08 Fakultät Mathematik und Physik

Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9

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Author: search.filters.author.Buchfink, PatrickSubject: search.filters.subject.510

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    Symplectic model order reduction with non-orthonormal bases
    (2019) Buchfink, Patrick; Bhatt, Ashish; Haasdonk, Bernard
    Parametric high-fidelity simulations are of interest for a wide range of applications. But the restriction of computational resources renders such models to be inapplicable in a real-time context or in multi-query scenarios. Model order reduction (MOR) is used to tackle this issue. Recently, MOR is extended to preserve specific structures of the model throughout the reduction, e.g. structure-preserving MOR for Hamiltonian systems. This is referred to as symplectic MOR. It is based on the classical projection-based MOR and uses a symplectic reduced order basis (ROB). Such a ROB can be derived in a data-driven manner with the Proper Symplectic Decomposition (PSD) in the form of a minimization problem. Due to the strong nonlinearity of the minimization problem, it is unclear how to efficiently find a global optimum. In our paper, we show that current solution procedures almost exclusively yield suboptimal solutions by restricting to orthonormal ROBs. As new methodological contribution, we propose a new method which eliminates this restriction by generating non-orthonormal ROBs. In the numerical experiments, we examine the different techniques for a classical linear elasticity problem and observe that the non-orthonormal technique proposed in this paper shows superior results with respect to the error introduced by the reduction.
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    Greedy kernel methods for approximating breakthrough curves for reactive flow from 3D porous geometry data
    (2024) Herkert, Robin; Buchfink, Patrick; Wenzel, Tizian; Haasdonk, Bernard; Toktaliev, Pavel; Iliev, Oleg
    We address the challenging application of 3D pore scale reactive flow under varying geometry parameters. The task is to predict time-dependent integral quantities, i.e., breakthrough curves, from the given geometries. As the 3D reactive flow simulation is highly complex and computationally expensive, we are interested in data-based surrogates that can give a rapid prediction of the target quantities of interest. This setting is an example of an application with scarce data, i.e., only having a few available data samples, while the input and output dimensions are high. In this scarce data setting, standard machine learning methods are likely to fail. Therefore, we resort to greedy kernel approximation schemes that have shown to be efficient meshless approximation techniques for multivariate functions. We demonstrate that such methods can efficiently be used in the high-dimensional input/output case under scarce data. Especially, we show that the vectorial kernel orthogonal greedy approximation (VKOGA) procedure with a data-adapted two-layer kernel yields excellent predictors for learning from 3D geometry voxel data via both morphological descriptors or principal component analysis.
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    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.
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    Dictionary-based online-adaptive structure-preserving model order reduction for parametric Hamiltonian systems
    (2024) Herkert, Robin; Buchfink, Patrick; Haasdonk, Bernard
    Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov n -widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities, and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e., the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach.