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Athanasios C. Antoulas

Publications and source records attributed to Athanasios C. Antoulas.

18 recordsLinked to original sources
arXiv2025

Rational approximation for Zolotarev sign and ratio problems

We study algorithms for Zolotarev (3rd and 4th) problems rational approximation. First, we show that the Loewner framework (LF) is appropriate to rapidly and with no iterations, approximate the Zolotarev problems by compressing the (numerous) interpolation conditions. Second, we compare the approximation properties (e.g. coefficients and poles) of LF with the standard AAA, AAA-sign and its AAA-Lawson variants. We concentrate the study on a canonical example, namely the symmetric two-circles one, for which the optimal solution is well documented. For this case, we highlight the numerical robustness of LF and its ability to recover the structure of the optimal solution. Additional non-trivial geometries are also reported, emphasizing that LF is fast, reliable, and yields accurate approximants with no user intervention.

math.NA↗
arXiv2025

Tensor-based multivariate function approximation: methods benchmarking and comparison

We evaluate some methods designed for tensor- (or data-) based multivariate model construction (approximation and compression). To this aim, a collection of multivariate functions and an evaluation methodology are suggested. First, these functions, with varying complexity (e.g., number and degree of the variables) and nature (e.g., rational, irrational, differentiable or not, symmetric, etc.) are used to build $n$-dimensional tensors, each of different dimension and memory size. Second, grounded on this tensor, we evaluate the performances of different methods and implementations leading to different types of surrogate models (e.g., rational functions, networks). The accuracy, the computational time, the parameter tuning impact, etc. are monitored and reported. One objective is to evaluate the different available strategies to guide users on the prospects, advantages, and limits of the various tools. The contributions are twofold: (i) to suggest a comprehensive benchmark collection together with a methodology for tensor approximation with a surrogate model and, in addition, (ii) to provide a digest and additional details of the multivariate Loewner Framework (mLF) approach [Antoulas et al., 2025], as well as detailed examples and code.

math.NA↗
arXiv2025

On the Loewner framework, the Kolmogorov superposition theorem, and the curse of dimensionality

The Loewner framework is an interpolatory approach for the approximation of linear and nonlinear systems. The purpose here is to extend this framework to linear parametric systems with an arbitrary number n of parameters. To achieve this, a new generalized multivariate rational function realization is proposed. Then, we introduce the n-dimensional multivariate Loewner matrices and show that they can be computed by solving a set of coupled Sylvester equations. The null space of these Loewner matrices allows the construction of the multivariate barycentric rational function. The principal result of this work is to show how the null space of the n-dimensional Loewner matrix can be computed using a sequence of 1-dimensional Loewner matrices, leading to a drastic reduction of the computational burden. Equally importantly, this burden is alleviated by avoiding the explicit construction of large-scale n-dimensional Loewner matrices of size $N \times N$. Instead, the proposed methodology achieves decoupling of variables, leading to (i) a complexity reduction from $O(N^3)$ to below $O(N^{1.5})$ when $n > 5$ and (ii) to memory storage bounded by the largest variable dimension rather than their product, thus taming the curse of dimensionality and making the solution scalable to very large data sets. This decoupling of the variables leads to a result similar to the Kolmogorov superposition theorem for rational functions. Thus, making use of barycentric representations, every multivariate rational function can be computed using the composition and superposition of single-variable functions. Finally, we suggest two algorithms (one direct and one iterative) to construct, directly from data, multivariate (or parametric) realizations ensuring (approximate) interpolation. Numerical examples highlight the effectiveness and scalability of the method.

math.NA↗
arXiv2022

Bilinear realization from input-output data with neural networks

We present a method that connects a well-established nonlinear (bilinear) identification method from time-domain data with neural network (NNs) advantages. The main challenge for fitting bilinear systems is the accurate recovery of the corresponding Markov parameters from the input and output measurements. Afterward, a realization algorithm similar to that proposed by Isidori can be employed. The novel step is that NNs are used here as a surrogate data simulator to construct input-output (i/o) data sequences. Then, classical realization theory is used to build a bilinear interpretable model that can further optimize engineering processes via robust simulations and control design.

math.DS↗
arXiv2021

A framework for fitting quadratic-bilinear systems with applications to models of electrical circuits

In this contribution, we propose a data-driven procedure to fit quadratic-bilinear surrogate models from data. Although the dynamics characterizing the original model are strongly nonlinear, we rely on lifting techniques to embed the original model into a quadratic-bilinear format. Here, data represent generalized transfer function values. This method is an extension of methods that do bilinear, or quadratic inference, separately. It is based on first fitting a linear model with the classical Loewner framework, and then on inferring the best supplementing nonlinear operators, in a least-squares sense. The application scope of this method is given by electrical circuits with nonlinear components (such as diodes). We propose various test cases to illustrate the performance of the method.

eess.SY↗
arXiv2021

Data-driven modeling and control of large-scale dynamical systems in the Loewner framework

In this contribution, we discuss the modeling and model reduction framework known as the Loewner framework. This is a data-driven approach, applicable to large-scale systems, which was originally developed for applications to linear time-invariant systems. In recent years, this method has been extended to a number of additional more complex scenarios, including linear parametric or nonlinear dynamical systems. We will provide here an overview of the latter two, together with time-domain extensions. Additionally, the application of the Loewner framework is illustrated by a collection of practical test cases. Firstly, for data-driven complexity reduction of the underlying model, and secondly, for dealing with control applications of complex systems (in particular, with feedback controller design).

