• Corpus ID: 1333656

A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations

@article{Buhr2014ANS,
title={A numerically stable a posteriori error estimator for reduced basis approximations of elliptic equations},
author={Andreas Buhr and Christian Engwer and Mario Ohlberger and Stephan Rave},
journal={arXiv: Numerical Analysis},
year={2014}
}
• Published 30 July 2014
• Computer Science
• arXiv: Numerical Analysis
AbstractThe Reduced Basis (RB) method is a well established method for the model order reductionof problems formulated as parametrized partial di erential equations. One crucial requirementfor the application of RB schemes is the availability of an a posteriori error estimator toreliably estimate the error introduced by the reduction process. However, straightforwardimplementations of standard residual based estimators show poor numerical stability, renderingthem unusable if high accuracy is…

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References

SHOWING 1-8 OF 8 REFERENCES
Accurate and efficient evaluation of the a posteriori error estimator in the reduced basis method
• Computer Science
• 2012
A new approximation of the error bound using the Empirical Interpolation Method (EIM), which achieves higher levels of accuracy and requires potentially less precomputations than the usual formula.
REDUCED BASIS METHOD FOR FINITE VOLUME APPROXIMATIONS OF PARAMETRIZED LINEAR EVOLUTION EQUATIONS
• Computer Science, Mathematics
• 2008
This work introduces a new offline basis-generation algorithm based on the derivation of rigorous a-posteriori error estimates in various norms for general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations.
Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations
• Computer Science
• 2007
The text is focused on formulation, analysis, and computational procedures for reduced basis approximation and a posteriori error estimation for parametrized PDEs in the real-time and many-query
Convergence Rates for Greedy Algorithms in Reduced Basis Methods
• Mathematics, Computer Science
SIAM J. Math. Anal.
• 2011
The reduced basis method was introduced for the accurate online evaluation of solutions to a parameter dependent family of elliptic PDEs by determining a “good” n-dimensional space to be used in approximating the elements of a compact set $\mathcal{F}$ in a Hilbert space $\ mathscal{H}$.
A Space-Time Petrov-Galerkin Certified Reduced Basis Method: Application to the Boussinesq Equations
• Masayuki Yano
• Computer Science, Mathematics
SIAM J. Sci. Comput.
• 2014
We present a space-time certified reduced basis method for long-time integration of parametrized parabolic equations with quadratic nonlinearity which admit an affine decomposition in parameter but
The Reduced Basis Method for Time‐Harmonic Maxwell's Equations
• Computer Science
• 2012
The Reduced Basis Method generates low‐order models of parametrized PDEs to allow for efficient evaluation of the input‐output behaviour in many‐query and real‐time contexts and plays a crucial role in the greedy process to generate the surrogate model.
Reduced basis method for finite volume approximations of parametrized linear evolution equations . M 2 AN ( Math
• . Model . Numer . Anal . )
• 2008