Chebyshev Approximation Research Articles

Effectiveness of the Chebyshev Approximation in Magnetic Field Line Tracking

The tracking of magnetic field lines can be very expensive, in terms of computational burden, when the field sources are numerous and have complex geometries, especially when accuracy is a priority, because an evaluation of the field is required in many situations. In some important applications, the computational cost can be significantly reduced by using a suitable approximation of the field in the integrated regions. This paper shows how Chebyshev polynomials are well-suited for field interpolation in magnetic field-line tracking, then discusses the conditions in which they are most appropriate, and quantifies the effectiveness of parallel computing in the approximation procedures.

An Extremal Problem Arising in the Dynamics of Two‐Phase Materials That Directly Reveals Information about the Internal Geometry

AbstractIn two‐phase materials, each phase having a non‐local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [‐1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [‐1,1] and, to simplify the analysis, Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [‐1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov‐type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time‐dependent boundary potentials are suitably tailored. © 2022 Wiley Periodicals, Inc.

A Hybrid Chebyshev Approximation Technique and Subdomain AIM Method for Wideband RCS Computation

An efficient hybrid method based on subdomain adaptive integral method (SAIM) and Chebyshev approximation technology (CAT) is proposed for analyzing the wideband radar cross section (RCS) of complex multiscale objects in this letter. The SAIM cannot only be used to effectively deal with the difficulty of slow convergence caused by the mutiscale problems, but also greatly reducing the computational memory. By introducing the CAT into SAIM, we can use the currents at a few frequency sampling points to predict the currents at any frequency point over the entire band. In addition, the application of the Maehly approximation can obtain a better accuracy. Numerical examples show that the proposed SAIM-CAT can significantly improve the calculation efficiency without loss of accuracy.

Linear Pseudospectral Method with Chebyshev Collocation for Optimal Control Problems with Unspecified Terminal Time

In this paper, a linear Chebyshev pseudospectral method (LCPM) is proposed to solve the nonlinear optimal control problems (OCPs) with hard terminal constraints and unspecified final time, which uses Chebyshev collocation scheme and quasi-linearization. First, Taylor expansion around the nonlinear differential equations of the system is used to obtain a set of linear perturbation equations. Second, the first-order necessary conditions for OCPs with these linear equations and unspecified terminal time are derived, which provide the successive correction formulas of control and terminal time. Traditionally, these formulas are linear time varying and cannot be solved in an analytical manner. Third, Lagrange interpolation, whose supporting points are orthogonal Chebyshev–Gauss–Lobatto (CGL), is employed to discretize the resulting problem. Therefore, a series of analytical correction formulas are successfully derived in approximating polynomial space. It should be noted that Chebyshev approximation is close to the best polynomial approximation, and CGL points can be solved in closed form. Finally, LCPM is applied to the air-to-ground missile guidance problem. The simulation results show that it has high computational efficiency and convergence rate. A comparison with the other typical OCP solvers is provided to verify the optimality of the proposed algorithm. In addition, the results of Monte Carlo simulations are presented, which show that the proposed algorithm has strong robustness and stability. Therefore, the proposed method has potential to be onboard application.

Time domain numerical modeling of wave propagation in poroelastic media with Chebyshev rational approximation of the fractional attenuation

In this work, we investigate the poroelastic waves by solving the time‐domain Biot‐JKD equation with an efficient numerical method. The viscous dissipation occurring in the pores depends on the square root of the frequency and is described by the Johnson‐Koplik‐Dashen (JKD) dynamic tortuosity/permeability model. The temporal convolutions of order 1/2 shifted fractional derivatives are involved in the time‐domain Biot‐JKD model, causing the problem to be stiff and challenging to be implemented numerically. Based on the best relative Chebyshev approximation of the square‐root function, we design an efficient algorithm to approximate and localize the convolution kernel by introducing a finite number of auxiliary variables that satisfy a local system of ordinary differential equations. The imperfect hydraulic contact condition is used to describe the interface boundary conditions and the Runge‐Kutta discontinuous Galerkin (RKDG) method together with the splitting method is applied to compute the numerical solutions. Several numerical examples are presented to show the accuracy and efficiency of our approach.

