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From ESPRIT to ESPIRA: estimation of signal parameters by iterative rational approximation

IMA JOURNAL OF NUMERICAL ANALYSIS 2023年第2期43卷 789-827页

作者： Derevianko, Nadiia Plonka, Gerlind Petz, Markus Gottingen Univ Inst Numer & Appl Math Lotzestr 16-18 D-37083 Gottingen Germany

We introduce a new method for Estimation of Signal Parameters based on Iterative Rational Approximation (ESPIRA) for sparse exponential sums. Our algorithm uses the aaa algorithm for rational approximation of the discrete Fourier transform of the given equidistant signal values. We show that ESPIRA can be interpreted as a matrix pencil method (MPM) applied to Loewner matrices. These Loewner matrices are closely connected with the Hankel matrices that are usually employed for signal recovery. Due to the construction of the Loewner matrices via an adaptive selection of index sets, the MPM is stabilized. ESPIRA achieves similar recovery results for exact data as ESPRIT and the MPM, but with less computational effort. Moreover, ESPIRA strongly outperforms ESPRIT and the MPM for noisy data and for signal approximation by short exponential sums.

关键词： sparse exponential sums matrix pencil method ESPRIT rational interpolation aaa algorithm Loewner matrices Hankel matrices

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Unitarity of some barycentric rational approximants

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IMA JOURNAL OF NUMERICAL ANALYSIS 2023年第4期44卷 2070-2089页

作者： Jawecki, Tobias Singh, Pranav TU Wien Inst Theoret Phys Wiedner Hauptstr 8-10 A-1040 Vienna Austria Univ Bath Dept Math Sci Bath BA2 7AY England

The exponential function maps the imaginary axis to the unit circle and for many applications this unitarity property is also desirable from its approximations. We show that this property is conserved not only by the $(k,k)$-rational interpolant of the exponential on the imaginary axis, but also by $(k,k)$-rational barycentric approximants that minimize a linearized approximation error. These results are a consequence of certain properties of singular vectors of Loewner-type matrices associated to linearized approximation errors. Prominent representatives of this class are rational approximants computed by the adaptive Antoulas-Anderson (aaa) method and the aaa-Lawson method. Our results also lead to a modified procedure with improved numerical stability of the unitarity property and reduced computational cost.

关键词： exponential unitary rational approximation barycentric formula Loewner matrix aaa algorithm aaa-Lawson algorithm

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Treating the Gibbs phenomenon in barycentric rational interpolation and approximation via the S-Gibbs algorithm

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APPLIED MATHEMATICS LETTERS 2020年第0期103卷 106196-000页

作者： Berrut, J-P De Marchi, S. Elefante, G. Marchetti, F. Univ Fribourg Dept Math Fribourg Switzerland Univ Padua Dipartimento Matemat Tullio Levi Civita Padua Italy Univ Padua Dipartimento Salute Donna & Bambino Padua Italy

In this work, we extend the so-called mapped bases or fake nodes approach to the barycentric rational interpolation of Floater-Hormann and to aaa approximants. More precisely, we focus on the reconstruction of discontinuous functions by the S-Gibbs algorithm introduced in De Marchi et al. (2020). Numerical tests show that it yields an accurate approximation of discontinuous functions. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Barycentric rational interpolation Gibbs phenomenon Floater-Hormann interpolant aaa algorithm Fake nodes

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AN algorithm FOR REAL AND COMPLEX RATIONAL MINIMAX APPROXIMATION

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2020年第5期42卷 A3157-A3179页

作者： Nakatsukasa, Yuji Trefethen, Lloyd N. Univ Oxford Math Inst Oxford OX2 6GG England

Rational minimax approximation of real functions on real intervals is an established topic, but when it comes to complex functions or domains, there appear to be no algorithms currently in use. Such a method is introduced here, the aaa-Lawson algorithm, available in Chebfun. The new algorithm solves a wide range of problems on arbitrary domains by a procedure consisting of two steps. First, the standard aaa algorithm is run to obtain a near-best approximation and a set of support points for a barycentric representation of the rational approximant. Then a "Lawson phase" of iteratively reweighted least-squares adjustment of the barycentric coefficients is carried out to improve the approximation to minimax.

