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2016 | OriginalPaper | Buchkapitel

Reduced Basis Exact Error Estimates with Wavelets

verfasst von : Mazen Ali, Karsten Urban

Erschienen in: Numerical Mathematics and Advanced Applications ENUMATH 2015

Verlag: Springer International Publishing

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Abstract

A (multi-)wavelet expansion is used to derive a rigorous bound for the (dual) norm Reduced Basis residual. We show theoretically and numerically that the error estimator is online efficient, reliable and rigorous. It allows to control the exact error (not only with respect to a “truth” discretization).

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Vorheriges Kapitel Output Error Estimates in Reduced Basis Methods for Time-Harmonic Maxwell’s Equations

Nächstes Kapitel Model Order Reduction for Pattern Formation in FitzHugh-Nagumo Equations

We would like to stress that most what is said here can also be extended to Petrov-Galerkin inf-sup-stable problems with different trial and test spaces.

If (6) does not hold, the Empirical Interpolation Method (EIM) determines an approximation with an upper interpolation bound [2], which, however, may not be accessible and which also reduces the sharpness of the error bound.

Note, that (9) amounts to take the square root, which is not a problem in terms of efficiency, but it is an issue with respect to accuracy – the well-known so-called “square root effect”.

Here A ∼ B abbreviates cA ≤ B ≤ CB with constants 0 < c ≤ C < ∞.

One might argue that f(⋅ ; μ) is extremely smooth and that the μ-dependence only enters through the right-hand side. Of course, the wavelet-based error estimator equally works in other situations as well. However, we want to do a comparison with the standard RB setting of a common truth. In order to do so, we need (1) a formula for the exact solution, (2) a parameter-dependence which causes local effects. For more complex situations, an even larger improvement is to be expected.

The computation is finished before C++’s ctime std::clock function manages to update the number of clocks.

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Titel: Reduced Basis Exact Error Estimates with Wavelets
verfasst von: Mazen Ali
Karsten Urban
Verlag: Springer International Publishing
Buch: Numerical Mathematics and Advanced Applications ENUMATH 2015
Print ISBN: 978-3-319-39927-0

Electronic ISBN: 978-3-319-39929-4

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-39929-4_34

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