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2016 | OriginalPaper | Chapter

Reduced Basis Exact Error Estimates with Wavelets

Authors : Mazen Ali, Karsten Urban

Published in: Numerical Mathematics and Advanced Applications ENUMATH 2015

Publisher: 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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previous chapter Output Error Estimates in Reduced Basis Methods for Time-Harmonic Maxwell’s Equations
next chapter Model Order Reduction for Pattern Formation in FitzHugh-Nagumo Equations
Footnotes
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
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.
 
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.
 
Literature
1.
M. Ali, K. Steih, K. Urban, Reduced Basis Methods Based Upon Adaptive Snapshot Computations (2014). arXiv:1407.1708
2.
M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An “empirical interpolation” method: application to efficient reduced-basis discretization of partial differential equations. CR Acad. Sci. Paris I 339 (9), 667–672 (2004)MathSciNetCrossRefMATH
3.
W. Dahmen, Wavelet and multiscale methods for operator equations. Acta Numer. 6, 55–228 (Cambridge University Press 1997)
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5.
J. Hesthaven, G. Rozza, B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations (Springer, Cham, 2016)CrossRefMATH
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D. Huynh, G. Rozza, S. Sen, A. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. CR Math. Acad. Sci. Paris 345 (8), 473–478 (2007)MathSciNetCrossRefMATH
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A. Quarteroni, A. Manzoni, F. Negri, Reduced Basis Methods for Partial Differential Equations. Unitext, vol. 92 (Springer, Cham, 2016)
8.
K. Steih, Reduced basis methods for time-periodic parametric partial differential equations. Ph.D. thesis, Ulm University (2014)
9.
K. Urban, Wavelet Methods for Elliptic Partial Differential Equations (Oxford University Press, Oxford, 2009)MATH
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M. Yano, A reduced basis method with exact-solution certificates for symmetric coercive equations. Comput. Meth. Appl. Mech. Eng. 287, 290–309 (2015)MathSciNetCrossRef
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M. Yano, A minimum-residual mixed reduced basis method: exact residual certification and simultaneous finite-element reduced-basis refinement. Math. Model. Numer. Anal. 50 (1), 163–185 (2015)MathSciNetCrossRefMATH
Metadata
Title
Reduced Basis Exact Error Estimates with Wavelets
Authors
Mazen Ali
Karsten Urban
Copyright Year
2016
Book
Numerical Mathematics and Advanced Applications ENUMATH 2015

Print ISBN: 978-3-319-39927-0

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

Copyright Year: 2016

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

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