Active learning strategy: Computer aided numerical class project on pole-zero plot and transfer function of five low pass filter approximation functions

The conversion of the concept and the teaching of approximation function for analog low pass filter to computer aided numerical class project as an active learning strategy is presented. Governing equations for five approximation functions are presented and discussed. It is recommended that students are tasked to develop computer programs in any convenient computer programming language to make the pole-zero plot, obtain the transfer function, and plot the frequency response of low pass filters. Steps for such programs are suggested. Typical programs were developed and tested. The pole-zero plots and frequency response graphs generated are good illustrative and innovative teaching aids. Linear-phase response over the passband is a characteristic of Bessel approximation function but suffers less amplitude discrimination in the stopband. Chebyshev and Inverse Chebyshev approximation functions have the same order requirement which is usually greater than that of Elliptic approximation function but less than that of Butterworth approximation function. Elliptic approximation function is the best choice although it is more complex than the others. Through this computer aided numerical class project, the authors themselves gained more insight in the subject matter; certainly, students will learn actively.

Solution to a Damped Duffing Equation Using He’s Frequency Approach

In this paper, we generalize He's frequency approach for solving the damped Duffing equation by introducing a time varying amplitude. We also solve this equation by means of the homotopy method and the Lindstedt–Poincaré method. High accurate formulas for approximating the Jacobi elliptic function cn are formally derived using Chebyshev and Pade approximation techniques.

Fourier–Chebyshev approximation of Kochin functions in the Fourier–Kochin approach to panel methods

The dominant computational task involved in the methods, widely called panel methods, currently used to compute 3D potential flows around large floating bodies such as ships and offshore structures (and other classes of dispersive waves such as flexural-gravity waves in very large floating structures or ice sheets) consists in evaluating N2 coefficients, called influence coefficients. These coefficients are associated with the flows created by distributions of singularities (sources, dipoles) over the N panels that approximate the surface of the body (ship, offshore structure) at every panel of the body surface. In contrast to the O(N2) computations required in existing panel methods, the dominant part of the computations of influence coefficients are independent of N in the Fourier–Kochin (FK) method if the Kochin functions in the FK flow representation are approximated by means of Fourier series and Chebyshev polynomials. The numerical analysis reported in this study shows that Fourier–Chebyshev approximations to the Kochin functions are feasible and indeed practical. In particular, the analysis shows that the number NF of basic Fourier integrals (the major and most difficult computational task) that must be evaluated within the approach to the numerical implementation of the FK method considered in the study is given by NF≈(600)2, i.e. is smaller than N2 if 600<N. Thus, one has NF≪N2 for typical panel numbers N=O(104), and the FK approach opens the way for a class of panel methods in which the dominant computations are independent of the number N of panels instead of O(N2) in existing panel methods. Another major advantageous feature of the FK method is that this approach circumvents the notorious difficulties involved in the numerical evaluation and the panel-surface-integration of the complicated Green functions associated with ship and offshore hydrodynamics.

Parallel Evaluation of Chebyshev Approximations: Applications in Astrodynamics

Approximations based on Chebyshev polynomials have several astrodynamic applications. The performance of these approximations can be improved by parallel implementations exploiting parallel architectures, such as OpenMP and CUDA. In this paper, we introduce the parallel implementation to two astrodynamic applications. The first is the gravitational finite element model (FEM): a piecewise Chebyshev approximation that replaces high degree and order gravitational spherical harmonic models (SHMs). Thus, much lower degree, locally valid functions can efficiently model and compute local gravity perturbations in parallel structure for efficient performance. For this model, the total gravity acceleration is split into a reference and disturbance term. The reference includes two-body plus J_2, which are relatively cheap to compute. The FEM approximates the higher-order gravity terms. It is developed from a 2D mesh grid covering a sphere of a specified radius, and a family of spherical shells is sampled using a cosine distribution in the radial direction. To reduce the required memory when seeking a specific accuracy, an adaptive version of the gravitational FEM is introduced. In addition, a parallel implementation of the FEM using OpenMP is preseneted. We show the runtime comparison for the 200 degree × 200 order EGM2008 SHM and the serial and parallel equivalent FEM algorithms. The other application is the Chebyshev-Picard method (CPM): a numerical integrator that solves an ordinary differential equation by approximating the integrand using a Chebyshev approximant and iterates over the trajectory via Picard iteration. A parallel CUDA implementation of the CPM method in conjunction with the EGM2008 SHM and the FEM is introduced. We present numerical examples for propagating four Earth-orbiting satellites considering both the 200times 200 EGM2008 SHM and the equivalent FEM representation to test the algorithm’s performance via parallel and serial computation (i.e., a single CPU thread).

