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arXiv 2303.18072math.NAcs.NA
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针对参数化哈密顿系统的基于字典的在线自适应保结构模型降阶

Dictionary-based Online-adaptive Structure-preserving Model Order Reduction for Parametric Hamiltonian Systems

Robin Herkert, Patrick Buchfink, Bernard Haasdonk

备注：
29 pages, 13 figures
发表信息：
Advances in Computational Mathematics, Volume 50, 2024
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AI总结：

本文提出一种基于字典的在线自适应保结构 MOR 方法，通过状态、非线性项和 DEIM 点字典实现与全维无关的局部基调整，并在波动方程和 Sine-Gordon 模型上验证其效率。

摘要：

针对参数化问题的经典模型降阶（MOR）可能因所需投影基的规模过大而导致计算效率低下，尤其是对于 Kolmogorov n-width 衰减缓慢的问题。此外，动力系统可能具有哈密顿结构，并且在降阶过程中应予以保持。在本报告中，我们通过提出一种相应的基于字典的在线自适应 MOR 方法来解决这两个方面的问题。该方法需要状态变量、非线性项以及离散经验插值（DEIM）点的字典。在在线仿真过程中，以在线高效的方式执行局部基扩展/简化，即基修改和降阶模型在线仿真的运行时复杂度不依赖于完整状态维度。在线性波动方程和非线性 Sine-Gordon 算例上的实验证明了该方法的高效性。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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