Quantifying the error that is induced by numerical approximation techniques is an important task in many fields of applied mathematics. Two characteristic properties of error bounds that are desirable are reliability and efficiency. In this article, we present an error estimation procedure for general nonlinear problems and, in particular, for parameter-dependent problems. With the presented auxiliary linear problem (ALP)-based error bounds and corresponding theoretical results, we can prove large improvements in the accuracy of the error predictions compared with existing error bounds. The application of the procedure in parametric model order reduction setting provides a particularly interesting setup, which is why we focus on the application in the reduced basis framework. Several numerical examples illustrate the performance and accuracy of the proposed method.
For certain dynamical systems it is possible to significantly simplify the study of stability by means of the center manifold theory. This theory allows to isolate the complicated asymptotic behavior of the system close to a non-hyperbolic equilibrium point, and to obtain meaningful predictions of its behavior by analyzing a reduced dimensional problem. Since the manifold is usually not known, approximation methods are of great interest to obtain qualitative estimates. In this work, we use a data-based greedy kernel method to construct a suitable approximation of the manifold close to the equilibrium. The data are collected by repeated numerical simulation of the full system by means of a high-accuracy solver, which generates sets of discrete trajectories that are then used to construct a surrogate model of the manifold. The method is tested on different examples which show promising performance and good accuracy.
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