矩阵方程，稀疏解算器：M-M. E.S.S. - 2.0.1 --（参数）模型的原理、特点及应用
Matrix Equations, Sparse Solvers: M-M.E.S.S.-2.0.1 -- Philosophy, Features and Application for (Parametric) Model
摘要：
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起着关键作用。当这些系统产生于演化偏微分方程的空间离散化时，它们的系数矩阵通常是大而稀疏的。此外，这些系统的输入和输出的数量通常远小于空间自由度的数量。然后，在许多情况下，相应的大型矩阵方程的解决方案，观察到有低（数值）秩。M-M.E.S.S.利用了此功能。找到连续更大的低秩因子分解近似的解决方案。这一贡献描述了背后的实施和功能包的基本理念，以及它的应用在大规模线性时不变（LTI）系统和参数LTI系统的模型降阶。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
