The AAA Algorithm for Rational Approximation

Nakatsukasa, Yuji; Sète, Olivier; Trefethen, Lloyd N.

doi:10.1137/16m1106122

Cited by 349 publications

(435 citation statements)

References 68 publications

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“…The primary method of this study is the Loewner framework [2] which addresses this problem in a natural and direct way. The other methods that were studied, vector fitting (VF) [5], and adaptive Antoulas-Anderson (AAA) [4], are instead based on an iterative and adaptive selection procedure. In order to present the adaptive selection as a feature in the Loewner framework and to reduce the time complexity at the same time, we introduced a Loewner CUR method based on the cross approximation algorithm [6,7].…”

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confidence: 99%

Data‐driven approximation methods applied to non‐rational functions

Karachalios

Gosea

Antoulas

2018

Proc Appl Math and Mech

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In this study, four data-driven interpolation-based methods are compared. The aim is to construct reduced-order models for which the corresponding rational transfer function matches the original non-rational one at selected interpolation points. The primary method of this study is the Loewner framework [2] which addresses this problem in a natural and direct way. The other methods that were studied, vector fitting (VF) [5], and adaptive Antoulas-Anderson (AAA) [4], are instead based on an iterative and adaptive selection procedure. In order to present the adaptive selection as a feature in the Loewner framework and to reduce the time complexity at the same time, we introduced a Loewner CUR method based on the cross approximation algorithm [6,7]. The performance of the above methods is tested by a classical example in approximation theory: the inverse of the Bessel function of the first kind. 1 SISO: Single Input Single Output. Where E, A ∈ C n×n and B, C T ∈ C n×1 , and a scalar D ∈ C. 2 real symmetryH(s) = H(s): we can obtain a real model [2]. Where(·) denotes the complex conjugate and (·) * the complex conjugate transpose.

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confidence: 99%

Data‐driven approximation methods applied to non‐rational functions

Karachalios

Gosea

Antoulas

2018

Proc Appl Math and Mech

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“…In this case, the approximant is sought to be an (approximant) interpolant as opposed to a LS fit. The recently developed Adaptive Anderson-Antoulas algorithm [29] is a hybrid approach where a rational interpolant is constructed to interpolate a subset of the data and to minimize LS error in the rest. As pointed out earlier, in this work, the focus is on the LS framework where the regular VF framework is used to solve the data-driven modeling problem.…”

Section: Data-driven Rational Approximationmentioning

confidence: 99%

Estimating dispersion curves from Frequency Response Functions via Vector-Fitting

Albakri

Malladi

Gugercin

et al. 2020

Mechanical Systems and Signal Processing

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Driven by the need for describing and understanding wave propagation in structural materials and components, several analytical, numerical, and experimental techniques have been developed to obtain dispersion curves. Accurate characterization of the structure (waveguide) under test is needed for analytical and numerical approaches. Experimental approaches, on the other hand, rely on analyzing waveforms as they propagate along the structure. Material inhomogeneity, reflections from boundaries, and the physical dimensions of the structure under test limit the frequency range over which dispersion curves can be measured.In this work, a new data-driven modeling approach for estimating dispersion curves is developed. This approach utilizes the relatively easy-to-measure, steady-state Frequency Response Functions (FRFs) to develop a state-space dynamical model of the structure under test. The developed model is then used to study the transient response of the structure and estimate its dispersion curves. This paper lays down the foundation of this approach and demonstrates its capabilities on a one-dimensional homogeneous beam using numerically calculated FRFs. Both in-plane and out-of-plane FRFs corresponding, respectively, to longitudinal (the first symmetric) and flexural (the first anti-symmetric) wave modes are analyzed. The effects of boundary conditions on the performance of this approach are also addressed.

