Abstract
In this chapter, we compare two ways to treat the time within reduced basis methods (RBMs) for parabolic problems: Time-stepping and space-time variational based methods. We briefly recall both concepts and review well-posedness, error control and model reduction in both cases as well as the numerical realization. In particular, we highlight the conceptual differences of the two approaches.
We provide numerical investigations focussing on the performance of the RBM in both variants regarding approximation quality, efficiency and reliability of the error estimator. Pro’s and Con’s of both approaches are discussed.
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Notes
- 1.
For simplicity, we restrict ourselves to Linear Time-Invariant (LTI) systems. However, much of what is said also holds for time-variant operators A(t).
- 2.
It can be seen that V h ⊂ V ↪ H is in fact sufficient [8].
- 3.
It is somehow misleading to use the term “effectivity” within the POD-Greedy framework, since usually effectivity is the ratio of error bound and error.
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Glas, S., Mayerhofer, A., Urban, K. (2017). Two Ways to Treat Time in Reduced Basis Methods. 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_1
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