Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the Iterative Rational Krylov Algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

翻译：插值方法是一种成熟且有效的技术，用于高效生成精确的降阶代理模型。这类方法面临的常见挑战包括自动选择良好乃至最优的插值点，以及确定降阶模型的适当规模。针对线性非结构化系统，迭代有理Krylov算法（IRKA）通过求解线性特征值问题进行迭代更新以计算最优插值点，从而解决了第一个问题。然而，在保持系统内部结构的情况下，最优插值点未知，而基于非线性特征值问题的启发式方法会导致潜在插值点数量通常超出降阶系统的合理规模。本研究提出一种受IRKA启发的基于投影的迭代插值方法，适用于一般结构化系统，能够自适应地计算近最优插值点并确定降阶系统的适当规模。此外，可选择插值点的迭代更新策略，使降阶模型在指定的感兴趣频率范围内提供精确近似。针对此类应用，我们的新方法在精度和计算效率方面均优于现有方法。我们通过不同结构的数值算例证明了这一结论。