We develop a unifying framework for interpolatory $\mathcal{L}_2$-optimal reduced-order modeling for a wide classes of problems ranging from stationary models to parametric dynamical systems. We first show that the framework naturally covers the well-known interpolatory necessary conditions for $\mathcal{H}_2$-optimal model order reduction and leads to the interpolatory conditions for $\mathcal{H}_2 \otimes \mathcal{L}_2$-optimal model order reduction of multi-input/multi-output parametric dynamical systems. Moreover, we derive novel interpolatory optimality conditions for rational discrete least-squares minimization and for $\mathcal{L}_2$-optimal model order reduction of a class of parametric stationary models. We show that bitangential Hermite interpolation appears as the main tool for optimality across different domains. The theoretical results are illustrated on two numerical examples.

翻译：我们针对从静态模型到参数化动态系统的广泛问题类别，发展了一种插值 $\mathcal{L}_2$ 最优降阶建模的统一框架。首先证明，该框架自然涵盖了经典的 $\mathcal{H}_2$ 最优模型降阶插值必要条件，并导出了多输入/多输出参数化动态系统 $\mathcal{H}_2 \otimes \mathcal{L}_2$ 最优模型降阶的插值条件。此外，我们推导了有理离散最小二乘最小化以及一类参数化静态模型 $\mathcal{L}_2$ 最优降阶的新颖插值最优性条件。研究表明，双侧切向埃尔米特插值成为跨领域最优性的主要工具。理论结果通过两个数值算例进行了验证。