Browsing by Author "Wittwar, Dominik"

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Open Access
Approximation with matrix-valued kernels and highly effective error estimators for reduced basis approximations
(2022) Wittwar, Dominik; Haasdonk, Bernard (Prof. Dr.)
This thesis can be summarized under the aspect of surrogate modelling for vector-valued functions and error quantification for those surrogate models. The thesis, in a broad sense, is split into two different parts. The first aspect deals with constructing surrogate models via matrix-valued kernels using both interpolation and regularization procedures. For this purpose, a new class of so called uncoupled separable matrix-valued kernels is introduced and heavy emphasis is placed on how suitable sample points for the construction of the surrogate can be chosen in such a way that quasi-optimal convergence rates can be achieved. In the second part, the focus does not lie on the construction of the surrogate itself, but on how existing a-posteriori error estimation can be improved to result in highly efficient error bounds. This is done in the context of reduced basis methods, which similar to the kernel surrogates, construct surrogate models by using data acquired from samples of the desired target function. Both parts are accompanied by numerical experiments which illustrate the effectiveness as well as verify the analytically derived properties of the presented methods.
Open Access
Improved a posteriori error bounds for reduced port-Hamiltonian systems
(2024) Rettberg, Johannes; Wittwar, Dominik; Buchfink, Patrick; Herkert, Robin; Fehr, Jörg; Haasdonk, Bernard
Projection-based model order reduction of dynamical systems usually introduces an error between the high-fidelity model and its counterpart of lower dimension. This unknown error can be bounded by residual-based methods, which are typically known to be highly pessimistic in the sense of largely overestimating the true error. This work applies two improved error bounding techniques, namely (a) a hierarchical error bound and (b) an error bound based on an auxiliary linear problem , to the case of port-Hamiltonian systems. The approaches rely on a secondary approximation of (a) the dynamical system and (b) the error system. In this paper, these methods are adapted to port-Hamiltonian systems. The mathematical relationship between the two methods is discussed both theoretically and numerically. The effectiveness of the described methods is demonstrated using a challenging three-dimensional port-Hamiltonian model of a classical guitar with fluid–structure interaction.
Open Access
Port-Hamiltonian fluid-structure interaction modelling and structure-preserving model order reduction of a classical guitar
(2023) Rettberg, Johannes; Wittwar, Dominik; Buchfink, Patrick; Brauchler, Alexander; Ziegler, Pascal; Fehr, Jörg; Haasdonk, Bernard
Open Access
Rigorous and effective a-posteriori error bounds for nonlinear problems : application to RB methods
(2020) Schmidt, Andreas; Wittwar, Dominik; Haasdonk, Bernard
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.