Simple Iterative Method Research Articles

A Polynomial Time Algorithm to Compute Geodesics in CAT(0) Cubical Complexes

This paper presents the first polynomial time algorithm to compute geodesics in a CAT(0) cubical complex in general dimension. The algorithm is a simple iterative method to update breakpoints of a path joining two points using Owen and Provan’s algorithm (IEEE/ACM Trans Comput Biol Bioinform 8(1):2–13, 2011) as a subroutine. Our algorithm is applicable to any path in any CAT(0) space in which geodesics between two close points can be computed, not limited to CAT(0) cubical complexes.

Analysis of a proposed connection for the two-winding single-phase self-excited induction generator operating at constant voltage and frequency

This paper introduces the steady-state and dynamic behaviors of a proposed connection for the two-winding single-phase self-excited induction generator (TWSPSEIG) equipped with an excitation capacitor and a compensation capacitor for operation at constant load voltage and frequency irrespective of the no-load or different load conditions. The performance equations at steady-state conditions are attained by applying loop impedance method via the exact equivalent circuit models of the TWSPSEIG based on the double revolving field theory. Keeping the values of the excitation capacitor and the compensation capacitor as unknowns, two second-order equations, for given values of generator parameters, prime mover speed, frequency and load impedance, are derived. These equations are solved using simple iterative method to calculate the optimum values of the two capacitors under the constraints that the load voltage and frequency are constant. The range of capacitor variations for variable load at variable prime mover speed is also calculated. The steady-state results are verified by developing the dynamic model of the proposed connection incorporating its nonlinearity behavior and various no-load and load conditions. The dynamic behavior of the proposed connection proves the capabilities of the proposed configuration and calculation method to maintain both the load voltage and frequency constants. A comparative study between the performances of the proposed connection and the traditional connection of the TWSPSEIG is presented to illustrate the merits of the proposed connection.

Combination of the variational iteration method and numerical algorithms for nonlinear problems

A very simple and efficient local variational iteration method (LVIM), or variational iteration method with local property, for solving problems of nonlinear science is proposed in this paper. The analytical iteration formula of this method is derived first using a general form of first order nonlinear differential equations, followed by straightforward discretization using Chebyshev polynomials and collocation method. The resulting numerical algorithm is very concise and easy to use, only involving highly sparse matrix operations of addition and multiplication, and no inversion of the Jacobian in nonlinear problems. Apart from the simple yet efficient iteration formula, another extraordinary feature of LVIM is that in each local domain, all the collocation nodes participate in the calculation simultaneously, thus each local domain can be regarded as one “node” in calculation through GPU acceleration and parallel processing. For illustration, the proposed algorithm of LVIM is applied to various nonlinear problems including Blasius equations in fluid mechanics, buckled bar equations in solid mechanics, the Chandrasekhar equation in astrophysics, the low-Earth-orbit equation in orbital mechanics, etc. Using the built-in highly optimized ode45 function of MATLAB as a comparison, it is found that the LVIM is not only very accurate, but also much faster by an order of magnitude than ode45 in all the numerical examples, especially when the nonlinear terms are very complicated and difficult to evaluate.

Measurement and Correction of Interface Shear Stiffness between Slab and Base for Concrete Pavement

Interface parameters could affect the adhesion and durability performance of cement concrete pavements. In order to improve the pavement performance, the interface shear stiffness between the slab and the base of concrete pavement were investigated accurately. A horizontal shear tester was developed which could simulate the vertical pressure. The bond-slip curve of the specimens that the interface treated by three different measures was determined, such as emulsified asphalt, geotextile and asphalt concrete. Then, the correction of interface vertical pressure was made for average shear stiffness based on simple iterative method, and the equivalent shear stiffness of three types of specimens was recommended. The results showed that the specimen treated by emulsified asphalt required the maximum shear force, and asphalt concrete was minimum during the static friction stage. The contact states between slab and base are significantly correlated with the interface treatments. So, the interface treatments selection needed to find a balance between flexural tensile stress and thermal stress of the slab. In the sliding friction stage, the shear resistances of three types of interface treatment methods were significantly weakened and the average shear stiffness declined dramatically. It is very different for interface shear stiffness when the pavement structure in different using stages. The interface parameter must consistent with the using state of the pavement during the analysis process of the pavement structure.

