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A Modified aaa algorithm for Learning Stable Reduced-Order Models from Data

JOURNAL OF SCIENTIFIC COMPUTING 2025年第1期103卷 1-28页

作者： Bradde, Tommaso Grivet-Talocia, Stefano Aumann, Quirin Gosea, Ion Victor Politecn Torino Dept Elect & Telecommun I-10129 Turin Italy Max Planck Inst Dynam Complex Tech Syst Sandtorstr 1 D-39106 Magdeburg Germany

In recent years, the Adaptive Antoulas-Anderson (aaa) algorithm has been applied extensively for generating Reduced Order Models (ROMs) of large scale Linear Time-Invariant systems starting from measurements of their transfer functions. When used for Model Order Reduction (MOR) of an asymptotically stable system, the ROMs generated applying aaa are not guaranteed to preserve this fundamental property, thus rendering them impractical in many relevant applications. To overcome this issue, we propose a novel algebraic characterization for the stability of ROMs represented under the aaa barycentric structure. We then translate such characterization into a set of convex semidefinite constraints that can be embedded into the aaa optimization routine to explicitly maximize the model accuracy while ensuring its stability. Finally, we generalize the resulting modeling framework to allow for efficient stable MOR of Multi-Input-Multi-Output systems. An extensive set of numerical experiments provides practical evidence for the effectiveness of the proposed approach and compares its performance with those of available state-of-the-art methods.

关键词： Reduced-order modeling Data-driven method Rational approximation aaa algorithm Frequency-domain data Stability enforcement Barycentric forms

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A aaa-type algorithm for the microwave duplexer filtering

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NUMERICAL algorithmS 2025年 1-25页

作者： Zhang, Lei-Hong Zhang, Ya-Nan Yang, Linyi Xiao, Ruwu Soochow Univ Sch Math Sci Suzhou 215006 Jiangsu Peoples R China Suzhou CHAHO Commun Technol Co Ltd Suzhou 215006 Jiangsu Peoples R China

In the duplexer filtering of the three-port network microwave application, three coupled rational functions need to be reconstructed by fitting sampled responses measured at a set of frequencies efficiently and effectively. Traditionally, the Cauchy method and its modifications are common approaches that find the underlying rational functions by solving the linearized system. However, with increasing degrees of polynomials, Cauchy methods encounter difficulties when applied to modern duplexer hardware, particularly due to the ill-conditioned Vandermonde matrices involved and a large variation of order of magnitude (dB) in the noise-contaminated and lossy responses. In this paper, by relying on recent achievements from the rational approximation literature using barycentric representations, namely the "adaptive Antoulas-Anderson" (aaa) algorithm and Lawson's iteration for scalar-valued as well as matrix-valued functions, we propose a vector-valued aaa-type approach (v-aaa-Lawson) for addressing the duplexer filtering problem, which is also applicable for the general matrix-valued rational approximations. Numerical experiments demonstrate that the new method is able to improve significantly the performance of the Cauchy methods both in accuracy and robustness.

关键词： Rational approximation Barycentric representation Duplexer aaa algorithm Lawson iteration Singular value decomposition Least squares

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THE aaa algorithm FOR RATIONAL APPROXIMATION

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2018年第3期40卷 A1494-A1522页

作者： Nakatsukasa, Yuji Sete, Olivier Trefethen, Lloyd N. Univ Oxford Math Inst Oxford OX2 6GG England TU Berlin Inst Math Str 17 Juni 136 D-10623 Berlin Germany

We introduce a new algorithm for approximation by rational functions on a real or complex set of points, implementable in 40 lines of MATLAB and requiring no user input parameters. Even on a disk or interval the algorithm may outperform existing methods, and on more complicated domains it is especially competitive. The core ideas are (1) representation of the rational approximant in barycentric form with interpolation at certain support points and (2) greedy selection of the support points to avoid exponential instabilities. The name aaa stands for "adaptive Antoulas-Anderson" in honor of the authors who introduced a scheme based on (1). We present the core algorithm with a MATLAB code and nine applications and describe variants targeted at problems of different kinds. Comparisons are made with vector fitting, RKFIT, and other existing methods for rational approximation.

