Browsing by Author "Haasdonk, Bernard (Prof. Dr.)"

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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
Deep and greedy kernel methods : algorithms, analysis and applications
(2023) Wenzel, Tizian; Haasdonk, Bernard (Prof. Dr.)
Open Access
Efficient schemes for parameterized multiscale problems
(2015) Alebrand, Sven; Haasdonk, Bernard (Prof. Dr.)
This thesis investigates efficient schemes for parameterized multiscale problems. The reduced basis method is a well-known technique for the reduction of computational effort for parameterized partial differential equations. Herein two extensions of the methodology are introduced. First, an extension to problems with high parameter dimension is suggested. This so-called online greedy basis construction approach relies on building parameter-dependent reduced-dimensional approximation spaces during the main computational phase, the so-called online-phase. Bases constructed using the online greedy basis construction are much smaller than those constructed using conventional greedy methods and runtime improvements can be observed for certain cases. Secondly, the reduced basis method is combined with ideas from so-called multiscale methods to make it applicable to problems with multiscale character by overcoming the lack of control over the runtime of its preparatory phase, the so-called offline-phase. It is established that a novel approach, the localized reduced basis multiscale method, allows to displace the computational effort between the two phases of the reduced basis method. This technique is applied to stationary heat diffusion problems and to two-phase flow in porous media. Additional performance can be reached by using multiscale methods as efficient solvers during the preparatory phase of the localized reduced basis method. A parallelization concept for these methods is introduced and scaling tests for an implementation of the concept are presented. Finally some details on our implementations of multiscale methods and of the aforementioned extensions to the reduced basis method are presented.
Open Access
Structure-preserving model reduction on subspaces and manifolds
(2024) Buchfink, Patrick; Haasdonk, Bernard (Prof. Dr.)
Mathematical models are a key enabler to understand complex processes across all branches of research and development since such models allow us to simulate the behavior of the process without physically realizing it. However, detailed models are computationally demanding and, thus, are frequently prohibited from being evaluated (a) multiple times for different parameters, (b) in real time or (c) on hardware with low computational power. The field of model (order) reduction (MOR) aims to approximate such detailed models with more efficient surrogate models that are suitable for the tasks (a-c). In classical MOR, the solutions of the detailed model are approximated in a problem-specific, low-dimensional subspace, which is why we refer to it as MOR on subspaces. The subspace is characterized by a reduced basis that can be computed from given data with a so-called basis generation technique. The two key aspects in this thesis are: (i) structure-preserving MOR techniques and (ii) MOR on manifolds. Preserving given structures throughout the reduction is important to obtain physically consistent reduced models. We demonstrate this for Lagrangian and Hamiltonian systems, which are dynamical systems that guarantee preservation of energy over time. MOR on manifolds, on the other hand, broadens the applicability of MOR to problems that cannot be treated efficiently with MOR on subspaces.
Open Access
Total variation minimization via dual-based methods and its discretization aspects
(2023) Hilb, Stephan; Haasdonk, Bernard (Prof. Dr.)