A robust ragged cloud detection algorithm for remote sensing image

Cloud detection plays a significant role in remote sensing (RS) image processing. Numbers of cloud detection algorithms have been developed in the literature. However, they suffer the weakness of omitting thin and small cloud, and poor ability of differentiating the cloud from confusing ground region (e.g. artificial building). In this study, a robust ragged cloud detection algorithm for RS image is proposed. First, the simple linear iterative clustering method is applied to segment ragged cloud. Then, the improved Qtsu's method is introduced to remove the redundant superpixel. Finally, the Natural Scene Statistic is designed to classify the cloud region. Finally, original image will be classified into thick cloud, thin cloud and non-cloud. Experimental results indicate that the proposed model outperforms the state-of-the-art methods for cloud detection.

Iterative Approach to Solving Boundary Integral Equations in the Two-Dimensional Vortex Methods of Computational Hydrodynamics

Under consideration are the issues of numerical solution of a. boundary integral equation describing the vorticity generation process on the streamlined airfoils in meshless vortex methods. The traditional approach based on the quadrature method leads to the necessity of solving a. system of linear algebraic equations with dense matrix. If we consider the system of airfoils moving relative to one another, this procedure has to be performed at each time step of the calculation, and its high computational complexity significantly reduces the efficiency of vortex methods. The transition from the traditional approach expressed by an integral equation of the first kind to an approach with the integral equation of the second kind makes it possible to apply the simple-iteration method for numerical solving the boundary integral equation. By examples of some model problems, we demonstrate that the iterative approach allows reducing the computational complexity of the problem by tens to hundreds times while providing an acceptable accuracy of the approximate solution.

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.

Calculation of Equilibrium for Multicomponent Mixtures Using the Parameters of Models for Binary Pairs of Pure Components

An algorithm and an application software package have been developed for calculating the vapor–liquid equilibrium of multicomponent mixtures using the optimal parameters of equilibrium models (namely, the Wilson, NRTL, and UNIQUAC models) for binary pairs of the pure components of a mixture. The parameters of the models have been optimized by extremum localization, simple iteration, and certain other methods at the minimum of the optimization criterion, i.e., the sum of the squared deviations between experimental and calculated values for the equilibrium compositions of the vapor phases (at checkpoints) for each of the binary pairs of pure components.

Incorporating spatial-anatomical similarity into the VGWAS framework for AD biomarker detection.

The detection of potential biomarkers of Alzheimer's disease (AD) is crucial for its early prediction, diagnosis and treatment. Voxel-wise genome-wide association study (VGWAS) is a commonly used method in imaging genomics and usually applied to detect AD biomarkers in imaging and genetic data. However, existing VGWAS methods entail large computational cost and disregard spatial correlations within imaging data. A novel method is proposed to solve these issues. We introduce a novel method to incorporate spatial correlations into a VGWAS framework for the detection of potential AD biomarkers. To consider the characteristics of AD, we first present a modification of a simple linear iterative clustering method for spatial grouping in an anatomically meaningful manner. Second, we propose a spatial-anatomical similarity matrix to incorporate correlations among voxels. Finally, we detect the potential AD biomarkers from imaging and genetic data by using a fast VGWAS method and test our method on 708 subjects obtained from an Alzheimer's Disease Neuroimaging Initiative dataset. Results show that our method can successfully detect some new risk genes and clusters of AD. The detected imaging and genetic biomarkers are used as predictors to classify AD/normal control subjects, and a high accuracy of AD/normal control classification is achieved. To the best of our knowledge, the association between imaging and genetic data has yet to be systematically investigated while building statistical models for classifying AD subjects to create a link between imaging genetics and AD. Therefore, our method may provide a new way to gain insights into the underlying pathological mechanism of AD. https://github.com/Meiyan88/SASM-VGWAS.