关键词： rational approximation barycentric formula analytic continuation aaa algorithm Froissart doublet vector fitting RKFIT

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Dosimetric study of the aaa algorithm for the VMAT technique using an anthropomorphic phantom in the pelvic region

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JOURNAL OF RADIOTHERAPY IN PRACTICE 2015年第2期14卷 162-170页

作者： Puchades, Vicente Serna, Alfredo Mata-Colodro, Fernando Ramos-Amores, Davis Casal, Emilio Alcaraz, Miguel Hosp Univ Santa Lucia Dept Med Phys Cartagena Spain CND Valencia Spain Univ Murcia Fac Med Dept Radiol & Med Fis Murcia Spain

Purpose: The objective of this work was to investigate the accuracy of aaa dose calculation algorithm for RapidArc volumetric modulated technique (VMAT) in the presence of anatomical heterogeneities in the pelvic region. Material and methods: An anthropomorphic phantom was used to simulate a prostate case, delineating planning target volumes (PTVs) and organs at risk. VMAT plans were optimised in eclipse (v10 center dot 0) treatment planning system (TPS). The dose distributions were calculated by the aaa dose calculation algorithm. A total of 49 thermoluminiscent dosimeters were inserted into the anthropomorphic phantom and dose measurements were compared with the predicted TPS doses. Results: The average dose variation was -1 center dot 5% for planning target volume corresponding to the prostate and -0 center dot 3% for planning target volume corresponding to the pelvic nodes, -0 center dot 2% for the rectum, +2 center dot 4% for the bladder, -2 center dot 0% for the femoral heads and +1 center dot 0% for the intestinal package. Conclusion: aaa is a reliable dose calculation for the treatment with VMAT in the anatomy of the pelvis.

关键词： aaa algorithm anthropomorphic phantom heterogeneities calculation volumetric arc therapy (VMAT)

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Efficient Rational Approximation of Optical Response Functions with the aaa algorithm

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LASER & PHOTONICS REVIEWS 2024年第11期18卷

作者： Betz, Fridtjof Hammerschmidt, Martin Zschiedrich, Lin Burger, Sven Binkowski, Felix Zuse Inst Berlin Takustr 7 D-14195 Berlin Germany JCMwave GmbH Bolivarallee 22 D-14050 Berlin Germany

A theoretical framework for the rational approximation of optical response functions in resonant photonic systems is introduced. The framework is based on the aaa algorithm and further allows to solve the underlying nonlinear eigenproblems and to efficiently model sensitivities. An adaptive sampling strategy exploits the predominance of resonances in the physical response. A chiral metasurface is investigated and it is shown that the chiroptical response on parameter variations can be accurately modeled in the vicinity of the relevant resonance frequencies.

关键词： aaa algorithm chiral metasurface nanophotonics rational approximation resonances sensitivities

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COMPUTATION OF TWO-DIMENSIONAL STOKES FLOWS VIA LIGHTNING AND aaa RATIONAL APPROXIMATION\ast

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2024年第2期46卷 A1214-A1234页

作者： Xue, Yidan Waters, Sarah l. Trefethen, Lloyd n. Univ Oxford Math Inst Oxford OX2 6GG England

Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen [SIAM J. Sci. Comput., 44 (2022), pp. A1205--A1226] recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polygonal domains. In this paper, we present the LARS algorithm (lightning-aaa rational Stokes) for computing two-dimensional (2D) Stokes flows in domains with smooth boundaries and multiply connected domains using lightning and aaa rational approximation [Y. Nakatsukasa, O. Se`\te, and L. against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy.

关键词： Stokes flow biharmonic equation lightning solver aaa algorithm rational approximation

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aaa RATIONAL APPROXIMATION ON A CONTINUUM

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2024年第2期46卷 A929-A952页

作者： Driscoll, Tobin A. Nakatsukasa, Yuji Trefethen, Lloyd N. Univ Delaware Dept Math Sci Newark DE 19716 USA Univ Oxford Math Inst Oxford OX4 4DY England Harvard Univ Sch Engn & Appl Sci Cambridge MA 02138 USA

aaa rational approximation has normally been carried out on a discrete set, typically hundreds or thousands of points in a real interval or complex domain. Here we introduce a continuum aaa algorithm that discretizes a domain adaptively as it goes. This enables fast computation of high-accuracy rational approximations on domains such as the unit interval, the unit circle, and the imaginary axis, even in some cases where resolution of singularities requires exponentially clustered sample points, support points, and poles. Prototype MATLAB (or Octave) and Julia codes aaax, aaaz, and aaai are provided for these three special domains;the latter two are equivalent by a Mobius transformation. Execution is very fast since the matrices whose SVDs are computed have only three times as many rows as columns. The codes include a aaa-Lawson option for improvement of a aaa approximant to minimax, so long as the accuracy is well above machine precision. The result returned is pole-free in the approximation domain.

