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Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs

Abdellah ChkifaAlbert CohenRonald DeVoreChristoph Schwab — 2013

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations...

Harmonic analysis of the space BV.

Albert CohenWolfgang DahmenIngrid DaubechiesRonald DeVore — 2003

Revista Matemática Iberoamericana

We establish new results on the space BV of functions with bounded variation. While it is well known that this space admits no unconditional basis, we show that it is almost characterized by wavelet expansions in the following sense: if a function f is in BV, its coefficient sequence in a BV normalized wavelet basis satisfies a class of weak-l1 type estimates. These weak estimates can be employed to prove many interesting results. We use them to identify the interpolation spaces between BV and Sobolev...

Approximation Classes for Adaptive Methods

Binev, PeterDahmen, WolfgangDeVore, RonaldPetrushev, Pencho — 2002

Serdica Mathematical Journal

* This work has been supported by the Office of Naval Research Contract Nr. N0014-91-J1343, the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation grants DMS-0221642 and DMS-0200665, the Deutsche Forschungsgemeinschaft grant SFB 401, the IHP Network “Breaking Complexity” funded by the European Commission and the Alexan- der von Humboldt Foundation. Adaptive Finite Element Methods (AFEM) are numerical procedures that approximate the solution to a partial...

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