Linear spatial stability analysis of particle-laden stratified shear layers

Hydrodynamic instabilities at the interface of stratified shear layers could occur in various modes. These instabilities have an important role in the mixing process. In this work, the linear stability analysis in spatial framework is used to study the stability characteristics of a particle-laden stratified two-layer flow. The effect of parameters such as velocity-to-density thickness ratio, bed slope, viscosity as well as particle size on the stability is considered. A simple iterative method applying the pseudospectral collocation method that employed Chebyshev polynomials is used to solve two coupled eigenvalue equations. Based on the results, the flow becomes stable for Richardson number larger than 0.25 (same as the result of temporal stability analysis); the stability is not affected by spatial wavenumber. The increase in bed slope makes the current more unstable as does in temporal framework. For 1% bed slope, the spatial growth rate increases by 70% in J = 0.23. For R = 5 (velocity-to-density thickness ratio) and zero bed slope, there are four zones: (a) two Kelvin–Helmholtz modes (0 < J < 0.09, J is local Richardson number), (b) two Holmboe modes (0.09 < J < 0.65), (c) no unstable mode (0.65 < J < 2.5) and (d) two Holmboe modes (2.5 < J < 4 where the second type of Holmboe modes appears). The second type of Holmboe modes does not appear in temporal framework in this condition. In spatial analysis, for nonzero bed slope there is no stable region. Also, just one type of Holmboe modes and two types of Kelvin–Helmholtz modes appear. Th existence of particles changes the instability characteristics of the flow. Particles increase the spatial growth rate. In temporal analysis, particles larger than a certain size (e.g., kaolin particles larger than 20 micron in Stokes’ law settling velocity) make the flow unstable, but in spatial framework particles with any size do this. As expected, the viscosity makes the current more stable like temporal analysis.

An Iterative Compensation Algorithm for Springback Control in Plane Deformation and Its Application

In order to solve the springback problem in sheet metal forming, the trial and error method is a widely used method in the factory, which is time-consuming and costly for its non-direction and non-quantitative. Finite element simulation is an effective method to predict the springback of complex shape parts, but its precision is sensitive to the simulation model, particularly material model and boundary conditions. In this paper, the simple iterative method is introduced to establish the iterative compensation algorithm, and the convergence criterion of iterative parameters is put forward. In addition, the new algorithm is applied to the V-free bending and stretch-bending processes, and the convergence of curvature and bending angle is proved theoretically and verified experimentally. At the same time, the iterative compensation experiments for plane bending show that, the new method can predict the next compensation value based on the springback of each test, so that the target bending angle with the error of less than ± 0.1% and the target curvature with the error of less than 0.5% are obtained after 2‒3 iterations. This research proposes a new iterative compensation algorithm to predict springback in sheet metal forming process, where each compensation value depends only on the iteration parameter difference before and after springback for the same forming process of same material.

New self-adaptive step size algorithms for solving split variational inclusion problems and its applications

In this paper, we study a special instance of the split inverse problem (SIP), which is the split variational inclusion problem (SVIP). Three simple iterative methods for solving it are introduced and weak and strong convergence theorems are established under mild and standard assumptions. As an application, the problem of minimizing two proper, convex, and lower semi-continuous functions is considered. We compare and illustrate the efficiency and applicability of our schemes for several numerical experiments as well as an example in the field of compressed sensing.

