Abstract
We consider parameterized parabolic partial differential equations (PDEs) with variable initial conditions, which are interpreted as a parameter function within the Reduced Basis Method (RBM). This means that we are facing an infinite-dimensional parameter space. We propose to use the space-time variational formulation of the parabolic PDE and show that this allows us to derive a two-step greedy method to determine offline separately the reduced basis for the initial value and the evolution. For the approximation of the initial value, we suggest to use an adaptive wavelet approximation. Online, for a given new parameter function, the reduced basis approximation depends on its (quasi-)best N-term approximation in terms of the wavelet basis. A corresponding offline-online decomposable error estimator is provided. Numerical experiments show the flexibility and the efficiency of the method.
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Notes
- 1.
Note, that we also allow for time-dependent bilinear forms, i.e., non-LTI problems as e.g. in [6].
- 2.
If we use a function in space, say \(q \in V ^{\mathcal{J}}\), as initial value, we “embed” it into \(\mathit{Q}^{\mathcal{J}}\), i.e., we set \(\tau ^{0} \otimes q \in \mathit{Q}^{\mathcal{J}}\) with the temporal basis function τ 0 at t = 0.
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Mayerhofer, A., Urban, K. (2017). A Reduced Basis Method for Parameter Functions Using Wavelet Approximations. In: Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban, K. (eds) Model Reduction of Parametrized Systems. MS&A, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-58786-8_5
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DOI: https://doi.org/10.1007/978-3-319-58786-8_5
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