Rational Polynomial Approximation Research Articles

Beyond [formula omitted]CDM with [formula omitted]CDM: Criticalities and solutions of Padé Cosmography

Recently, cosmography emerged as a valuable tool to effectively describe the vast amount of astrophysical observations without relying on a specific cosmological model. Its model-independent nature ensures a faithful representation of data, free from theoretical biases. Indeed, the commonly assumed fiducial model, the ΛCDM, shows some shortcomings and tensions between data at late and early times that need to be further investigated. In this paper, we explore an extension of the standard cosmological model by adopting the f(z)CDM approach, where f(z) represents the cosmographic series characterizing the evolution of recent universe driven by dark energy. To construct f(z), we take into account the Padé series, since this rational polynomial approximation offers a better convergence at high redshifts than the standard Taylor series expansion. Several orders of such an approximant have been proposed in previous works, here we want to answer the questions: What is the impact of the cosmographic series choice on the parameter constraints? Which series is the best for the analysis? So, we analyze the most promising ones by identifying which order is preferred in terms of stability and goodness of fit. Theoretical predictions of the f(z)CDM model are obtained by the Boltzmann solver code and the posterior distributions of the cosmological and cosmographic parameters are constrained by a Monte Carlo Markov Chains analysis. We consider a joint data set of cosmic microwave background temperature measurements from the Planck collaboration, type Ia supernovae data from the latest Pantheon+ sample, baryonic acoustic oscillations and cosmic chronometers data. In conclusions, we state which series can be used when only late time data are used, while which orders has to be considered in order to achieve the necessary stability when large redshifts are considered.

Spectral Laplace Transform of Signals on Arbitrary Domains

The Laplace transform is a central mathematical tool for analysing 1D/2D signals and for the solution to PDEs; however, its definition and computation on arbitrary data is still an open research problem. We introduce the Laplace transform on arbitrary domains and focus on the spectral Laplace transform, which is defined by applying a 1D filter to the Laplacian spectrum of the input domain. The spectral Laplace transform satisfies standard properties of the 1D and 2D transforms, such as dilation, translation, scaling, derivation, localisation, and relations with the Fourier transform. The spectral Laplace transform is enough general to be applied to signals defined on different discrete data, such as graphs, 3D surface meshes, and point sets. Working in the spectral domain and applying polynomial and rational polynomial approximations, we achieve a stable computation of the spectral Laplace Transform. As main applications, we discuss the computation of the spectral Laplace Transform of functions defined on arbitrary domains (e.g., 2D and 3D surfaces, graphs), the solution of the heat diffusion equation, and graph signal processing.

Predictors for High Frequency Signals Based on Rational Polynomial Approximation of Periodic Exponentials

The paper presents linear integral predictors for continuous time high-frequency signals with a a finite spectrum gap. The predictors are based on approximation of a complex valued periodic exponential (complex sinusoid) by rational polynomials.

Data-Driven Fuzzy Transform

The Fuzzy transform is applied mainly to 1-D signals and 2-D data organized as a regular grid (e.g., 2-D images), thus, limiting its potential application to arbitrary data in terms of dimensionality and structure. This article defines and analyzes the properties of the data-driven F-transform, with a focus on the construction of the class of data-driven membership functions, which are multiscale, local, linearly independent, intrinsic, and robust to data discretization. Data-driven membership functions are defined by applying a 1-D filter to the Laplace–Beltrami operator, which encodes the geometric and topological properties of the input data. Then, we address the efficient computation of the data-driven F-transform through a polynomial or a rational polynomial approximation of the input filter. In this way, the computation of the data-driven F-transform is independent of the evaluation of the membership functions at any point of the input domain and reduces to the solution of a small set of sparse and symmetric linear systems. Finally, the data-driven F-transform is efficiently evaluated on large and arbitrary data, in terms of dimensionality, structure, and size.

Calculation of effective atomic numbers using a rational polynomial approximation method with a dual-energy X-ray imaging system.

The study aims to develop a rational polynomial approximation method for improving the accuracy of the effective atomic number calculation with a dual-energy X-ray imaging system. This method is based on a multi-materials calibration model with iterative optimization, which can improve the calculation accuracy of the effective atomic number by adding a rational term without increasing the computation time. The performance of the proposed rational polynomial approximation method is demonstrated and validated by both simulated and experimental studies. The twelve reference materials are used to establish the effective atomic number calibration model, and the value of the effective atomic numbers are between 5.444 and 22. For the accuracy of the effective atomic number calculation, the relative differences between calculated and experimental values are less than 8.5%for all sample cases in this study. The average calculation accuracy of the method proposed in this study can be improved by about 40%compared with the conventional polynomial approximation method. Additionally, experimental quality assurance phantom imaging result indicates that the proposed method is compliant with the international baggage inspection standards for detecting the explosives. Moreover, the experimental imaging results reveal that the difference of color between explosives and the surrounding materials is in significant contrast for the dual-energy image with the proposed method.

