论文标题
通过学习参数域中的误差估计器来自适应插值
Adaptive Interpolatory MOR by Learning the Error Estimator in the Parameter Domain
论文作者
论文摘要
插值方法提供了一个有力的框架，用于生成具有随时间变化输入的非参数或参数系统的简化级模型（ROM）。自适应选择插值点仍然是一个积极关注的领域。 Feng等人引入了贪婪的框架。 [Esaim：数学。模型。 numer。肛门。 51（6），2017年]和冯和本纳[IEEE Trans。微值。理论技术。 67（12），2019年]使用后验误差估计器自动选择插值点。然而，当参数范围较大或参数空间维度大于两个时，贪婪算法可能需要大量时间，因为训练集需要包括相当数量的参数。作为一种补救措施，我们通过学习有效的后验误差估计器对参数域进行了自适应训练技术。快速学习过程是通过使用径向基函数（RBF）在良好的参数训练集上插值误差估计器来创建的，代表整个参数域。仅在包括一些参数样本的粗糙训练集上评估误差估计器。该算法是Chellappa等人的作品的扩展。 [Arxiv e-Prints 1910.00298]在频域中的插值模型降低（MOR）。除了这项工作之外，我们在频域中使用了新提出的无INF-SUP-SUP-SUP-cONTANT误差估计器，它通常比使用INF-SUP常数的误差估计器更紧得多。
Interpolatory methods offer a powerful framework for generating reduced-order models (ROMs) for non-parametric or parametric systems with time-varying inputs. Choosing the interpolation points adaptively remains an area of active interest. A greedy framework has been introduced in Feng et al. [ESAIM: Math. Model. Numer. Anal. 51(6), 2017] and in Feng and Benner [IEEE Trans. Microw. Theory Techn. 67(12), 2019] to choose interpolation points automatically using a posteriori error estimators. Nevertheless, when the parameter range is large or if the parameter space dimension is larger than two, the greedy algorithm may take considerable time, since the training set needs to include a considerable number of parameters. As a remedy, we introduce an adaptive training technique by learning an efficient a posteriori error estimator over the parameter domain. A fast learning process is created by interpolating the error estimator using radial basis functions (RBF) over a fine parameter training set, representing the whole parameter domain. The error estimator is evaluated only on a coarse training set including a few parameter samples. The algorithm is an extension of the work in Chellappa et al. [arXiv e-prints 1910.00298] to interpolatory model order reduction (MOR) in frequency domain. Beyond this work, we use a newly proposed inf-sup-constant-free error estimator in the frequency domain, which is often much tighter than the error estimator using the inf-sup constant.
