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：我们针对从稳态模型到参数化动力系统等广泛问题，建立了一种用于插值 $\mathcal{L}_2$ 最优降阶建模的统一框架。首先，我们证明该框架自然涵盖了经典的 $\mathcal{H}_2$ 最优模型降阶的插值必要条件，并导出了多输入/多输出参数化动力系统 $\mathcal{H}_2 \otimes \mathcal{L}_2$ 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稳态模型 $\mathcal{L}_2$ 最优降阶的新型插值最优性条件。研究表明，双切向Hermite插值是跨领域最优性的主要工具。最后通过两个数值算例验证了理论结果。