Iterative Approximation Research Articles

Improved Least Square Progressive Iterative Approximation Format for Triangular B-B Surfaces

Classical progressive iterative approximation method is simple, intuitive, and effective for data fitting, but it is not enough to process mass data. The least square progressive iterative approximation (LSPIA) could make up for the limited data, and be suitable for fitting mass data. To improve the convergence rate of LSPIA, the Schulz iteration of Moore-Penrose generalized inverse is combined with LSPIA iterative format. The improved least square progressive iteration approximation format is proposed for triangular B-B surfaces. And the iterative surface sequence converges to the least square fitting result to the given data points in second-order for 2,3,4-degree triangular B-B surfaces. Furthermore, the weight is calculated with the fastest convergence rate, and some numerical examples are presented to validate the correctness and efficiency of the improved LSPIA method.

High precision measurement of blade profile curve using iterative normal vector approximation

The precision measurement of blades has always been a complex problem in free-form surface inspection. For contact inspection of blades by coordinate measuring machine (CMM), the probe radius compensation has a critical influence on the measurement accuracy. In this study, an iterative approach based on the normal vector approximation is proposed to measure the blade profile curve, which aims to reduce the measurement error caused by the conventional method relying on a computer aided design model for compensation. The proposed method combines the local reconstruction of blade surface and CMM measurement software. It first takes the measured results of conventional method as a base, then the resulting corrected points of several sectional profiles are fitted to a B-spline surface to provide an alternative nominal model for CMM software, leading to the next same measurement task. Thus, the measurement-fitting process iterates, in which the current nominal model in CMM software is updated by the reconstructed model of foregoing iteration, until a preset condition is reached. Consequently, the proposed method is implemented on a four-axis CMM, and two blade samples are designed to verify the approach’s effectiveness. Experiments show that our approach is superior to the conventional one in blade profile measurement accuracy.

Sparse and Incomplete Signal Dictionaries for Reconstruction of MR Images

Compressed Sensing(CS) is a mathematical approach for data acquisition in which the signals are compressible and sparse w.r.t. to an orthonormal basis. These sparse signals are reconstructed from very less measurements. CS technique Is widely used in Magnetic Resonance Imaging (MRI) where the doctors suggest the patients to undergo MRI scans for diagnosing their body parts. During the prolonged MRI Scan, the exact slice of the MRI cannot be achieved due to the difficulties faced by the patient or irregular changes in the body position of the patient. The idea is to reduce the exposure time of the patient’s body against the MRI scan by considering only fewer samples. Is it possible to Reconstruct the signal by making use of a fewer number of samples that are less than the Nyquist rate? Yes, it is possible to reconstruct the signal by making use of the Compressed Sensing or sampling Technique. Compressed sensing is a new framework for signal acquisition and representation in a compressible manner less below the Nyquist sampling rate. In this article, Sampling and reconstruction are dealt here thoroughly as part of the research activity. Compressive Sensing Matching pursuit (CoSaMP) is a novel technique for optimization. It is an iterative approximation method for sparse and incomplete signal recovery. CoSaMP method along with Different transform techniques is used for reconstruction. The FFT_CoSaMP, DCT_CoSaMP and DWT_CoSaMP are proposed methods for MR Image Reconstruction, where DWT-based CoSaMP along with different wavelet families give the best results when compared to other CS-based techniques w.r.t. PSNR, SSIM and RMSE analysis.

Simulation of stationary non-Gaussian stochastic vector processes using an eigenvalue-based iterative translation approximation method

The simulation of stationary non-Gaussian stochastic vector processes is of great importance in the time-domain response analysis of structures under non-Gaussian excitations. When incompatibility between the target cross power spectral density matrix and the marginal probabilistic density function of the stochastic vector process occurs in the sense of translation processes, an iterative simulation scheme must be required. Currently, the Cholesky decomposition-based iterative translation approximation method is widely utilized. However, the use of this method still encounters some difficulties. First, it requires performing Cholesky decompositions three times at each iteration and is therefore cumbersome. Second, a complicated random shuffling procedure is necessary, which renders the final iteration result non-unique. Third, it is very difficult for this method to handle cases with complex-valued spectral matrices. Lastly, it can generally be used to simulate stochastic vector processes with very limited components. To overcome these difficulties, an eigenvalue-based iterative translation approximation method is proposed in this study. This method can not only avoid the cumbersome Cholesky decomposition and complex random shuffling, but can also handle cases with complex-valued and large spectral matrices more easily. Four numerical examples with no incompatibility, significant incompatibility, complex-valued coherence and a large number of component processes are respectively employed to demonstrate the effectiveness of the proposed method. The results show that the proposed method has similar accuracy to the Cholesky decomposition-based iterative method in the first two examples. For the last two examples, the simulation accuracy is also satisfactory. The above results verify that the proposed method is superior to the Cholesky decomposition-based iterative translation approximation method overall.

