A least-squares approximation of partial differential equations with high-dimensional random inputs

By Alireza Doostan, Gianluca Iaccarino. 2009

158 citations recorded by the source corpus.

Subfields: Civil and Structural Engineering, Environmental Engineering, Statistics and Probability

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Citations in the corpus

A literature survey of low‐rank tensor approximation techniques (2013, neutral): “It is probably most natural to combine ALS with the CP decomposition, see [27,68] for an application of this idea to linear systems.”
Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics (2012, background): “Note that other definitions and algorithms have been recently proposed for the construction of such separated representations of multiparametric stochastic functions [27, 28].”
Approximation of high-dimensional parametric PDEs * (2015, methodology): “Other non-intrusive methods are based on least-square regression as discussed in [16, 34, 35, 69], or on pseudo-spectral approximation as discussed in [91, 26].”
A Proper Generalized Decomposition for the solution of elliptic problems in abstract form by using a functional Eckart–Young approach (2011, background): “Still in the context of stochastic PDEs, a further separation of parameters have also been introduced, by exploiting the tensor product structure of stochastic function spaces [9, 22].”
Numerical construction of Green's functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations (2012, background): “This is particularly useful in the context of solving stochastic ordinary/partial differential equations (SODE’s/SPDE’s): in such problems, the presence of random inputs can lead lead to rapid growth in the formal dimension [150, 80, 46].”
A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations (2010, methodology): “…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] [8] 5 .”
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”
Non-intrusive low-rank separated approximation of high-dimensional stochastic models (2012, background): “A separated representation of the form (1) for the solution of PDEs and ODEs with high-dimensional random inputs has been first studied in [26, 24, 25].”
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]).”
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): “Remark 5 - This minimal residual formulation (or least-square formulation) has been proposed in [5] for the solution of algebraic equations in nite dimensional tensor product spaces and applied to the solution of stochastic algebraic equations in [13].”