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：我们针对从平稳模型到参数动力系统的广泛问题类别，发展了一个用于插值法 $\mathcal{L}_2$ 最优降阶建模的统一框架。首先证明该框架自然涵盖了 $\mathcal{H}_2$ 最优模型降阶中众所周知的插值必要条件，并推导出多输入/多输出参数动力系统在 $\mathcal{H}_2 \otimes \mathcal{L}_2$ 最优模型降阶中的插值条件。此外，我们为有理离散最小二乘最小化问题，以及一类参数平稳模型的 $\mathcal{L}_2$ 最优模型降阶问题，推导出新的插值最优性条件。研究表明，双侧切向埃尔米特插值是跨领域实现最优性的主要工具。通过两个数值算例展示了理论结果。