Some Iterative Approximation Results of F Iteration Process in Banach Spaces

In this research, we suggest some convergence results for operators having (RCSC) condition in Banach space setting under F iterative scheme. We establish weak convergence under Opials condition and also establish some important strong convergence results under some appropriate assumptions on the domain or on the mapping. We furnish a non-trivial example of mappings having (RCSC) condition and show that its F iterative scheme is more effective than the corresponding well known iterative schemes on this particular example.

Optimization of low-thrust multi-debris removal mission via an efficient approximation model of orbit rendezvous

An efficient iterative approximation model for low-thrust rendezvous in low-earth orbit is proposed to be applied in optimization of multi-debris removal missions. Based on an existing method for impulsive orbit rendezvous and a novel iteration process, the propellant cost and thrust time of a low-thrust rendezvous can be quickly evaluated. Moreover, another iterative method is also designed to help find the minimal transfer duration required for orbit rendezvous. Then, its application in the max-reward optimization of debris removal mission with a large-scale targets to be selected is studied. In the three-steps optimization including target selection, sequence optimization, and low-thrust trajectory optimization, the iterative approximation model is used to deal with the constraints and evaluate the objective functions more efficiently. The problem of the eighth Chinese trajectory optimization competition was used to test the performance, and a leading solution including 93 targets was found. Comparisons with previous results were made to prove the advantages of the method in this paper.

A new iterative approximation scheme for Reich–Suzuki-type nonexpansive operators with an application

In this article, we propose a faster iterative scheme, called the AH iterative scheme, for approximating fixed points of contractive-like mappings and Reich–Suzuki-type nonexpansive mappings. We show that the AH iterative scheme converges faster than a number of existing iterative schemes for contractive-like mappings. The w^{2}-stability result of the new iterative scheme is established and a supportive example is provided to illustrate the notion of w^{2}-stability. Then, we prove weak and several strong convergence results of our new iterative scheme for fixed points of Reich–Suzuki-type nonexpansive mappings. Further, we carry out a numerical experiment to illustrate the efficiency of our new iterative scheme. As an application, we use our main result to prove the existence of a solution of a mixed-type nonlinear integral equation. An illustrative example to validate the result in our application is also given. Our results extend and generalize several related results in the existing literature.

Numerical modelling of coronavirus pandemic in Peru

Abstract Objectives The main objective of this research is to demonstrate the effectiveness of non-pharmaceutical interventions (social isolation and quarantine) and of vaccination. Methods The SIR epidemiological numerical model has been revised to obtain a new model (SAIRDQ), which involves additional variables: the population that died due to the disease (D), the isolated (A), quarantined population (Q) and the effect of vaccination. We have obtained the epidemiological parameters from the data, which are not constant during the evolution of the pandemic, using an iterative approximation method. Results Analysis of the data of infected and deceased suggest that the evolution of the coronavirus epidemic in Peru has arrived at the end of the second wave (around October 2021). We have simulated the effect of quarantine and vaccination, which are effective measures to reduce the impact of the pandemic. For a variable infection and isolation rate, due to the end of the quarantine, the death toll would be around 200 thousand; if the isolation and quarantine were relaxed since March 01, 2021, there could be more than 280 thousand deaths. Conclusions Without non-pharmaceutical interventions and vaccination, the number of deaths would be much higher than 280 thousand.

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.

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

AbstractWe 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.

Multiscale scattering in nonlinear Kerr-type media

We propose a multiscale approach for a nonlinear Helmholtz problem with possible oscillations in the Kerr coefficient, the refractive index, and the diffusion coefficient. The method does not rely on structural assumptions on the coefficients and combines the multiscale technique known as Localized Orthogonal Decomposition with an adaptive iterative approximation of the nonlinearity. We rigorously analyze the method in terms of well-posedness and convergence properties based on suitable assumptions on the initial data and the discretization parameters. Numerical examples illustrate the theoretical error estimates and underline the practicability of the approach.

Optimal power scheduling of seaport microgrids with flexible logistic loads

Seaports are progressively electrified to harness various energy resources to provide green logistic services to ships. They essentially act as multi-energy microgrids to enhance the energy efficiency and environmental sustainability of the maritime industry. This trend necessitates the coordinated operation of multiple sectors in seaport microgrids, which is a challenging problem due to the complex couplings of power, thermal, and fluid flows and the heterogeneity of logistic loads. An optimal power scheduling framework for seaport microgrids is developed. Specifically, a non-linear, non-convex optimisation problem is formulated to integrate various logistic loads, including cold ironing, quay and yard cranes, and reefer areas, in an unbalanced multi-phase power distribution network connected with a thermal network. The optimisation problem is solved in a scalable decomposed manner based on iterative convex relaxation and approximation. Simulation results on a real-world port model verify the efficacy of the proposed framework, as well as the potential of the flexible logistic loads in improving port energy efficiency.