A novel method for breast mass segmentation: from superpixel to subpixel segmentation

In this paper, an effective method is proposed for breast mass segmentation using a superpixel generation and curve evolution method. The simple linear iterative clustering method and density-based spatial clustering of applications with noise method are applied to generate superpixels in mammograms at first. Thereafter, a region of interesting (ROI) that contains the breast mass is built on the superpixel generation results. Finally, the image patch and the position of the manual labeled seed are used to build the prior knowledge for the level set method driven by the local Gaussian distribution fitting energy and evolve the curve to capture the edge of breast mass in ROI. Experimental results on mammogram data set demonstrate that the proposed method shows superior performance in contrast to some well-known methods in breast mass segmentation.

Three‐dimensional‐based landmark tracker employing a superpixels method for neuroscience, biomechanics, and biology studies

AbstractExamining locomotion has improved our basic understanding of motor control and aided in treating motor impairment. Mice and rats are premier models of human disease and increasingly the model systems of choice for basic neuroscience. High frame rates (250 Hz) are needed to quantify the kinematics of these running rodents. Manual tracking, especially for multiple markers, becomes time‐consuming and impossible for large sample sizes. Therefore, the need for automatic segmentation of these markers has grown in recent years. Here, we address this need by presenting a method to segment the markers using the simple linear iterative clustering superpixel method. The two‐dimensional coordinates on the image plane are projected to a three‐dimensional (3D) domain using direct linear transform and a 3D Kalman filter has been used to predict the position of markers based on the speed and position of markers from the previous frames. A probabilistic function is used to find the best match from among superpixels. The method is evaluated for different difficulties for tracking of the markers, and it achieves 95% correct labeling of markers. Finally, the method was used to analyze data from an ongoing study seeking to improve recovery from spinal cord injury in rats

An Effective Subsuperpixel-Based Approach for Background Subtraction

How to achieve competitive accuracy and less computation time simultaneously for background estimation is still an intractable task. In this paper, an effective background subtraction approach for video sequences is proposed based on a subsuperpixel model. In our algorithm, the superpixels of the first frame are constructed using a simple linear iterative clustering method. After transforming the frame from a color format to gray level, the initial superpixels are divided into K smaller units, i.e., subsuperpixels, via the k -means clustering algorithm. The background model is then initialized by representing each subsuperpixel as a multidimensional feature vector. For the subsequent frames, moving objects are detected by the subsuperpixel representation and a weighting measure. In order to deal with ghost artifacts, a background model updating strategy is devised, based on the number of pixels represented by each cluster center. As each superpixel is refined via the subsuperpixel representation, the proposed method is more efficient and achieves a competitive accuracy for background subtraction. Experimental results demonstrate the effectiveness of the proposed method.

Extrapolation methods for the numerical solution of nonlinear Fredholm integral equations

In this paper, we want to exemplify the use of extrapolation methods (namely, Shanks transformations, the recursive algorithms for their implementation, and the freely available corresponding MATLAB software) in the solution of nonlinear Fredholm integral equations of the second kind. Extrapolations methods are well known in some domains of numerical analysis and applied mathematics, but, unfortunately, they are not frequently used in other domains. Thus, after presenting the most simple iterative method for the solution of Fredholm equations, we will show how the sequence it produces can be accelerated (under some assumptions) and also how the underlying system of nonlinear equations generated by it can be solved quite efficiently by a restarting method. Numerical examples and comparisons with other methods demonstrate the usefulness of these procedures.

On Some Families of New Constructed Polynomials

The purpose of this paper is to introduce some new polynomials obtained from second- and third-order algebraic equations by using a simple iterative method. One-variable polynomials obtained in this study deal with special form of Poschl–Teller potential with constant energy, and the two-variable polynomials are related to time-dependent wave equations. We present some recurrence relations, Binet formula and get various families of linear, multilinear and multilateral generating functions for these polynomials. In addition, we derive some special cases. At the end of the paper we also give an extension to the multidimensional case of our results.