A Resolver-to-Digital Conversion Method Based on Third-Order Rational Fraction Polynomial Approximation for PMSM Control

In this paper, a cost-effective and highly accurate resolver-to-digital conversion method is presented. The core of the idea is to apply a third-order rational fraction polynomial approximation for the conversion of sinusoidal signals into the pseudo linear signals, which are extended to the range 0°–360° in four quadrants. Then, the polynomial least squares method is used to achieve compensation to acquire the final angles. The presented method shows better performance in terms of accuracy and rapidity compared with the commercial available techniques in simulation results. This paper describes the implementation details of the proposed method and the way to incorporate it in digital signal processor-based permanent magnet synchronous motor drive system. Experimental tests under different conditions are carried out to verify the effectiveness for the proposed method. The obtained maximum error is about 0.0014° over 0°–360°, which can usually be ignored in most industrial applications.

Analytic Continuation with Padé Decomposition**Supported by the National Natural Science Foundation of China under Grant Nos 11474331 and 11190024.

The ill-posed analytic continuation problem for Green’s functions or self-energies can be carried out using the Padé rational polynomial approximation. However, to extract accurate results from this approximation, high precision input data of the Matsubara Green function are needed. The calculation of the Matsubara Green function generally involves a Matsubara frequency summation, which cannot be evaluated analytically. Numerical summation is requisite but it converges slowly with the increase of the Matsubara frequency. Here we show that this slow convergence problem can be significantly improved by utilizing the Padé decomposition approach to replace the Matsubara frequency summation by a Padé frequency summation, and high precision input data can be obtained to successfully perform the Padé analytic continuation.

Generalized Patched Potential Density and Thermodynamic Neutral Density: Two New Physically Based Quasi-Neutral Density Variables for Ocean Water Masses Analyses and Circulation Studies

AbstractIn this paper, two new quasi-neutral density variables—generalized patched potential density (GPPD) and thermodynamic neutral density γT—are introduced, which are showed to approximate Jackett and McDougall empirical neutral density γn significantly better than the quasi-material rational polynomial approximation γa previously introduced by McDougall and Jackett. In contrast to γn, γT is easily and efficiently computed for arbitrary climatologies of temperature and salinity (both realistic and idealized), has a clear physical basis rooted in the theory of available potential energy, and does not suffer from nonmaterial effects that make γn so difficult to use in water masses analysis. In addition, γT is also significantly more neutral than all known quasi-material density variables, such as σ2, while remaining less neutral than γn. Because unlike γn, γT is mathematically explicit, it can be used for theoretical as well as observational studies, as well as a generalized vertical coordinate in isopycnal models of the ocean circulation. On the downside, γT exhibits inversions and degraded neutrality in the polar regions, where the Lorenz reference state is the furthest away from the actual state. Therefore, while γT represents progress over previous approaches, further work is still needed to determine whether its polar deficiencies can be corrected, an essential requirement for γT to be useful in Southern Ocean studies, for instance.

Conjugate functions on the closed interval and their relationship with uniform rational and piecewise polynomial approximations

Earlier the second author showed that, in the periodic case, the rate of best uniform rational approximations of a function is well described in terms of the rates of best uniform piecewise polynomial approximations of the function itself and its conjugate. In the present paper, a similar result is obtained for a closed interval.

An Analytic Expression for the Binormal Partial Area under the ROC Curve

The partial area under the receiver operating characteristic (ROC) curve (pAUC) is a useful summary measure for diagnostic studies. Unlike most summary measures that are functions of the ROC curve, researchers have not been aware of an analytic expression available for computing the pAUC for an ROC curve based on a latent binormal model. Instead, the pAUC has been computed using numerical integration or a rational polynomial approximation. Our purpose is to provide analytic expressions for the two forms of pAUC. We discuss the two fundamentally different types of pAUC. We present analytic expressions for both types, provide corresponding proofs, and illustrate their application with an example comparing the performances of spin echo and cine magnetic resonance imaging for detecting thoracic aortic dissection. We provide an example of using the pAUC as the outcome in a multireader multicase analysis. We find that using the pAUC results in a more significant finding. We have provided analytic expressions for both types of pAUC, making it easier to compute the pAUCs corresponding to binormal ROC curves.

