Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients

By Radu Alexandru Todor, Christoph Schwab. 2007

252 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Finance, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/fabea4bf06c39eef1fc1d3483a928bf0d33060d8

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In Computational Theory and Mathematics

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In Numerical Analysis

Citations in the corpus

Proper Generalized Decompositions and Separated Representations for the Numerical Solution of High Dimensional Stochastic Problems (2010, methodology): “They lead to accurate predictions and allow for a better control on numerical simulations through a posteriori error estimation and adaptive approximation [23,46,45,33,48].”
Fast Numerical Methods for Stochastic Computations: A Review (2009, methodology): “The current work includes adaptive choice of polynomial basis [19, 79], adaptive element selection in multi-element gPC [80], and adaptive sparse grid collocation [20, 78].”
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].”
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data (2007, methodology): “A related approach due to Schwab and coworkers (see, for instance, [55]) uses sparse wavelet chaos subspaces.”
ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS (2012, background): “[1, 15, 19, 27, 30]), while Collocation is a sum of Lagrangian interpolants over the probability space (see e.”
Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats (2014, background): “In [18, 17] based on [3, 31] this leads to several data-sparse representations of the objects encountered in discretisation of the model problem as well as the obtained solution.”
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems (2011, unknown): “, [44] for more details.”
Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients (2009, methodology): “[11, 21, 33, 2, 28]) which consists in projecting the original equation onto a polynomial subspace which could be constructed as the span of tensor product polynomials or polynomials having total degree smaller that a given integer p.”
An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data (2008, background): “A related approach is proposed in [26], where sparse wavelet chaos subspaces are constructed for the approximation of the solution of linear scalar elliptic PDEs with random coefficients.”
Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison (2011, background): “Other global polynomial spaces has been considered recently, see for instance [5, 24 ], as well as different approximation spaces such as piecewise polynomials [2, 12, 25].”
A non-adapted sparse approximation of PDEs with stochastic inputs (2010, methodology): “We here briefly summarize the results of [60,8] on the sparsity of the Legendre PC expansion of the solution u(x,y) to the problem (2).”
Stochastic collocation and mixed finite elements for flow in porous media (2008, background): “Choices of collocation points include tensor product of zeros of orthogonal polynomials [2,41], sparse grid approximations [15,28,30,38,41], and probabilistic collocation [25].”