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：我们为从固定模型到参数动态系统等一系列广泛的问题制定了一个统一的内插 $mathcal{L ⁇ 2$-最佳减序模型框架。我们首先表明,这个框架自然包括众所周知的内插必要条件,用于减少美元mathcal{H ⁇ 2$-最佳模式订单,并导致美元mathcal{H ⁇ 2\otimes\mathcal{L ⁇ 2$-最佳减序模型,用于减少多输入/多输出参数动态系统。此外,我们从两个数字例子中说明了理论结果。
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