Development of burnup methods and capabilities in Monte Carlo code RMC

The Monte Carlo burnup calculation has always been a challenging problem because of its large time consumption when applied to full-scale assembly or core calculations, and thus its application in routine analysis is limited. Most existing MC burnup codes are usually external wrappers between a MC code, e.g. MCNP, and a depletion code, e.g. ORIGEN. The code RMC is a newly developed MC code with an embedded depletion module aimed at performing burnup calculations of large-scale problems with high efficiency. Several measures have been taken to strengthen the burnup capabilities of RMC. Firstly, an accurate and efficient depletion module called DEPTH has been developed and built in, which employs the rational approximation and polynomial approximation methods. Secondly, the Energy-Bin method and the Cell-Mapping method are implemented to speed up the transport calculations with large numbers of nuclides and tally cells. Thirdly, the batch tally method and the parallelized depletion module have been utilized to better handle cases with massive amounts of burnup regions in parallel calculations. Burnup cases including a PWR pin and a 5×5 assembly group are calculated, thereby demonstrating the burnup capabilities of the RMC code. In addition, the computational time and memory requirements of RMC are compared with other MC burnup codes.

Modelling room transfer functions using the decimated Padé approximant

The numerical issues involved in modelling measured room transfer functions (RTFs) are examined. This is explored using a nonlinear parametric estimation technique known as the decimated Padé approximant (DPA). The DPA combines the well-known methodology of Padé rational polynomial approximation with beamspace windowing. This combination helps to overcome the severe numerical instabilities encountered in calculations with large data records. The aim is to accurately extract the parameters that reliably quantify the analytic structure of signals composed of decaying sinusoidal oscillations. DPA parameter estimation provides the ability to construct a high-resolution spectral estimate of such signals for either specific spectral regions or the entire Nyquist interval. As demonstrated in the authors' previous work (O'Sullivan and Cowan, 2006), this technique, developed in quantum chemistry, readily cross-fertilises to the field of acoustics, where it can fully reconstruct the complicated spectra of experimental RTFs. A noise-filtering technique using the removal of Froissart doublets to obtain an irreducible rational model is investigated. This noise filtering can be used to find an order of the parametric model that is inherent in the data. Additionally, an example is shown suggesting that such a process may be useful in room spatialisation problems.

Mathematical Optimization of in vivo NMR Chemistry Through the Fast Padé Transform: Potential Relevance for Early Breast Cancer Detection by Magnetic Resonance Spectroscopy

Mathematical advances in signal processing through the fast Pade transform (FPT) can greatly improve the information extracted via in vivo nuclear magnetic resonance (NMR) chemistry. The FPT is a frequency-dependent, non-linear rational polynomial approximation of the exact Maclaurin series, which dramatically improves resolution and signal-to-noise ratio in a stable manner with robust error analysis and provides precise numerical data for all the peak parameters (position, height, width and phase) for every true resonance including those that are weak and/or overlapping. The concentrations of many of the chemical constituents of tissues can thereby be accurately determined. These advantages of the FPT are particularly germane for in vivo NMR detection and quantification of a number of molecular markers of breast cancer, such as phosphocholine, as well as lactate, which cannot be assessed using standard Fourier data analytical techniques applied to in vivo NMR in the clinical setting.

RATIONAL POLYNOMIAL APPROXIMATION MODELLING FOR ANALYSIS OF STRUCTURES WITH VE DAMPERS

The objective of this paper is to introduce a Rational Polynomial Approximation (RPA) method for modelling the response of structures that contain discrete elements with linear frequency-dependent stiffness and damping characteristics. The RPA method consists of two steps: First, system identification is performed to obtain a rational polynomial approximation for the system's transfer functions. Then, a time-domain model for the system is realised. The main advantage of the RPA method is that the resulting model is a system of ordinary differential equations, facilitating time history analysis of both linear and nonlinear structures using standard time-step integration algorithms and procedures. Viscoelastic (VE) dampers comprise one of the primary classes of frequency-dependent dampers with both frequency-dependent stiffness and damping. VE dampers are used for mitigation of seismic- and wind-induced structural vibration. When using VE dampers in analysis, effective modelling of the frequency-dependent characteristics of the VE damper plays a key role in accurate simulation of structural responses. Following a description of the theory behind the RPA method, the efficacy of the method is verified through several numerical examples employing VE damped structures. The results are compared with Kasai's fractional derivative model for VE dampers. Application of the RPA method to nonlinear structures is also given. The RPA method is shewn to be effective and efficient for modelling VE damped structure.