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“…There are various ways to fit this frequency domain data. One can enforce H(s) to interpolate the data at every sampling point using the Loewner framework [5,46], or construct H(s) to fit the data in a least-squares (LS) sense [20,25,39], or force H(s) to interpolate some of the data and while minimizing the LS fit in the rest [50]. In this paper, we will fit data solely in a LS sense.…”

Section: Data-driven Modeling From Transfer Function Samplesmentioning

confidence: 99%

“…that due to the nonlinear dependence on the poles of H(s), this is a nonlinear LS problem. There are various approaches for solving this problem, see, e.g., [20,25,32,39,41,50,60]. Our approach employs the Vector Fitting (VF) framework of [39] even though one can easily adapt any of the other LS methods.…”

Section: Data-driven Modeling From Transfer Function Samplesmentioning

confidence: 99%

Sampling-free parametric model reduction of systems with structured parameter variation

Beattie

Gugercin

Tomljanović

2018

Preprint

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We consider a parametric linear time invariant dynamical systems represented in state-space form as $$E \dot x(t) = A(p) x(t) + Bu(t), \\ y(t) = Cx(t),$$ where $E, A(p) \in \mathbb{R}^{n\times n}$, $B\in \mathbb{R}^{n\times m} $ and $C\in \mathbb{R}^{l\times n}$. Here $x(t)\in \mathbb{R}^{n} $ denotes the state variable, while $u(t)\in \mathbb{R}^{m}$ and $y(t)\in \mathbb{R}^{l}$ represent, respectively, the inputs and outputs of the system. We assume that $A(p)$ depends on $k\ll n$ parameters $p=(p_1, p_2, \ldots, p_k)$ such that we may write $$A(p)=A_0+U\,\diag (p_1, p_2, \ldots, p_k)V^T,$$ where $U, V \in \mathbb{R}^{n\times k}$ are given fixed matrices.We propose an approach for approximating the full-order transfer function $H(s;p)=C(s E -A(p))^{-1}B$ with a reduced-order model that retains the structure of parametric dependence and (typically) offers uniformly high fidelity across the full parameter range. Remarkably, the proposed reduction process removes the need for parameter sampling and thus does not depend on identifying particular parameter values of interest. Our approach is based on the classic Sherman-Morrison-Woodbury formula and allows us to construct a parameterized reduced order model from transfer functions of four subsystems that do not depend on parameters, allowing one to apply well-established model reduction techniques for non-parametric systems. The overall process is well suited for computationally efficient parameter optimization and the study of important system properties. One of the main applications of our approach is for damping optimization: we consider a vibrational system described by $$ \begin{equation}\label{MDK} \begin{array}{rl} M\ddot q(t)+(C_{int} + C_{ext})\dot q(t)+Kq(t)&=E w(t),\\ z(t)&=Hq(t), \end{array} \end{equation} $$ where the mass matrix, $M$, and stiffness matrix, $K$, are real, symmetric positive-definite matrices of order $n$. Here, $q(t)$ is a vector of displacements and rotations, while $ w(t) $ and $z(t) $ represent, respectively, the inputs (typically viewed as potentially disruptive) and outputs of the system. Damping in the structure is modeled as viscous damping determined by $C_{int} + C_{ext}$ where $C_{int}$ and $C_{ext}$ represent contributions from internal and external damping, respectively. Information regarding damper geometry and positioning as well as the corresponding damping viscosities are encoded in $C_{ext}= U\diag{(p_1, p_2, \ldots, p_k)} U^T$ where $U \in \mathbb{R}^{n\times k}$ determines the placement and geometry of the external dampers. The main problem is to determine the best damping matrix that is able to minimize influence of the disturbances, $w$, on the output of the system $z$. We use a minimization criteria based on the $\mathcal{H}_2$ system norm. In realistic settings, damping optimization is a very demanding problem. We find that the parametric model reduction approach described here offers a new tool with significant advantages for the efficient optimization of damping in such problems.

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The AAA Algorithm for Rational Approximation

Cited by 349 publications

References 68 publications

Data‐driven approximation methods applied to non‐rational functions

Data‐driven approximation methods applied to non‐rational functions

Estimating dispersion curves from Frequency Response Functions via Vector-Fitting

Sampling-free parametric model reduction of systems with structured parameter variation

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