Multivariate approximation by polynomial and generalized rational functions

In this paper, we develop an optimization-based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalized rational approximation. In the second case, the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimization problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood. In this paper we demonstrate that in the case of multivariate generalized rational approximation the corresponding optimization problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimization. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalized rational approximation with the same number of decision variables.

Analytic equation of state for the Buckingham Exp(α, m) fluids at high temperatures and densities based on the Ross variation perturbation theory

High temperatures and densities analytical thermodynamic properties are derived for the general Buckingham Exp(α, m) fluids by: a) using the Ross analytical (RA) variation perturbation theory developed by Sun [Mol. Phys. 105 (2007) 3139] for multi-Yukawa potentials; b) using a two-Yukawa potential matching the Exp(α, m), and c) introducing correcting terms which account for the discrepancies between the original Exp(α, m) potential and its two-Yukawa representation. We call this method the Ross analytical corrected theory (RAC), which leads to analytic expressions for the excess Helmholtz free energy, excess internal energy and the compressibility factor Z (i.e. the equation of state). They are valid at high temperatures and densities, since they are obtained from the Ross theory, which extends the range of validity of perturbation theories to higher densities. The matching two-Yukawa potential can also be considered as a reference potential for a first-order perturbation theory, which after further simplification leads to an analytic augmented van der Waals (avdW) theory for the above thermodynamic properties of the Exp(α, m) fluids. For the Exp(11.5, 6) and Exp(14.5, 6) fluids, the excess internal energy, compressibility factor Z and reduced pressure obtained from the RAC and avdW theories are compared with Monte Carlo (MC) simulation results corresponding to both the Exp and TY potentials. The overall accuracy of the RAC and avdW theories for the Exp(11.5, 6) fluid is also assessed in comparison to the Ross numerical (RN) theory (called also MCRSR for Mansoori, Canfield, Rasaiah, Stell and Ross) and to more complex methods such as the KLRR-T (Kang, Lee, Ree, Ree-Theostar) and the HMSA/C (hypernetted mean spherical approximation with Chebyshev approximants) theories, which require more extensive numerical calculations.

A generalisation of de la Vallée-Poussin procedure to multivariate approximations

The theory of Chebyshev approximation has been extensively studied. In most cases, the optimality conditions are based on the notion of alternance or alternating sequence (that is, maximal deviation points with alternating deviation signs). There are a number of approximation methods for polynomial and polynomial spline approximation. Some of them are based on the classical de la Vallée-Poussin procedure. In this paper we demonstrate that under certain assumptions the classical de la Vallée-Poussin procedure, developed for univariate polynomial approximation, can be extended to the case of multivariate approximation. There are two main advantages of our approach. First of all, it provides an elegant geometrical interpretation of the procedure. Second, the corresponding basis functions are not restricted to be monomials and therefore can be extended to a larger family of functions.

Chebyshev approximation of multivariable functions with the interpolation

A method of constructing a Chebyshev approximation of multivariable functions by a generalized polynomial with the exact reproduction of its values at a given points is proposed. It is based on the sequential construction of mean-power approximations, taking into account the interpolation condition. The mean-power approximation is calculated using an iterative scheme based on the method of least squares with the variable weight function. An algorithm for calculating the Chebyshev approximation parameters with the interpolation condition for absolute and relative error is described. The presented results of solving test examples confirm the rapid convergence of the method when calculating the parameters of the Chebyshev approximation of tabular continuous functions of one, two and three variables with the reproduction of the values of the function at given points.

