Iterative Approximation Technique Research Articles

New Progressive Iterative Approximation Techniques for Shepard-Type Curves

Progressive iterative approximation (PIA) technique is an efficient and intuitive method for data fitting. In CAGD modeling, if the given data points are taken as initial control points, PIA process generates a series of shaping curves by adjusting the control points iteratively, while the limit curve interpolates the data points. Such format was used successfully for Shepard-type curves. The aim of the paper is to construct simple variants of the PIA method for Shepard-type curves producing novel curves modeling data points, so the designer can choose among several pencils of shapes outlining original control polygon. Matrix formulations, convergence results, error estimates, algorithmic formulations, critical comparisons, and numerical tests are shown. An application to a progressive modeling format by truncated wavelet transform is also presented, improving in some sense analogous process by truncated Fourier transform. By playing on two shapes handles—the number of base wavelet transform functions and the iteration level of PIA algorithm—several new contours modeling the given control points are constructed.

The minimum zone fitting and error evaluation for the logarithmic curve based on geometry optimization approximation algorithm

The minimum zone fitting and error evaluation for the logarithmic curve has important applications. Based on geometry optimization approximation algorithm whilst considering geometric characteristics of logarithmic curves, a new fitting and error evaluation method for the logarithmic curve is presented. To this end, two feature points, to serve as reference, are chosen either from those located on the least squares logarithmic curve or from amongst measurement points. Four auxiliary points surrounding each of the two reference points are then arranged to resemble vertices of a square. Subsequently, based on these auxiliary points, a series of auxiliary logarithmic curves (16 curves) are constructed, and the normal distance and corresponding range of values between each measurement point and all auxiliary logarithmic curves are calculated. Finally, by means of an iterative approximation technique consisting of comparing, evaluating, and changing reference points; determining new auxiliary points; and constructing corresponding auxiliary logarithmic curves, minimum zone fitting and evaluation of logarithmic curve profile errors are implemented. The example results show that the logarithmic curve can be fitted, and its profile error can be evaluated effectively and precisely using the presented method.

Iterative rational least squares fitting

Abstract A progressive iterative approximation technique for rational least squares fitting curves is developed. The format is interesting in CAGD (Computer Aided Geometric Design) and improves the recent algorithms. An improved chord method for the root finding based on rational operators is also presented.

Optimal placement of Charging Stations for Electric Vehicles in large-scale Transportation Networks

This paper presents a new practical approach to optimally allocate charging stations in large-scale transportation networks for electric vehicles (EVs). The problem is of particular importance to meet the charging demand of the growing fleet of alternative fuel vehicles. Considering the limited driving range of EVs, there is need to supply EV owners with accessible charging stations to reduce their range anxiety. The aim of the Route Node Coverage (RNC) problem, which is considered in the current paper, is to find the minimum number of charging stations, and their locations in order to cover the most probable routes in a transportation network. We propose an iterative approximation technique for RNC, where the associated Integer Problem (IP) is solved by exploiting a probabilistic random walk route selection, and thereby taking advantage of the numerical stability and efficiency of the standard IP software packages. Furthermore, our iterative RNC optimization procedure is both pertinent and straightforward to implement in computer coding and the design technique is therefore highly applicable. The proposed optimization technique is applied on the Sioux-Falls test transportation network, and in a large-scale case study covering the southern part of Sweden, where the focus is on reaching the maximum coverage with a minimum number of charging stations. The results are promising and show that the flexibility, smart route selection, and numerical efficiency of the proposed design technique, can pick out strategic locations for charging stations from thousands of possible locations without numerical difficulties.

Weighting Shepard-type operators

Uniform approximation error estimates for weighted Shepard-type operators more flexible than the unweighted analogues are given. Error estimates for a linear combination of their iterates faster converging than previous ones are also showed. The results are applied in CAGD to construct Shepard-type curves useful in modeling and a weighted progressive iterative approximation technique exponentially converging.

Design of two-channel low-delay IIR nonuniform-division filter banks using L1 error criteria

The design of a two-channel nonuniform-division filter (NDF) bank with infinite impulse response (IIR) analysis/synthesis filters and low group delay in the sense of L1 error criteria is considered. The problem formulation results in a nonlinear optimisation problem. Based on a variant of Karmarkar's algorithm, the optimisation problem is solved through a frequency sampling and iterative approximation technique to find the tap coefficients and the reflection coefficients for the numerator and the denominator of the IIR analysis filters. An efficient stabilisation procedure ensures that the reflection coefficients lie in (−1, 1). Simulation results are provided for illustration and comparison.

Design of two-channel IIR filter banks with arbitrary group delay

The nonlinear optimisation problem that results from considering the design of a two-channel nonuniform division filter bank is solved. This is through a frequency sampling and iterative approximation technique to find the tap coefficients and the reflection coefficients for the numerator and the denominator of the IIR analysis filters. An efficient stabilisation procedure ensures that the reflection coefficients lie in (-1,1). Simulation examples are provided for illustration.

Analysis of an Oscillatory Oil Squeeze Film Including Effects of Fluid Inertia

This paper presents an analytic solution for the flow of a laminar incompressible Newtonian fluid between two parallel circular plates undergoing oscillatory motion. The solution uses an iterative approximation technique and includes both convective and unsteady fluid inertia terms. The extent to which these terms influence the pressure and load carrying capacity for various flow parameters is analyzed. The results show a significant deviation from the inertialess (viscous) solution at low excursion ratio, e, and moderate modified Reynolds number, Reo. For a purely harmonic (sinusoidal) motion of the plates, a steady-state force is generated in addition to the dynamic transient force. This is due to the inclusion of the inertia terms and is shown to be a function of the excursion ratio, e, and another dimensionless parameter, the squeeze velocity number, N.

Iterative technique for designing IIR filters with arbitrary log‐magnitude function

AbstractThis paper proposes an iterative approximation technique of log‐magnitude response for IIR filters. Minimization of the mean‐square error between a desired log‐frequency response and that of an IIR filter leads to a nonlinear problem. In this paper, the least squares problem on the logarithmic scale is approximated by the iteration of a weighted least squares problem on the linear scale. the weighting function is updated using the result of the previous iteration so that the weighted error on the linear scale approximates the squared error on the logarithmic scale. the filter coefficients at each iteration step can efficiently be obtained using a fast recursive algorithm for a set of linear equations. Several design examples are presented, which show that the iterative algorithm converges after one or two iterations in practice and the weighted error at convergence gives a good estimate of the mean‐square error on the logarithmic scale. an example is also presented, where both log‐magnitude and phase responses are approximated simultaneously.

Transient response of a ferroresonant circuit

The transient response of a partical ferroresonant circuit with quintic nonlinearity has been obtained. A suitable closed loop block digram for this circuit is used to allow an iterative approximation technique to be used for the determination of the transient response due to the sudden application of a sinusoidal voltage. The effect of the variation of the amplitude and the angle of the applied voltage and of the various circuit parameters on the voltage across and the current through the nonlinear inductor of the circuit has been investigated.

On the frequency weighted least-square design of finite duration filters

The frequency weighted least-square design of finiteduration filters, in one and two dimensions, continuous and discrete is considered. Some new theoretical results and some practical design techniques for conventional and unconventional filters are presented. In some cases optimum discrete filters can be found conveniently by matrix inversion. In many cases a simple, iterative approximation technique using FFT can be used to carry out the design or to adjust the frequency response of filters.