用于大规模斯坦方程的新型广义低秩乔列斯基因子ADI算法
A New Generalized Low-Rank Cholesky Factor ADI Algorithm for Large-Scale Stein Equations
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中文总结 AI 辅助
本文提出一种新型广义低秩乔列斯基因子ADI算法，用于求解大规模斯坦方程，该算法可自动选择移位，支持单位圆上的移位，还可实现限频率斯坦方程的数值积分与数据驱动模型降阶，经大规模模型验证了其有效性。
中文摘要 AI 辅助
低秩交替方向隐式（ADI）方法是求解具有低秩解的大规模斯坦方程的高效方法。本文表明，与连续时间李雅普诺夫方程的情况类似，用于斯坦方程的低秩乔列斯基因子ADI（LRCF-ADI）方法隐式地对离散时间系统执行H₂伪最优模型降阶。这一观察结果催生了一种自动移位生成策略，使LRCF-ADI无需用户干预即可选择后续移位。标准LRCF-ADI方法要求移位在单位圆外，我们将该方法推广为允许移位位于复平面上的任意位置，包括单位圆上。该扩展通过在单位圆上的点对被积函数进行插值，实现了限频率斯坦方程的数值积分；还利用单位圆上可实验测量的传递函数样本，实现了非侵入式、数据驱动的平衡截断和限频率平衡截断，无需访问状态空间实现。大规模模型的数值结果证明了所提方法作为低秩斯坦方程求解器和数据驱动模型降阶方法的有效性。
英文摘要
The low-rank alternating direction implicit (ADI) method is an efficient solver for large-scale Stein equations with low-rank solutions. This paper shows that, as in the continuous-time Lyapunov equation case, the low-rank Cholesky factor ADI (LRCF-ADI) method for Stein equations implicitly performs $\mathcal{H}_2$-pseudo-optimal model order reduction for discrete-time systems. This observation leads to an automatic shift-generation strategy, allowing LRCF-ADI to select subsequent shifts without user intervention. The standard LRCF-ADI method requires shifts outside the unit circle. We generalize the method to allow shifts anywhere in the complex plane, including on the unit circle. This extension enables numerical integration for frequency-limited Stein equations by interpolating the integrand at points on the unit circle. It also enables non-intrusive, data-driven balanced truncation and frequency-limited balanced truncation using experimentally measurable transfer function samples on the unit circle, without requiring access to a state-space realization. Numerical results for large-scale models demonstrate the effectiveness of the proposed methods as low-rank Stein equation solvers and data-driven model order reduction methods.
