We develop an optimization-based algorithm for parametric model order reduction (PMOR) of linear time-invariant dynamical systems. Our method aims at minimizing the $\mathcal{H}_\infty \otimes \mathcal{L}_\infty$ approximation error in the frequency and parameter domain by an optimization of the reduced order model (ROM) matrices. State-of-the-art PMOR methods often compute several nonparametric ROMs for different parameter samples, which are then combined to a single parametric ROM. However, these parametric ROMs can have a low accuracy between the utilized sample points. In contrast, our optimization-based PMOR method minimizes the approximation error across the entire parameter domain. Moreover, due to our flexible approach of optimizing the system matrices directly, we can enforce favorable features such as a port-Hamiltonian structure in our ROMs across the entire parameter domain. Our method is an extension of the recently developed SOBMOR-algorithm to parametric systems. We extend both the ROM parameterization and the adaptive sampling procedure to the parametric case. Several numerical examples demonstrate the effectiveness and high accuracy of our method in a comparison with other PMOR methods.

翻译：我们开发了一种优化算法，用于线性时不变动态系统的参数模型降阶（PMOR）。我们的方法旨在通过对降阶模型（ROM）矩阵进行优化来在频率和参数域中最小化$\mathcal{H}_\infty \otimes \mathcal{L}_\infty$逼近误差。最先进的PMOR方法经常计算几个不同参数样品的非参数ROM，然后将它们组合成一个单一的参数ROM。然而，这些参数ROM在使用的样品点之间的精度可能很低。相反，我们基于优化的PMOR方法通过在整个参数域内最小化逼近误差来提高精度。此外，由于我们采用了直接优化系统矩阵的灵活方法，我们可以在整个参数域内强制实现有利的特性，例如哈密顿端口结构。我们的方法是最近开发的SOBMOR算法向参数系统的扩展。我们扩展了ROM参数化和自适应采样过程以适应参数情况。我们通过与其他PMOR方法的比较，在几个数值例子中展示了我们方法的有效性和高精度。