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：我们提出了一种基于优化的参数化模型降阶（PMOR）算法，用于线性时不变动力系统。该方法旨在通过优化降阶模型（ROM）矩阵，最小化频率域和参数域中的$\mathcal{H}_\infty \otimes \mathcal{L}_\infty$近似误差。现有PMOR方法通常针对不同参数样本计算多个非参数化ROM，再将其组合成单一参数化ROM。然而，这类参数化ROM在采样点之间的参数区间内可能精度较低。相比之下，我们提出的基于优化的PMOR方法可在整个参数域内最小化近似误差。此外，由于采用直接优化系统矩阵的灵活策略，我们能够在整个参数域的ROM中强制引入理想特性（如端口-哈密顿结构）。本方法是对近期开发的SOBMOR算法在参数化系统上的扩展，我们将ROM参数化策略与自适应采样流程均推广至参数化情形。多个数值算例表明，与其他PMOR方法相比，本方法在有效性和高精度方面具有显著优势。