Practical challenges in data-driven interpolation: dealing with noise, enforcing stability, and computing realizations

Abstract

In this contribution, we propose a detailed study of interpolation-based datadriven methods that are of relevance in the model reduction and also in the systems and control communities. The data are given by samples of the transfer function of the underlying (unknown) model, i.e., we analyze frequency-response data. We also propose novel approaches that combine some of the main attributes of the established methods, for addressing particular issues. This includes placing poles and hence, enforcing stability of reduced-order models, robustness to noisy or perturbed data, and switching from different rational function representations. We mention here the classical state-space format and also various barycentric representations of the fitted rational interpolants. We show that the newly-developed approaches yield, in some cases, superior numerical results, when comparing to the established methods. The numerical results include a thorough analysis of various aspects related to approximation errors, choice of interpolation points, or placing dominant poles, which are tested on some benchmark models and data-sets.

Plug‐and‐play adaptive surrogate modeling of parametric nonlinear dynamics in frequency domain

“…13 The number S j and the locations of the support frequencies 𝜔 j,i are crucial for the approximation properties and efficiency of the MRI surrogate: too few samples yield an inaccurate approximation, whereas too many make the training of the local surrogate overly expensive, and can even result in numerical instabilities. 20,35 A practical and effective way of choosing the support frequencies is the greedy MRI (gMRI) method, introduced in Reference 19.…”

Leveraging the Hankel norm approximation and data‐driven algorithms in reduced order modeling

“…For the output of the AAA algorithm to be useful for the HNA algorithm, we must find a system realization of the rational approximation. The system generated by the canonical block-AAA algorithm 26,29 has a well-conditioned transfer function G but a system realization (Ã, B, C, D) that is hard to represent in floating-point arithmetic, causing numerical issues in the second stage (see Subsection 5.2). We propose a modification of the block-AAA algorithm that sacrifices some accuracy for numerical stability.…”