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A hierarchical a posteriori error estimator for the Reduced Basis Method

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Abstract

In this contribution, we are concerned with tight a posteriori error estimation for projection-based model order reduction of \(\inf \)-\(\sup \) stable parameterized variational problems. In particular, we consider the Reduced Basis Method in a Petrov-Galerkin framework, where the reduced approximation spaces are constructed by the (weak) greedy algorithm. We propose and analyze a hierarchical a posteriori error estimator which evaluates the difference of two reduced approximations of different accuracy. Based on the a priori error analysis of the (weak) greedy algorithm, it is expected that the hierarchical error estimator is sharp with efficiency index close to one, if the Kolmogorov N-with decays fast for the underlying problem and if a suitable saturation assumption for the reduced approximation is satisfied. We investigate the tightness of the hierarchical a posteriori estimator both from a theoretical and numerical perspective. For the respective approximation with higher accuracy, we study and compare basis enrichment of Lagrange- and Taylor-type reduced bases. Numerical experiments indicate the efficiency for both, the construction of a reduced basis using the hierarchical error estimator in a greedy algorithm, and for tight online certification of reduced approximations. This is particularly relevant in cases where the \(\inf \)-\(\sup \) constant may become small depending on the parameter. In such cases, a standard residual-based error estimator—complemented by the successive constrained method to compute a lower bound of the parameter dependent \(\inf \)-\(\sup \) constant—may become infeasible.

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Funding

M.R. was supported by the European Union within the EU-MORNet project.

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Authors and Affiliations

  1. Institute for Numerical Mathematics, Ulm University, Helmholtzstr. 20, 89081, Ulm, Germany

    Stefan Hain, Mladjan Radic & Karsten Urban

  2. Applied Mathematics, University of Münster, Einsteinstr. 62, 48149, Münster, Germany

    Mario Ohlberger

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  1. Stefan Hain
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  2. Mario Ohlberger
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Correspondence to Karsten Urban.

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Communicated by: Anthony Nouy

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Hain, S., Ohlberger, M., Radic, M. et al. A hierarchical a posteriori error estimator for the Reduced Basis Method. Adv Comput Math 45, 2191–2214 (2019). https://doi.org/10.1007/s10444-019-09675-z

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