ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF PARAMETRIC AND STOCHASTIC ELLIPTIC PDE'S

By Albert Cohen, Ronald A. DeVore, Christoph Schwab. 2011

309 citations recorded by the source corpus.

Subfields: Numerical Analysis, 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): “General nonlinear best R -term approximation schemes [49, 50, 171] represent another important technique, which we cannot cover in detail.”
The Bayesian Approach to Inverse Problems (2013, background): “.i.d. random variables may be found in [28]. • Subsection 2.2. These uniform priors have been extensively studied in the context of the ﬁeld of Uncertainty Quantiﬁcation and the reader is directed to [19, 20] for more details. Uncertainty Quantiﬁcation in this context does not concern inverse proble”
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”
Model Reduction and Neural Networks for Parametric PDEs (2020, methodology): “This construction ensures that there exists p ∈ (0, 1) such that (‖φj‖L∞)j≥1 ∈ `(N;R) which is required for the theory in [14].”
Low-rank methods for high-dimensional approximation and model order reduction (2015, unknown): “ity for the approximation of a tensor in the corresponding low-dimensional space XJ = span{ψk}k∈J. The construction of index sets J can be based on a priori analyses [62, 6] or on adaptive algorithms [8, 77, 28, 29, 26, 23, 25]. Sparse and low-rank methods exploit diﬀerent low-dimensional structures”
Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs* (2011, methodology): “We have the following analogue to Lemma 3.5 from Cohen et al. (2011). at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0962492911000055 Downloaded from https:/www.cambridge.org/core.”
Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs (2010, background): “A different avenue of their proof is through techniques of several complex variables; this is explored in the forthcoming work [6].”
Adaptive stochastic Galerkin FEM (2014, methodology): “The observed convergence rate of 1/2 with respect to the total number of degrees of freedom, which is the convergence rate for a single deterministic problem, coincides with the approximation rates predicted by [9, 17].”
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems (2011, background): “The following lemma, attributed to Stechkin in [11], will prove very useful for this purpose.”