Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs

By Albert Cohen, Ronald A. DeVore, Christoph Schwab. 2010

384 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): “General nonlinear best R -term approximation schemes [49, 50, 171] represent another important technique, which we cannot cover in detail.”
ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS (2012, background): “Again, a similar result is given in [8] for the special case a = a0 + ∑N n=1 bn(x)yn.”
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.”
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* (2011, methodology): “Proof of analyticity of u(z) There are several approaches to proving analyticity of u(z) as a V -valued function: a power series approach would require a bootstrapping argument for real parameter values as in Cohen et al. (2010).”
Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs (2010, background): “23) Similar to the results in [9], this theorem reveals that the sequence (‖tν‖V )ν∈F inherits the summability properties of the sequence (‖ψj‖L∞(D))j≥1, however now under the weaker assumption UEA(r,R).”
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs (2011, background): “By assumption UEA(r,R), for any closed subspace Vh ⊂ V the Finite Element approximation is uniquely defined and the analysis in [9, 10] and all results of the present paper also apply to the discretized problem.”
Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances (2012, background): “Notice that the rank estimate for the Tucker-type approximation of uM (y, x) in (M + 1)dimensional space can be derived based on the analyticity results in [10].”
Quantics-TT Collocation Approximation of Parameter-Dependent and Stochastic Elliptic PDEs (2010, neutral): “AMS Subject Classification: 65F30, 65F50, 65N35, 65F10 Key words: elliptic equations, stochastic PDEs, the Karhunen-Loève expansion, separable approximation, quantics-TT tensors, preconditioners, tensor-truncated iteration.”
Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs (2015, methodology): “The strategy developed in [5, 6] for proving Theorem 1.”
N-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs (2014, background): “As in [8, 7], in the analysis of the spatial regularity we only aim at bounds for the second weak derivatives of the parametric solution u(x, y) which are required for convergence rate estimates of continuous, piecewise linear Finite Element Methods, and exploit moreover that the stochastic coeffici”