Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs

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

123 citations recorded by the source corpus.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/992221c65185d02f59e62eb5bf015586fea6591c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Albert Cohen
2. Ronald A. DeVore
3. Christoph Schwab
4. Tony Lelièvre
5. É. Cancès
6. Virginie Ehrlacher
7. Abdellah Chkifa
8. Nikola B. Kovachki

In Computational Theory and Mathematics

1. Ronald A. DeVore
2. Albert Cohen
3. Christoph Schwab
4. Andrew M. Stuart
5. Abdellah Chkifa
6. Tony Lelièvre
7. É. Cancès
8. Virginie Ehrlacher

In Statistics and Probability

1. Christoph Schwab
2. Albert Cohen
3. Ronald A. DeVore
4. Andrew M. Stuart
5. Nikola B. Kovachki
6. Kaushik Bhattacharya
7. Abdellah Chkifa
8. Bamdad Hosseini

Citations in the corpus

• Model Reduction and Neural Networks for Parametric PDEs (2020, methodology): “The form (2.16) of the approximate solution operator can also be related to the Taylor approximations developed in [14, 18] where a particular form of the input x is considered, namely x = ¯ x + j ≥ 1 a j ˜ x j where ¯ x ∈ X is fixed, { a j } j ≥ 1 ∈ ∞ ( N ; R ) are uniformly bounded, and { ˜ x j }…”
• Greedy algorithms for high-dimensional non-symmetric linear problems (2012, background): “Let us mention among others sparse grids [21], tensor formats [11], reduced bases [4] and adaptive polynomial approximations [6].”
• Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs (2015, background): “12), a property that appears to be useful for numerical computation [4, 7].”
• Approximation of high-dimensional parametric PDEs * (2015, methodology): “The second strategy, discussed in §6, performs an explicit computation of the truncated Taylor series, up to the spatial discretization of the coefficients tν , by a recursive method introduced in [17].”
• 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”