关键词： aaa algorithm rational approximation minimax MATLAB Julia

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Solving time-fractional differential equations via rational approximation

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IMA JOURNAL OF NUMERICAL ANALYSIS 2023年第3期43卷 1263-1290页

作者： Khristenko, Ustim Wohlmuth, Barbara Tech Univ Munich Dept Math Boltzmannstr 3 D-85748 Garching Germany

Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Their nonlocal structure requires taking into account the whole evolution history during the time integration, which then possibly causes additional memory use to store the history, growing in time. An alternative to a quadrature for the history integral is to approximate the fractional kernel with a sum of exponentials, which is equivalent to considering the FDE solution as a sum of solutions to a system of ordinary differential equations. One possibility to construct this system is to approximate the Laplace spectrum of the fractional kernel with a rational function. In this paper we use the adaptive Antoulas-Anderson algorithm for the rational approximation of the kernel spectrum, which yields only a small number of real-valued poles. We propose a numerical scheme based on this idea and study its stability and convergence properties. In addition, we apply the algorithm to a time-fractional Cahn-Hilliard problem.

关键词： time-fractional differential equations rational approximation aaa algorithm

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THE aaatrig algorithm FOR RATIONAL APPROXIMATION OF PERIODIC FUNCTIONS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2021年第5期43卷 A3372-A3392页

作者： Baddoo, Peter J. MIT Dept Math 77 Massachusetts Ave Cambridge MA 02139 USA

We present an extension of the aaa (adaptive Antoulas-Anderson) algorithm for periodic functions, called ``aaatrig."" The algorithm uses the key steps of aaa approximation by (i) representing the approximant in (trigonometric) barycentric form and (ii) selecting the support points greedily. Accordingly, aaatrig inherits all the favorable characteristics of aaa and is thus extremely flexible and robust, being able to consider quite general sets of sample points in the complex plane. We consider a range of applications with particular emphasis on solving Laplace's equation in periodic domains and compressing periodic conformal maps. These results reproduce the tapered exponential clustering effect observed in other recent studies. The algorithm is implemented in Chebfun.

关键词： rational functions trigonometric rational functions aaa algorithm harmonic functions conformal mapping

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Modelling permanent magnet excited uniform fields with rational approximations

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COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING 2024年第3期43卷 604-619页

作者： Costa, Stefano Costamagna, Eugenio Di Barba, Paolo IEEE Italy Sect Piacenza Italy Univ Pavia Pavia Italy Univ Pavia Dept Elect Engn Pavia Italy

PurposeA novel method for modelling permanent magnets is investigated based on numerical approximations with rational functions. This study aims to introduce the aaa algorithm and other recently developed, cutting-edge mathematical tools, which provide outstandingly fast and accurate numerical computation of potentials and vector ***/methodology/approachFirst, the aaa algorithm is briefly introduced along with its main variants and other advanced mathematical tools involved in the modelling. Then, the analysis of a circular Halbach array with a one-pole pair is carried out by means of the aaa-least squares method, focusing on vector potential and flux density in the bore and validating results by means of classic finite element software. Finally, the investigation is completed by a finite difference *** methods for field analysis prove to be strikingly fast and accurate. Results are in excellent agreement with those provided by the finite element model, and the very good agreement with those from finite differences suggests future improvements. They are also easy programming;the MATLAB code is less than 200 lines. This indicates they can provide an effective tool for rapid *** limitations/implicationsaaa methods in magnetostatics are novel, but their extension to analogous physical problems seems straightforward. Being a meshless method, it is unlikely that local non-linearities can be considered. An aspect of particular interest, left for future research, is the capability of handling inhomogeneous domains, i.e. solving general interface ***/valueThe authors use cutting-edge mathematical tools for the modelling of complex physical objects in magnetostatics.

关键词： Finite differences Permanent magnets Finite elements Function approximation Uniform fields aaa algorithm

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