Multivariate n-term rational and piecewise polynomial approximation

We study nonlinear approximation in L p( R d) (0<p<∞, d>1) from (a) n-term rational functions, and (b) piecewise polynomials generated by different anisotropic dyadic partitions of R d . To characterize the rates of each such piecewise polynomial approximation we introduce a family of smoothness spaces (B-spaces) which can be viewed as an anisotropic variation of Besov spaces. We use the B-spaces to prove Jackson and Bernstein estimates and then characterize the piecewise polynomial approximation by interpolation. Our main estimate relates n-term rational approximation with piecewise polynomial approximation in L p( R d) . This result enables us to obtain a direct estimate for n-term rational approximation in terms of a minimal B-norm (over all dyadic partitions). We also show that the Haar bases associated with anisotropic dyadic partitions of R d can be successfully utilized for nonlinear approximation. We give an effective algorithm for best Haar basis or best B-space selection.

Automated Discovery of Numerical Approximation Formulae via Genetic Programming

This paper describes the use of genetic programming to perform automated discovery of numerical approximation formulae. We present results involving rediscovery of known approximations for Harmonic numbers, discovery of rational polynomial approximations for functions of one or more variables, and refinement of existing approximations through both approximation of their error function and incorporation of the approximation as a program tree in the initial GP population. Evolved rational polynomial approximations are compared to Pade approximations obtained through the Maple symbolic mathematics package. We find that approximations evolved by GP can be superior to Pade approximations given certain tradeoffs between approximation cost and accuracy, and that GP is able to evolve approximations in circumstances where the Pade approximation technique cannot be applied. We conclude that genetic programming is a powerful and effective approach that complements but does not replace existing techniques from numerical analysis.

Chebyshev rational matrix approximation with a priori error bounds for linear and Riccati matrix equations

This paper deals with initial value problems for Lipschitz continuous coefficient matrix Riccati equations. Using Chebyshev polynomial matrix approximations the coefficients of the Riccati equation are approximated by matrix polynomials in a constructive way. Then using the Fröbenius method developed in [1], given an admissible error ϵ > 0 and the previously guaranteed existence domain, a rational matrix polynomial approximation is constructed so that the error is less than ϵ in all the existence domain. The approach is also considered for the construction of matrix polynomial approximations of nonhomogeneous linear differential systems avoiding the integration of the transition matrix of the associated homogeneous problem.

The effect of large structural perturbations on the response of structural-acoustic systems

Due to high sensitivity, first-order perturbation analysis of structural-acoustic systems can be inaccurate for large perturbations. Using high-order derivatives to create a Taylor series approximation for the perturbed solution can result in slow convergence or even divergence. A rational polynomial or Pade approximation may overcome the poor convergence of the Taylor series by canceling out the pole causing the poor convergence. In this study, a finite element framework is used to describe the structural-acoustic system. External radiation and scattering problems are accommodated by truncating the infinite fluid region using exponential decay infinite elements. Changes to a nominal model are introduced through a structural perturbation that modifies the nominal structural stiffness matrix. An efficient method for calculating the solution derivatives with respect to the structural perturbation is presented. A Taylor series expansion is constructed using the derivative information and the convergence criteria for the series is examined. The local solution and derivatives are then used to construct a Pade approximation. The method is illustrated using several numerical examples. The approximation is shown to be quite accurate for large perturbations even when there are one or more nearby poles and the Taylor series fails to converge.

Rational Approximation with Locally Geometric Rates

We investigate the rate of pointwise rational approximation of functions from two classes. The distinguishing feature of these classes is the essentially faster convergence of the best uniform rational approximants versus best uniform polynomial approximants. It is known that for piecewise analytic functions “near best” polynomials converging geometrically fast at every point of analyticity of the function exist. Here we construct rational approximants enjoying similar properties. We also show that our construction yields rates of convergence that are, in a certain sense, best possible.

Characterization of embedded passives using macromodels in LTCC technology

This paper discusses the frequency and time domain response of embedded passive components in a multilayered structure fabricated using low temperature co-fired ceramic (LTCC) technology. A rational polynomial approximation that combines the accuracy of EM solvers with interpolation methods has been used to capture the frequency dependent losses and parasitics of embedded passives in a macro-model. This method allows for a significant speed-up in computation time while using commercial EM solvers. The macromodel with suitable modification has been used to compute the time domain response in SPICE for typical embedded passive structures. Simulation results show good correlation with time domain reflectometry/time-domain-transmission (TDR/TDT) measurements. The behavior of embedded passives in the high frequency operation of transmission lines and voltage divider networks has also been discussed.