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Description:
Reduced basis methods (RBMs) have become an established model reduction method for parameterized PDEs. We introduce new RBMs for numerically expensive parameterized time-periodic parabolic equations. Three kinds of RBMs are developed in this thesis, distinguished by the type of numerical scheme that is employed for the solution of the high-dimensional (offline) quantities. Rigorous a posteriori error estimators are developed in all settings and tested in numerical experiments. The first is a further development of time-stepping RB methods. It provides a spatial reduction of the problem, using fixed-point iterations to ensure periodicity. We derive error estimators and discuss the reduction of basis construction costs. The approach is easy to implement, yet suffers from long computation times for the reduced solutions. As a second method, we consider space-time formulations. This avoids the need for fixed-point iterations through the choice of periodic basis functions and enables a reduction in the full space-time domain, leading to fast reduced models. The error bounds are similar to those in stationary elliptic settings. Yet, the additional temporal dimension causes a high numerical cost in the computation of the high-dimensional solutions. As a third scheme, we thus employ space-time adaptive wavelet Galerkin methods to compute snapshots and error bounds. This leads to a new framework for RBMs in which the adaptively generated discretizations allow to bound the error with respect to the exact solution. We illustrate why multiple snapshot selection can occur and what strategies avoid a subsequent early termination of the Greedy training. We show that rigorous and equivalent error bounds can be constructed and discuss implementable variants. The results are not restricted to a periodic setting. To enable the adaptive computations, a multitree-based adaptive wavelet scheme is extended to an implementable version in the (space-time) Petrov-Galerkin setting.