SIAM Journal on Scientific Computing, Volume 42, Issue 6, Page A4017-A4045, January 2020.
In many applications throughout science and engineering, model reduction plays an important role replacing expensive large-scale linear dynamical systems by inexpensive reduced order models that capture key features of the original, full order model. One approach to model reduction finds reduced order models that are locally optimal approximations in the $\mathcal{H}_2$-norm, an approach taken by the iterative rational Krylov algorithm (IRKA), among others. Here we introduce a new approach for $\mathcal{H}_2$-optimal model reduction using the projected nonlinear least squares framework previously introduced in [J. M. Hokanson, SIAM J. Sci. Comput., 39 (2017), pp. A3107--A3128]. At each iteration, we project the $\mathcal{H}_2$ optimization problem onto a finite-dimensional subspace yielding a weighted least squares rational approximation problem. Subsequent iterations append this subspace such that the least squares rational approximant asymptotically satisfies the first order necessary conditions of the original, $\mathcal{H}_2$ optimization problem. This enables us to build reduced order models with similar error in the $\mathcal{H}_2$-norm but using far fewer evaluations of the expensive, full order model compared to competing methods. Moreover, our new algorithm only requires access to the transfer function of the full order model, unlike IRKA, which requires a state-space representation, or TF-IRKA, which requires both the transfer function and its derivative. Applying the projected nonlinear least squares framework to the $\mathcal{H}_2$-optimal model reduction problem opens new avenues for related model reduction problems.

中文翻译：

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投影非线性最小二乘法的$ \ mathcal {H} _2 $-最优模型约简

SIAM科学计算杂志，第42卷，第6期，第A4017-A4045页，2020年1月。
在整个科学和工程中的许多应用中，模型简化在用便宜的降阶模型代替昂贵的大规模线性动力学系统方面起着重要的作用，该模型可以捕获原始的全阶模型的关键特征。一种模型简化方法可以找到降阶模型，该模型是$ \ mathcal {H} _2 $范数中的局部最优近似值，是迭代有理Krylov算法（IRKA）采取的一种方法。在这里，我们采用先前在[JM Hokanson，SIAM J. Sci。计算（39）（2017），A3107--A3128页。在每次迭代中，我们将$ \ mathcal {H} _2 $优化问题投影到一个有限维子空间上，从而产生一个加权最小二乘有理逼近问题。随后的迭代附加此子空间，以使最小二乘有理逼近渐近满足原始$ \ mathcal {H} _2 $优化问题的一阶必要条件。这使我们能够建立降阶模型，在$ \ mathcal {H} _2 $范数中具有相似的误差，但与竞争方法相比，使用昂贵的全订单模型的评估少得多。而且，我们的新算法只需要访问全阶模型的传递函数，而不像IRKA需要状态空间表示或TF-IRKA既需要传递函数又需要其导数。将投影的非线性最小二乘框架应用于$ \ mathcal {H} _2 $-最优模型归约问题，为相关的模型归约问题开辟了新途径。