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

TL;DR

The paper addresses non-intrusive model order reduction for ADI-based low-rank balanced truncation by leveraging the interpolatory Loewner framework. It presents a data-driven two-step approach: first build an interim ROM through interpolation at mirror ADI shifts using transfer-function samples (and derivatives when needed), then apply the balanced square-root algorithm with low-rank Cholesky factors derived from the interpolation data. Key contributions include a rigorous non-intrusive formulation that requires only samples of $G(s)$ (and $G'(s)$ at common shifts) and a demonstration that, when the reduced order equals the product of shifts, the process reduces to standard interpolation at those mirror shifts. An illustrative example shows the data-driven method reproduces the top Hankel singular values and a matching transfer function to intrusive LRCF-ADI, highlighting its practical potential for data-driven MOR without access to the state-space realization.

Abstract

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.