On the numerical solution of the iterative parabolic equations by ETDRK pseudospectral methods in linear and nonlinear media

Iterative parabolic equations (IPEs) were recently introduced as a promising tool for solving wave propagation problems in complex linear and non-linear media. In this study a general approach to the numerical solution of iterative parabolic equations on the basis of powerful ETD pseudospectral method is developed. The solution of nth IPE requires the computation of the input term that is obtained by differentiation of the solution of n−1th IPE. The numerical noise resulting from such differentiation spoils the solutions of higher-order IPEs. It is shown that this noise can be suppressed via a filtering procedure involving Chebyshev polynomials of a discrete variable. In this study Padé type iterative parabolic approximations are introduced both for the case of linear and nonlinear media. It is shown that Padé-type approximations are more suitable for solving the problems of wave propagation.

A Local Iterative Linear Approximation Framework to Integrate Discrete Choice and Mathematical Optimization Models

A Local Iterative Linear Approximation Framework to Integrate Discrete Choice and Mathematical Optimization Models

Maximin Joint Design of Transmit Waveform and Receive Filter Bank for MIMO-STAP Radar Under Target Uncertainties

This letter deals with the joint design of transmit waveform and receive filter bank for airborne multiple-input multiple-output (MIMO) radar under the target uncertainties. Assuming that the spatial angle and the Doppler frequency of the target are unknown, we formulate the maximin joint design problem by maximizing the worst-case signal-to-interference-plus-noise ratio (SINR) under the energy constraint, flexible modulus constraint, and similarity constraint on the transmit waveform. To tackle this problem, we develop a computationally efficient algorithm based on iterative feasible point pursuit successive convex approximation (FPP-SCA). Numerical results are provided to demonstrate the effectiveness and robustness of the proposed algorithm.

Graph Convergence and Iterative Approximation of Solution of a Set‐Valued Variational Inclusions

In this article, first, we introduce a class of proximal‐point mapping associated with generalized αiβj‐Hp(.,.,…)‐accretive mapping. Further, we discuss the graph convergence of generalized αiβj‐Hp(.,.,…)‐accretive mapping. As an application, we consider a set‐valued variational inclusion problem (SVIP) in real Banach spaces. Furthermore, we propose an iterative scheme involving the above class of proximal‐point mapping to find a solution of SVIP and discuss its convergence under some convenient assumptions. An example is constructed and demonstrated few graphics in support of our main results.

Privacy-Preserving Object Detection for Medical Images With Faster R-CNN

In this paper, we propose a lightweight privacy-preserving Faster R-CNN framework (SecRCNN) for object detection in medical images. Faster R-CNN is one of the most outstanding deep learning models for object detection. Using SecRCNN, healthcare centers can efficiently complete privacy-preserving computations of Faster R-CNN via the additive secret sharing technique and edge computing. To implement SecRCNN, we design a series of interactive protocols to perform the three stages of Faster R-CNN, namely feature map extraction, region proposal and regression and classification. To improve the efficiency of SecRCNN, we improve the existing secure computation sub-protocols involved in SecRCNN, including division, exponentiation and logarithm. The newly proposed sub-protocols can dramatically reduce the number of messages exchanged during the iterative approximation process based on the coordinate rotation digital computer algorithm. Moreover, the effectiveness, efficiency and security of SecRCNN are demonstrated through comprehensive theoretical analysis and extensive experiments. The experimental findings show that the communication overhead in computing division, logarithm and exponentiation decreases to 36.19%, 73.82% and 43.37%, respectively.

Hermitian and skew-Hermitian splitting method on a progressive-iterative approximation for least squares fitting

<abstract><p>To solve the problems of curves and surfaces approximation with normalized totally positive bases, a new progressive and iterative approximation for least square fitting method called HSS-LSPIA is proposed, which is based on the HSS iterative approach for solving linear equations of LSPIA. The HSS-LSPIA format includes two iterations with iterative difference vectors, each of which is distinct from the other. The approximate optimal positive constant, as well as convergence analyses, are provided. Furthermore, the HSS-LSPIA method can be faster than the ELSPIA, LSPIA, and WHPIA methods in terms of convergence speed. Numerical results verify this phenomenon.</p></abstract>

Existence and Monotone Iterative Approximation of Solutions for Neutral Differential Equations with Generalized Fractional Derivatives

We study the existence and monotone iterative approximation of mild solutions of fractional‐order neutral differential equations involving a generalized fractional derivative of order 0 < α < 1 which can be reduced to Riemann–Liouville or Hadamard fractional derivatives. The existence of mild solutions is obtained via fixed point techniques in a partially ordered space. The approach is constructive and can be applied numerically. In particular, we construct a monotone sequence of functions converging to a solution which is illustrated by a numerical example.

Genetic Decompilation Concept of the Telecommunication Devices Machine Code

Reverse engineering correct source code from a machine code to find and neutralize vulnerabilities is the most pressing problem for the field of telecommunications equipment. The decompilation techniques applicable for this have potentially reached their evolutionary limit. As a result, new concepts are required that can make a quantum leap in problem solving. Proceeding from this, the paper proposes the concept of genetic decompilation, which is a solution to the problem of multiparameter optimization in the form of iterative approximation of instances of the source code to the "original" one which will compile to the given machine code. This concept is tested by conducting a series of experiments with the developed software prototype using a basic example of machine code. The results of the experiments prove the proof of the concept, thereby suggesting new innovative directions for ensuring information security in this subject area.