关键词： rational approximation barycentric formula aaa algorithm aaa-Lawson algorithm iteratively reweighted least-squares

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Numerical analytic continuation

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JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 2023年第3期40卷 1587-1636页

作者： Trefethen, Lloyd N. N. Univ Oxford Math Inst Woodstock Rd Oxford OX2 6GG England

Let f be an analytic function on a simply-connected compact continuum E of the complex z-plane. This might be an interval of the real line, where f might be real analytic. How can we calculate good estimates of the analytic continuation of f to other points z ? C? How can we estimate the locations of real or complex singularities of f ? We review both the theory and the practice of some existing methods for these problems and propose that excellent results can be obtained from the computation of rational approximations off by the aaa algorithm. In the case of analytic functions of two or more variables, the rational approximations are applied along line segments or other analytic arcs.

关键词： Analytic continuation Real analytic function Singularities Rational approximation aaa algorithm Fat branch cuts

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The effect of bladder contrast on dose calculation in volumetric modulated arc therapy planning in patients treated for postoperative prostate cancer

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JAPANESE JOURNAL OF RADIOLOGY 2016年第5期34卷 376-382页

作者： Ercan, Tulay Igdem, Sefik Alco, Gul Florence Nightingale Gayrettepe Hosp Dept Radiat Oncol Cemil Aslan Guder Sok 8Floor 1 TR-34349 Istanbul Turkey

The aim of this study was to evaluate the effect of contrast agent on dose calculation in volumetric modulated arc therapy (VMAT) in the post-prostatectomy setting. Ten patients were studied. Each patient received planning computed tomography (CT) images with contrast agent. All of the plans were done on virtually simulated contrast-free CT scans. The plan approved by the radiation oncologist was replicated to the contrast CT series. In both of the plans the same monitor unit was used. The doses calculated from the two plans were compared in regard to target volumes and organs at risk. A paired sample t-test was used to evaluate the differences in cumulative dose volume histogram between the two plans. We showed that the use of contrast agent may cause significant differences in dose distribution. There was a significant decrease in doses received by planning target volume (PTV70), rectum V65 Gy, rectum V40 Gy, bladder V65 Gy, penile bulb V40 Gy in plans with contrast-enhanced CT sets. The decrease in mean, maximum and minimum doses received by PTV70 also contributed to the significant decrease in conformity index. Using a contrast agent at the time of CT simulation may cause significant differences in dose distribution. For this reason, the plan should always be carried out on non-contrast CT data sets to avoid additional errors in the treatment planning process.

关键词： VMAT Contrast agent aaa algorithm Dose calculation Treatment planning

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Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients

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ANALYSIS AND APPLICATIONS 2022年第3期20卷 543-577页

作者： Derevianko, Nadiia Plonka, Gerlind Gottingen Univ Inst Numer & Appl Math Lotzestr 16-18 D-37083 Gottingen Germany

In this paper, we derive a new recovery procedure for the reconstruction of extended exponential sums of the form y(t) = Sigma(M)(j=1) (Sigma(nj)(m=0) gamma(j,m) t(m))e(2 pi lambda jt), where the frequency parameters lambda(j) is an element of C are pairwise distinct. In order to reconstruct y(t) we employ a finite set of classical Fourier coefficients of y with regard to a finite interval (0, P) subset of R with P > 0. For our method, 2N + 2 Fourier coefficients c(k)(y) are sufficient to recover all parameters of y, where N := Sigma(M)(j=1) (1 + n(j)) denotes the order of y(t). The recovery is based on the observation that for lambda(j )is not an element of i/PZ the terms of y(t) possess Fourier coefficients with rational structure. We employ a recently proposed stable iterative rational approximation algorithm in [Y. Nakatsukasa, O. Sete and L. N. Trefethen, The aaa algorithm for rational approximation, SIAM J. Sci. Comput. 40(3) (2018) A1494A1522]. If a sufficiently large set of L Fourier coefficients of y is available (i.e. L >= 2N + 2), then our recovery method automatically detects the number M of terms of y, the multiplicities n(j) for j = 1, ..., M, as well as all parameters lambda(j), j = 1, ..., M, and gamma(j,m), j = 1, ..., M, m = 0, ..., n(j), determining y(t). Therefore, our method provides a new stable alternative to the known numerical approaches for the recovery of exponential sums that are based on Prony's method.

