arXiv subjects

Michael S. Ackermann

Publications and source records attributed to Michael S. Ackermann.

6 recordsLinked to original sources

arXiv2026

A refined nonlinear least-squares method for the rational approximation problem

The adaptive Antoulas-Anderson (AAA) algorithm for rational approximation is a widely used method for the efficient construction of highly accurate rational approximations to given data. While AAA can often produce rational approximations accurate to any prescribed tolerance, these approximations may have degrees larger than what is actually required to meet the given tolerance. In this work, we consider the adaptive construction of interpolating rational approximations while aiming for the smallest feasible degree to satisfy a given error tolerance. To this end, we introduce refinement approaches to the linear least-squares step of the classical AAA algorithm that aim to minimize the true nonlinear least-squares error with respect to the given data. Furthermore, we theoretically analyze the derived approaches in terms of the corresponding gradients from the resulting minimization problems and use these insights to propose a new greedy framework that ensures monotonic error convergence. Numerical examples from function approximation and model order reduction verify the effectiveness of the proposed algorithm to construct accurate rational approximations of small degrees.

math.NA↗

arXiv2025

$L^2$ and $L^\infty$ rational approximation

Using recently developed algorithms, we compute and compare best $L^2$ and $L^\infty$ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first computational study. To compute the $L^2$ best approximations, we employ a new formulation of TF-IRKA in barycentric form.

math.NA↗

arXiv2025

Second-order AAA algorithms for structured data-driven modeling

The data-driven modeling of dynamical systems has become an essential tool for the construction of accurate computational models from real-world data. In this process, the inherent differential structures underlying the considered physical phenomena are often neglected making the reinterpretation of the learned models in a physically meaningful sense very challenging. In this work, we present three data-driven modeling approaches for the construction of dynamical systems with second-order differential structure directly from frequency domain data. Based on the second-order structured barycentric form, we extend the well-known Adaptive Antoulas-Anderson algorithm to the case of second-order systems. Depending on the available computational resources, we propose variations of the proposed method that prioritize either higher computation speed or greater modeling accuracy, and we present a theoretical analysis for the expected accuracy and performance of the proposed methods. Three numerical examples demonstrate the effectiveness of our new structured approaches in comparison to classical unstructured data-driven modeling.

math.NA↗

arXiv2024

Time-Domain Iterative Rational Krylov Method

The Realization Independent Iterative Rational Krylov Algorithm (TF-IRKA) is a frequency-based data-driven reduced order modeling (DDROM) method that constructs $\mathcal H_2$ optimal DDROMs. However, as the $\mathcal H_2$ optimal approximation theory dictates, TF-IRKA requires repeated sampling of frequency data, that is, values of the system transfer function and its derivative, outside the unit circle. This repeated evaluation of frequency data requires repeated full model computations and may not be feasible. The data-informativity framework for moment matching provides a method for obtaining such frequency data from a single time-domain simulation. However, this framework usually requires solving linear systems with prohibitively ill-conditioned matrices, especially when recovering frequency data from off the unit circle as required for optimality. In this paper, building upon our previous work with the data informativity framework for moment matching, we provide a formula for the nonzero extreme eigenvalues of a symmetric rank-$1$ perturbation to an orthogonal projection, which then leads to an optimal scaling of the aforementioned linear systems. We also establish connections between the underlying dynamical system and conditioning of these linear systems. This analysis then leads to our algorithmic development, time-domain IRKA, which allows us to implement a time-domain variant of TF-IRKA, constructing $\mathcal H_2$ optimal DDROMs from a single time-domain simulation without requiring repeated frequency data evaluations. The numerical examples illustrate the effectiveness of the proposed algorithm.

math.NA↗

arXiv2024

Frequency-Based Reduced Models from Purely Time-Domain Data via Data Informativity

Frequency-based methods have been successfully employed in creating high fidelity data-driven reduced order models (DDROMs) for linear dynamical systems. These methods require access to values (and sometimes derivatives) of the frequency-response function (transfer function) in the complex plane. These frequency domain values can at times be costly or difficult to obtain (especially if the method of choice requires resampling); instead one may have access to only time-domain input-output data. The data informativity approach to moment matching provides a powerful new framework for recovering the required frequency data from a single time-domain trajectory. In this work, we analyze and extend upon this framework, resulting in vastly improved conditioning of the associated linear systems, an error indicator, and removal of an assumption that the system order is known. This analysis leads to a robust algorithm for recovering frequency information from time-domain data, suitable for large scale systems. We demonstrate the effectiveness of our algorithm by forming frequency based DDROMs from time-domain data of several dynamical systems.

math.NA↗

arXiv2018

Switching of biaxial synthetic antiferromagnets: a micromagentic study

We simulate the switching behavior of nanoscale synthetic antiferromagnets (SAFs), inspired by recent experimental progress in spin-orbit-torque switching of crystal antiferromagnets. The SAF consists of two ferromagnetic thin films with in-plane biaxial anisotropy and interlayer exchange coupling. Staggered field-like Rashba spin-orbit torques from the opposite surfaces of the SAF induce a canted net magnetization, which triggers an orthogonal torque that drives 90$^\circ$ switching of the Néel vector. Such dynamics driven by the field-like spin-orbit torque allows for faster switching with increased Gilbert damping, without a significant detrimental increase of the threshold switching current density. Our results point to the potential of SAFs as model systems, based on simple ferromagnetic metals, to mimic antiferromagnetic device physics.

cond-mat.mes-hall↗