论文标题
$ \ MATHCAL {L} _2 $ - 最佳的减少订单建模，使用参数可分配的表单
$\mathcal{L}_2$-optimal Reduced-order Modeling Using Parameter-separable Forms
论文作者
论文摘要
我们为$ \ Mathcal {l} _2 $ - 最佳的减少订单建模提供了一个统一的框架，用于线性时间不变的动力学系统和固定参数问题。使用还原模型数量的参数分离形式，我们将$ \ Mathcal {l} _2 $成本函数得出相对于降低的矩阵的梯度，然后该矩阵允许仅使用输出样本构建最佳的最佳近似值，从而允许使用非梯度，基于数据驱动的，基于数据驱动的，基于梯度的下降算法。通过选择适当的度量，该框架涵盖了连续（Lebesgue）和离散成本功能。我们通过各种数值示例显示了所提出的算法的功效。此外，我们分析了可以通过投影获得数据驱动的近似值的条件。
We provide a unifying framework for $\mathcal{L}_2$-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the $\mathcal{L}_2$ cost function with respect to the reduced matrices, which then allows a non-intrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection.
