SPDEs) involving a nite dimensional noise.”