标题： 基于非侵入式数据驱动的ADI低秩平衡截断 显示英文标题

标题： Non-intrusive Data-driven ADI-based Low-rank Balanced Truncation

摘要： 在本简短的注释中，提供了一种基于ADI的低秩平衡截断的非侵入式数据驱动公式。 所提出的算法仅需要在ADI移位的镜像位置处的传递函数样本。 如果某些移位同时用于近似可控性Gramian和可观性Gramian，则还需要这些移位处的传递函数导数的样本，以在Loewner框架中强制Hermite插值。 需要注意的是，基于ADI的低秩平衡截断可以视为一个两步过程。 第一步涉及在ADI移位的镜像位置处构建原始模型的插值，这可以在Loewner框架内非侵入式地完成。 第二步涉及使用与插值数据相关的Gramian的低秩因子，通过平衡平方根算法来降低此插值的阶数。 这一步不需要任何系统信息，使得整个过程是非侵入式的，唯一需要的信息是在ADI移位的镜像位置处的传递函数和/或其导数的样本。 此外，还表明当在基于ADI的低秩平衡截断中选择降阶模型的阶数以匹配Gramian的低秩因子的数值秩时，它会有效地退化为在ADI移位的镜像位置处的标准插值。 提供了一个示例来解释所提出的方法。 显示英文摘要

摘要： In this short note, a non-intrusive data-driven formulation of ADI-based low-rank balanced truncation is provided. The proposed algorithm only requires transfer function samples at the mirror images of ADI shifts. If some shifts are used in both approximating the controllability Gramian and the observability Gramian, then samples of the transfer function's derivative at these shifts are also needed to enforce Hermite interpolation in the Loewner framework. It is noted that ADI-based low-rank balanced truncation can be viewed as a two-step process. The first step involves constructing an interpolant of the original model at the mirror images of the ADI shifts, which can be done non-intrusively within the Loewner framework. The second step involves reducing this interpolant using low-rank factors of Gramians associated with the interpolation data through the balanced square-root algorithm. This second step does not require any system information, making the overall process non-intrusive with the only required information being samples of the transfer function and/or its derivative at the mirror images of ADI shifts. Furthermore, it is shown that when the order of the reduced model in ADI-based low-rank balanced truncation is selected to match the numerical rank of the low-rank factors of the Gramians, it effectively reduces to standard interpolation at the mirror images of the ADI shift. An illustrative example is provided to explain the proposed approach.