AI 中文总结
研究复平面上ADI移位位置受限的问题，提出新的LRCF-ADI方法使移位可在任意位置，还扩展到多种方程求解。基于此构建非侵入性降阶模型，并提出仅需虚轴上传递函数样本的数据驱动低秩平衡截断算法，突破实验测量限制。
AI 中文摘要
低秩Cholesky因子交替方向隐式（LRCF-ADI）迭代方法是计算大规模Lyapunov方程低秩解的有效方法，其形式为\(P\approx ZZ^\top\)。这种形式对平衡截断很有用。标准LRCF-ADI方法要求所有ADI移位的实部为负，这对某些应用有局限性。本文提出一种新的LRCF-ADI方法，ADI移位可位于复平面任何位置，包括虚轴。该方法还扩展到求解频率受限、时间受限的Lyapunov方程及Riccati方程。基于LRCF-ADI的平衡截断降阶模型可从ADI移位镜像处的传递函数样本非侵入性构建。标准LRCF-ADI方法需复平面右半部分样本，实验中无法测量，而新方法的ADI移位可在虚轴上，据此提出仅需虚轴上传递函数样本的数据驱动低秩平衡截断算法，可通过实验测量。
英文摘要
The low-rank Cholesky factor alternating direction implicit (LRCF-ADI) iteration method is an effective and efficient approach for computing low-rank solutions to large-scale Lyapunov equations in the form \(P\approx ZZ^\top\). This form is useful for balanced truncation, as the square-root algorithm requires computing the controllability and observability Gramians in this form. The standard LRCF-ADI method requires all ADI shifts to have negative real parts, which can be restrictive for applications like frequency-limited and data-driven balanced truncation, where purely imaginary ADI shifts are a more suitable choice. This paper proposes a new LRCF-ADI method where the ADI shifts can be located anywhere in the complex plane, including on the imaginary axis. The proposed generalized LRCF-ADI algorithm reduces to the standard LRCF-ADI algorithm as a special case. The new method is also extended to solve frequency-limited Lyapunov equations, time-limited Lyapunov equations, and Riccati equations. Approximations of matrix logarithm and matrix exponential products using the proposed method are also discussed. LRCF-ADI-based reduced models for balanced truncation can be constructed non-intrusively from transfer function samples at the mirror images of the ADI shifts, without accessing the state-space realization of the original model. Since the standard LRCF-ADI method requires all ADI shifts to have negative real parts, its non-intrusive implementation requires samples in the right half of the complex plane, which cannot be measured in an experimental setting. However, the ADI shifts in the proposed method can lie on the imaginary axis. Exploiting this property, we also propose a data-driven low-rank balanced truncation algorithm that requires only transfer function samples on the imaginary axis, which can be measured experimentally.
