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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Notes
- 1.
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.
- 2.
- 3.
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”.
- 4.
Here A ∼ B abbreviates cA ≤ B ≤ CB with constants 0 < c ≤ C < ∞.
- 5.
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.
- 6.
The computation is finished before C++’s ctime std::clock function manages to update the number of clocks.
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Ali, M., Urban, K. (2016). Reduced Basis Exact Error Estimates with Wavelets. In: Karasözen, B., Manguoğlu, M., Tezer-Sezgin, M., Göktepe, S., Uğur, Ö. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2015. Lecture Notes in Computational Science and Engineering, vol 112. Springer, Cham. https://doi.org/10.1007/978-3-319-39929-4_34
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DOI: https://doi.org/10.1007/978-3-319-39929-4_34
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