Method of iteration of solving invariant equations with an approximate operator in the case of an arbitrary choice of the regularization parameter

In the introduction, the object of investigation is indicated – incorrect problems described by first-kind operator equations. The subject of the study is an explicit iterative method for solving first-kind equations. The aim of the paper is to prove the convergence of the proposed method of simple iterations with an alternating step alternately and to obtain error estimates in the original norm of a Hilbert space for the cases of self-conjugated and non self-conjugated problems. The a priori choice of the regularization parameter is studied for a source-like representable solution under the assumption that the operator and the right-hand side of the equation are given approximately. In the main part of the work, the achievement of the stated goal is expressed in four reduced and proved theorems. In Section 1, the first-kind equation is written down and a new explicit method of simple iteration with alternating steps is proposed to solve it. In Section 2, we consider the case of the selfconjugated problem and prove Theorem 1 on the convergence of the method and Theorem 2, in which an error estimate is obtained. To obtain an error estimate, an additional condition is required – the requirement of the source representability of the exact solution. In Section 3, the non-self-conjugated problem is solved, the convergence of the proposed method is proved, which in this case is written differently, and its error estimate is obtained in the case of an a priori choice of the regularization parameter. In sections 2 and 3, the error estimates obtained are optimized, that is, a value is found – the step number of the iteration, in which the error estimate is minimal. Since incorrect problems constantly arise in numerous applications of mathematics, the problem of studying them and constructing methods for their solution is topical. The obtained results can be used in theoretical studies of solution of first-kind operator equations, as well as applied ill-posed problems encountered in dynamics and kinetics, mathematical economics, geophysics, spectroscopy, systems for complete automatic processing and interpretation of experiments, plasma diagnostics, seismic and medicine.

Temperature dependent iterative model of thermoelectric generator including thermal losses in passive elements

Design and development of a thermoelectric module and in its use in real application require an accurate simulation tool, which provides electrical and thermal characterizations as a function of temperature. The problem of correctly solving the basic thermoelectric equations originates from the fact that junctions temperatures are unknown and normally cannot be measured, whereas only external temperatures are available in testing and real applications. Due to thermal losses in passive layers, the internal temperatures can be significantly different. At the same time all the equations contains temperature dependent parameters. Many approximations are usually introduced to achieve results. Here we propose a simple iterative method to obtain the temperature losses and calculate all thermoelectric performances with corrected temperatures. The method, developed in Matlab language, takes into account temperature dependent material properties, thermal and electrical resistance of passive elements (electrodes, ceramics, interfaces, contact pad, etc.). Joule heating, Peltier and Thomson effects are considered in determine the temperatures. The effectiveness of the proposed procedure and the differences between approximated methods are investigated. The accuracy is proved with two commercial modules: the deviation were found to be within 4% and 3%. The code demonstrates to converge in few iterations in both cases. Moreover, the robustness has been investigated as a function of different parameters confirming that the code is suitable also for parametric simulations.

On Optimal Parameter Not Only for the SOR Method

The Jacobi, Gauss-Seidel and SOR methods belong to the class of simple iterative methods for linear systems. Because of the parameter , the SOR method is more effective than the Gauss-Seidel method. Here, a new approach to the simple iterative methods is proposed. A new parameter q can be introduced to every simple iterative method. Then, if a matrix of a system is positive definite and the parameter q is sufficiently large, the method is convergent. The original Jacobi method is convergent only if the matrix is diagonally dominated, while the Jacobi method with the parameter q is convergent for every positive definite matrix. The optimality criterion for the choice of the parameter q is given, and thus, interesting results for the Jacobi, Richardson and Gauss-Seidel methods are obtained. The Gauss-Seidel method with the parameter q, in a sense, is equivalent to the SOR method. From the formula for the optimal value of q results the formula for optimal value of . Up to present, this formula was known only in special cases. Practical useful approximate formula for optimal value  is also given. The influence of the parameter q on the speed of convergence of the simple iterative methods is shown in a numerical example. Numerical experiments confirm: for very large scale systems the speed of convergence of the SOR method with optimal or approximate parameter  is near the same (in some cases better) as the speed of convergence of the conjugate gradients method.