Matrix equations are omnipresent in (numerical) linear algebra and systems theory. Especially in model order reduction (MOR) they play a key role in many balancing based reduction methods for linear dynamical systems. When these systems arise from spatial discretizations of evolutionary partial differential equations, their coefficient matrices are typically large and sparse. Moreover, the numbers of inputs and outputs of these systems are typically far smaller than the number of spatial degrees of freedom. Then, in many situations the solutions of the corresponding large-scale matrix equations are observed to have low (numerical) rank. This feature is exploited by M-M.E.S.S. to find successively larger low-rank factorizations approximating the solutions. This contribution describes the basic philosophy behind the implementation and the features of the package, as well as its application in the model order reduction of large-scale linear time-invariant (LTI) systems and parametric LTI systems.

中文翻译：

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矩阵方程，稀疏求解器：MM.ESS-2.0.1——（参数）模型的哲学、特征和应用

矩阵方程在（数值）线性代数和系统理论中无处不在。特别是在模型降阶 (MOR) 中，它们在许多基于平衡的线性动力系统降阶方法中发挥着关键作用。当这些系统由进化偏微分方程的空间离散化产生时，它们的系数矩阵通常很大而且很稀疏。此外，这些系统的输入和输出数量通常远小于空间自由度的数量。然后，在许多情况下，观察到相应的大规模矩阵方程的解具有低（数值）秩。MM.ESS 利用此功能来寻找近似解的连续较大的低秩分解。