关键词： Sparse exponential sums extended exponential sums rational approximation aaa algorithm barycentric representation Fourier coefficients

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EXPONENTIAL ASYMPTOTICS USING NUMERICAL RATIONAL APPROXIMATION IN LINEAR DIFFERENTIAL EQUATIONS

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ANZIAM JOURNAL 2024年第4期65卷 285-307页

作者： Lustri, Christopher J. Crew, Samuel Chapman, S. Jonathan Univ Sydney Sch Math & Stat Camperdown NSW 2006 Australia Imperial Coll London Dept Math South Kensington Campus London SW7 2AZ England Ruhr Univ Bochum Fac Comp Sci Univ Str D-44799 Bochum Germany Grad Univ Okinawa Inst Sci & Technol Theoret Sci Visiting Program Onna 9040495 Japan Univ Oxford Math Inst Oxford OX2 6GG England

Singularly perturbed ordinary differential equations often exhibit Stokes' phenomenon, which describes the appearance and disappearance of oscillating exponentially small terms across curves in the complex plane known as Stokes lines. These curves originate at singular points in the leading-order solution to the differential equation. In many important problems, it is impossible to obtain a closed-form expression for these leading-order solutions, and it is therefore challenging to locate these singular points. We present evidence that the analytic leading-order solution of a linear differential equation can be replaced with a numerical rational approximation using the adaptive Antoulas-Anderson (aaa) method. Despite such an approximation having completely different singularity types and locations, we show that the subsequent exponential asymptotic analysis accurately predicts the exponentially small behaviour present in the solution. For sufficiently small values of the asymptotic parameter, this approach breaks down;however, the range of validity may be extended by increasing the number of poles in the rational approximation. We present a related nonlinear problem and discuss the challenges that arise due to nonlinear effects. Overall, our approach allows for the study of exponentially small asymptotic effects without requiring an exact analytic form for the leading-order solution;this permits exponential asymptotic methods to be used in a much wider range of applications.

关键词： exponential asymptotics Stokes' phenomenon numerical rational approximation aaa algorithm

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ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

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NUMERICAL algorithmS 2023年第1期92卷 437-470页

作者： Derevianko, Nadiia Plonka, Gerlind Razavi, Raha Gottingen Univ Inst Numer & Appl Math Lotzestr 16-18 D-37083 Gottingen Germany

In this paper we introduce two algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz + Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case.

关键词： Sparse cosine sums Prony method Toeplitz plus Hankel matrices Rational interpolation aaa algorithm Loewner matrices

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Advanced kernel methods vs. Monte Carlo-based dose calculation for high energy photon beams

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RADIOTHERAPY AND ONCOLOGY 2009年第3期93卷 645-653页

作者： Fotina, Irina Winkler, Peter Kuenzler, Thomas Reiterer, Jochen Simmat, Isabell Georg, Dietmar Med Univ Vienna AKH Wien Dept Radiotherapy A-1090 Vienna Austria Kaiser Franz Josef Spital Dept Radiotherapy Vienna Austria

Purpose. The aim of this study was to compare the dose calculation accuracy of advanced kernel-based methods and Monte Carlo algorithms in commercially available treatment planning systems. Materials and methods: Following dose calculation algorithms and treatment planning (TPS) systems were compared: the collapsed cone (CC) convolution algorithm available in Oncentia Masterplan, the XVMC Monte Carlo algorithm implemented in;Plan and Monaco. and the analytical anisotropic algorithm (aaa) implemented in Eclipse. Measurements were performed with a calibrated ionization chamber and radiochromic EBT type films in a homogenous polystyrene phantom and in heterogeneous lung phantoms Single beam tests, conformal treatment plans and IMRT plans were validated Dosimetric evaluations included absolute close measurements. 1D gamma-evaluation of depth-close curves and profiles using 2 mm and 2% dose difference criteria for single beam tests, and gamma-evaluation of axial planes for composite treatment plans applying 3 mm and 3% dose difference criteria Results' Absolute dosimetry revealed no large differences between MC and advanced kernel dose calculations 1D-evaluation showed significant discrepancies between depth-dose curves in different phantom geometries For the CC algorithm gamma(mean) values were 0 90 +/- 0 74 vs 0 43 +/- 0 41 in heterogeneous vs. homogeneous conditions and for the aaa gamma(mean) values were 1.13 +/- 0.91 vs 041 +/- 0.28, respectively. In general, 1D results obtained with both MC TPS were similar in both phantoms and on average equal to 0.5 both for profiles and depth-dose curves The results obtained with the CC algorithm in Heterogeneous phantoms were slightly better in comparison to the aaa algorithm. The 2D gamma-evaluation results of IMRT plans and four-field plans showed smaller mean gamma-values for MC dose calculations compared to the advanced kernel algorithms (gamma(mean) for four-field plan and IMRT obtained with Monaco MC were 0 28 and 0.5. re

关键词： Monte Carlo Collapsed cone algorithm aaa algorithm EBT film dosimetry

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