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Reduced basis methods with adaptive snapshot computations

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Abstract

We use asymptotically optimal adaptive numerical methods (here specifically a wavelet scheme) for snapshot computations within the offline phase of the Reduced Basis Method (RBM). The resulting discretizations for each snapshot (i.e., parameter-dependent) do not permit the standard RB ‘truth space’, but allow for error estimation of the RB approximation with respect to the exact solution of the considered parameterized partial differential equation. The residual-based a posteriori error estimators are computed by an adaptive dual wavelet expansion, which allows us to compute a surrogate of the dual norm of the residual. The resulting adaptive RBM is analyzed. We show the convergence of the resulting adaptive greedy method. Numerical experiments for stationary and instationary problems underline the potential of this approach.

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Authors and Affiliations

  1. Institute for Numerical Mathematics, Ulm University, Helmholtzstrasse 20, D-89069, Ulm, Germany

    Mazen Ali, Kristina Steih & Karsten Urban

Authors
  1. Mazen Ali
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  2. Kristina Steih
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  3. Karsten Urban
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Correspondence to Karsten Urban.

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Communicated by: Helge Holden

This work has partly been supported by the Deutsche Forschungsgemeinschaft (DFG) within the Research Training Group (Graduiertenkolleg) GrK1100 Modellierung, Analyse und Simulation in der Wirtschaftsmathematik at Ulm University.

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Ali, M., Steih, K. & Urban, K. Reduced basis methods with adaptive snapshot computations. Adv Comput Math 43, 257–294 (2017). https://doi.org/10.1007/s10444-016-9485-9

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  • DOI: https://doi.org/10.1007/s10444-016-9485-9

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