Computationally Efficient Approximations for Matrix-Based Rényi's Entropy

The recently developed matrix based Renyi's entropy enables measurement of information in data simply using the eigenspectrum of symmetric positive semi definite (PSD) matrices in reproducing kernel Hilbert space, without estimation of the underlying data distribution. This intriguing property makes the new information measurement widely adopted in multiple statistical inference and learning tasks. However, the computation of such quantity involves the trace operator on a PSD matrix $G$ to power $\alpha$(i.e., $tr(G^\alpha)$), with a normal complexity of nearly $O(n^3)$, which severely hampers its practical usage when the number of samples (i.e., $n$) is large. In this work, we present computationally efficient approximations to this new entropy functional that can reduce its complexity to even significantly less than $O(n^2)$. To this end, we leverage the recent progress on Randomized Numerical Linear Algebra, developing Taylor, Chebyshev and Lanczos approximations to $tr(G^\alpha)$ for arbitrary values of $\alpha$ by converting it into matrix-vector multiplications problem. We also establish the connection between the matrix-based Renyi's entropy and PSD matrix approximation, which enables exploiting both clustering and block low-rank structure of $G$ to further reduce the computational cost. We theoretically provide approximation accuracy guarantees and illustrate the properties of different approximations. Large-scale experimental evaluations on both synthetic and real-world data corroborate our theoretical findings, showing promising speedup with negligible loss in accuracy.

ЧЕБИШОВСЬКЕ НАБЛИЖЕННЯ ФУНКЦIЙ БАГАТЬОХ ЗМIННИХ РАЦIОНАЛЬНИМ ВИРАЗОМ З IНТЕРПОЛЮВАННЯМ

A method for constructing the Chebyshev approximation by the rational expression of the multivariable functions with the interpolation is proposed. The method is based on the construction of the ultimate mean-power approximation by a rational expression with the interpolation condition in the norm of space $L_p$ at $p \to \infty$. To construct such an approximation, an iterative scheme based on the least squares method with two variable weight functions was used.

A multilevel approach to stochastic trace estimation

This article presents a randomized matrix-free method for approximating the trace of f(A), where A is a large symmetric matrix and f is a function analytic in a closed interval containing the eigenvalues of A. Our method uses a combination of stochastic trace estimation (i.e., Hutchinson's method), Chebyshev approximation, and multilevel Monte Carlo techniques. We establish general bounds on the approximation error of this method by extending an existing error bound for Hutchinson's method to multilevel trace estimators. Numerical experiments are conducted for common applications such as estimating the log-determinant, nuclear norm, and Estrada index. We find that using multilevel techniques can substantially reduce the variance of existing single-level estimators.

Bounds on strong unicity for Chebyshev approximation with bounded coefficients

AbstractWe obtain new effective results in best approximation theory, specifically moduli of uniqueness and constants of strong unicity, for the problem of best uniform approximation with bounded coefficients, as first considered by Roulier and Taylor. We make use of techniques from the field of proof mining, as introduced by Kohlenbach in the 1990s. In addition, some bounds are obtained via the Lagrangian interpolation formula as extended through the use of Schur polynomials to cover the case when certain coefficients are restricted to be zero.

Structural-Parametric Synthesis of the RoboMech Class Parallel Mechanism with Two Sliders

This paper addresses the structural-parametric synthesis and kinematic analysis of the RoboMech class of parallel mechanisms (PM) having two sliders. The proposed methods allow the synthesis of a PM with its structure and geometric parameters of the links to obtain the given laws of motions of the input and output links (sliders). The paper outlines a possible application of the proposed approach to design a PM for a cold stamping technological line. The proposed PM is formed by connecting two sliders (input and output objects) using one passive and one negative closing kinematic chain (CKC). The passive CKC does not impose a geometric constraint on the movements of the sliders and the geometric parameters of its links are varied to satisfy the geometric constraint of the negative CKC. The negative CKC imposes one geometric constraint on the movements of the sliders and its geometric parameters are determined on the basis of the Chebyshev and least-square approximations. Problems of positions and analogues of velocities and accelerations of the considered PM are solved to demonstrate the feasibility and effectiveness of the proposed formulations and case of study.

Chebyshev approximation of functions of two variables by a rational expression with interpolation

A method for constructing a Chebyshev approximation by a rational expression with interpolation for functions of two variables is proposed The idea of the method is based on the construction of the ultimate mean-power approximation in the norm of space Lp at p° . An iterative scheme based on the least squares method with two variable weight functions was used to construct such a Chebyshev approximation. The results of test examples confirm the effectiveness of the proposed method for constructing the Chebyshev approximation by a rational expression with interpolation.

An Optimal Design of Non-Causal Recursive Digital Filters with Zero Phase Shift using Chebyshev approximation and Linear Programming

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