Here, we focus on Model Order Reduction (MOR) of non-parametric second-order dynamical systems. In these MOR algorithms, sequences of large and sparse linear systems arise during the model reduction process. Solving such linear systems is the main computational bottleneck in efficient scaling of these MOR algorithms for reducing extremely large dynamical systems. Preconditioned iterative methods are often used for solving such linear systems. These iterative methods introduce errors because they solve the linear systems up to a certain tolerance. Hence, our focus is to analyze the stability of these MOR algorithms when using inexact linear solves. Adaptive Iterative Rational Global Arnoldi (AIRGA) is a popular MOR algorithm belonging to this category. We prove that, under four mild conditions, the AIRGA algorithm is backward stable with respect to the errors introduced by these inexact linear solves. Our results easily extend to other MOR algorithms belonging to this category. Our first condition enforces the use of a Ritz-Galerkin based linear solver, where the residual of a linear system is made orthogonal to the corresponding Krylov subspace. Our second condition requires satisfying few extra orthogonalities. We show how to modify the underlying linear solver to achieve these extra orthogonalities. We further demonstrate that using a recycling variant of the underlying linear solver helps us achieve these orthogonalities cheaply and with no code changes. Our third condition involves existence and invertibility of a matrix mostly dependent upon the input dynamical system, with the norm of this matrix bounded by one. Our fourth and final condition involves being able to compute a perturbation from the derived expression and bounding its norm by one as well. The last two conditions are easily satisfied by all our models.

翻译：本文聚焦于非参数二阶动力系统的模型降阶。在这些模型降阶算法中，降阶过程中会产生一系列大型稀疏线性系统。求解此类线性系统是这些模型降阶算法在高效缩减极大规模动力系统时的主要计算瓶颈。通常采用预条件迭代法来求解这些线性系统。由于迭代法仅以特定容差求解线性系统，因此会引入误差。因此，我们重点分析在使用非精确线性求解时这些模型降阶算法的稳定性。自适应迭代有理全局Arnoldi算法是此类中一种流行的模型降阶算法。我们证明，在四个温和条件下，AIRGA算法对于这些非精确线性求解引入的误差是向后稳定的。我们的结果可轻松推广至属于此类的其他模型降阶算法。第一个条件强制使用基于Ritz-Galerkin的线性求解器，其中线性系统的残差被正交化到相应的Krylov子空间。第二个条件要求满足若干额外的正交性。我们展示了如何修改底层线性求解器以实现这些额外正交性。我们进一步证明，使用底层线性求解器的循环变体有助于以低成本且无需代码修改的方式实现这些正交性。第三个条件涉及一个主要依赖于输入动力系统的矩阵的存在性与可逆性，且该矩阵的范数以1为界。第四个也是最后一个条件涉及能够根据推导表达式计算一个扰动，并使其范数同样以1为界。后两个条件在我们所有模型中均易满足。