eess.SY↗
arXiv2021

Reduced-order modeling of LPV systems in the Loewner framework

We propose a model reduction method for LPV systems. We consider LPV state-space representations with an affine dependence on the scheduling variables. The main idea behind the proposed method is to compute the reduced order model in such a manner that its frequency domain transfer function coincides with that of the original model for some frequencies. The proposed method uses Loewner-like matrices, which can be calculated from the frequency domain representation of the system. The contribution of the paper represents an extension of the well-established Loewner framework to LPV models.

eess.SY↗
arXiv2021

Factorization of the Loewner matrix pencil and its consequences

This paper starts by deriving a factorization of the Loewner matrix pencil that appears in the data-driven modeling approach known as the Loewner framework and explores its consequences. The first is that the associated quadruple constructed from the data yields a model without requiring further processing. The second consequence is related to how sensitive the eigenvalues of the Loewner pencil are to perturbations. Based on an explicit generalized eigenvalue decomposition of this pencil and by making use of perturbation theory of matrix pencils, we explore two types of eigenvalue sensitivities. The first one is defined with respect to unstructured perturbations of the Loewner pencil, while the second one is defined for structured perturbations. We also discuss how the choice of data affects the two sensitivities.

math.NA↗
arXiv2020

Learning reduced-order models of quadratic control systems from input-output data

In this paper, we address an extension of the Loewner framework for learning quadratic control systems from input-output data. The proposed method first constructs a reduced-order linear model from measurements of the classical transfer function. Then, this surrogate model is enhanced by incorporating a term that depends quadratically on the state. More precisely, we employ an iterative procedure based on least squares fitting that takes into account measured or computed data. Here, data represent transfer function values inferred from higher harmonics of the observed output, when the control input is purely oscillatory.

math.OC↗
arXiv2020

On Loewner data-driven control for infinite-dimensional systems

In this paper, we address extensions of the Loewner Data-Driven Control (L-DDC) methodology. First, this approach is extended by incorporating two alternative approximation methods known as Adaptive-Antoulas-Anderson (AAA) and Vector Fitting (VF). These algorithms also include least squares fitting which provides additional flexibility and enables possible adjustments for control tuning. Secondly, the standard model reference data-driven setting is extended to handle noise affecting the data and uncertainty in the closed-loop objective function. These proposed adaptations yield a more robust data-driven control design.

math.OC↗
arXiv2020

Rational approximation of the absolute value function from measurements: a numerical study of recent methods

In this work, we propose an extensive numerical study on approximating the absolute value function. The methods presented in this paper compute approximants in the form of rational functions and have been proposed relatively recently, e.g., the Loewner framework and the AAA algorithm. Moreover, these methods are based on data, i.e., measurements of the original function (hence data-driven nature). We compare numerical results for these two methods with those of a recently-proposed best rational approximation method (the minimax algorithm) and with various classical bounds from the previous century. Finally, we propose an iterative extension of the Loewner framework that can be used to increase the approximation quality of the rational approximant.

math.NA↗
arXiv2020

Model reduction of linear hybrid systems

The paper proposes a model reduction algorithm for linear hybrid systems, i.e., hybrid systems with externally induced discrete events, with linear continuous subsystems, and linear reset maps. The model reduction algorithm is based on balanced truncation. Moreover, the paper also proves an analytical error bound for the difference between the input-output behaviors of the original and the reduced order model. This error bound is formulated in terms of singular values of the Gramians used for model reduction.

math.DS↗
arXiv2019

Data-driven Spatio-temporal Prediction of High-dimensional Geophysical Turbulence using Koopman Operator Approximation

We show the skills of a data-driven low-dimensional linear model in predicting the spatio-temporal evolution of turbulent Rayleigh-Bénard convection. The model is based on dynamic mode decomposition with delay-embedding, which provides a data-driven finite-dimensional approximation to the system's Koopman operator. The model is built using vector-valued observables from direct numerical simulations, and can provide accurate predictions. Similar high prediction skills are found for the Kuramoto-Sivashinsky equation in the strongly-chaotic regimes.

physics.flu-dyn↗
arXiv2017

Approximation of a damped Euler-Bernoulli beam model in the Loewner framework

The Loewner framework for model order reduction is applied to the class of infinite-dimension systems. The transfer function of such systems is irrational (as opposed to linear systems, whose transfer function is rational) and can be expressed as an infinite series of rational functions. The main advantage of the method is the fact that reduced orders models are constructed using only input-output measurements. The procedure can be directly applied to the original transfer function or to the one obtained from the finite element discretization of the PDE. Significantly better results are obtained when using it directly, as it is presented in the experiments section.

math.NA↗
arXiv2017

Data-driven model order reduction of linear switched systems

The Loewner framework for model reduction is extended to the class of linear switched systems. One advantage of this framework is that it introduces a trade-off between accuracy and complexity. Moreover, through this procedure, one can derive state-space models directly from data which is related to the input-output behavior of the original system. Hence, another advantage of the framework is that it does not require the initial system matrices. More exactly, the data used in this framework consists in frequency domain samples of input-output mappings of the original system. The definition of generalized transfer functions for linear switched systems resembles the one for bilinear systems. A key role is played by the coupling matrices, which ensure the transition from one active mode to another.

math.NA↗
arXiv2017

Balanced truncation for linear switched systems

We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians.

math.NA↗
arXiv2012

Interpolatory Weighted-H2 Model Reduction

This paper introduces an interpolation framework for the weighted-H2 model reduction problem. We obtain a new representation of the weighted-H2 norm of SISO systems that provides new interpolatory first order necessary conditions for an optimal reduced-order model. The H2 norm representation also provides an error expression that motivates a new weighted-H2 model reduction algorithm. Several numerical examples illustrate the effectiveness of the proposed approach.

math.NA↗