Fast and Accurate Predictor-Corrector Methods Using Feedback-Accelerated Picard Iteration for Strongly Nonlinear Problems

Although predictor-corrector methods have been extensively applied,they might not meet the requirements of practical applications and engineering tasks,particularly when high accuracy and efficiency are necessary.A novel class of correctors based on feedback-accelerated Picard iteration(FAPI)is proposed to further enhance computational performance.With optimal feedback terms that do not require inversion of matrices,significantly faster convergence speed and higher numerical accuracy are achieved by these correctors compared with their counterparts;however,the computational complexities are comparably low.These advantages enable nonlinear engineering problems to be solved quickly and accurately,even with rough initial guesses from elementary predictors.The proposed method offers flexibility,enabling the use of the generated correctors for either bulk processing of collocation nodes in a domain or successive corrections of a single node in a finite difference approach.In our method,the functional formulas of FAPI are discretized into numerical forms using the collocation approach.These collocated iteration formulas can directly solve nonlinear problems,but they may require significant computational resources because of the manipulation of high-dimensionalmatrices.To address this,the collocated iteration formulas are further converted into finite difference forms,enabling the design of lightweight predictor-corrector algorithms for real-time computation.The generality of the proposed method is illustrated by deriving new correctors for three commonly employed finite-difference approaches:the modified Euler approach,the Adams-Bashforth-Moulton approach,and the implicit Runge-Kutta approach.Subsequently,the updated approaches are tested in solving strongly nonlinear problems,including the Matthieu equation,the Duffing equation,and the low-earth-orbit tracking problem.The numerical findings confirm the computational accuracy and efficiency of the derived predictor-corrector algorithms.

Another SSOR Iteration Method

Kellogg gave a version of the Peaceman-Radford method. In this paper, we introduce a SSOR iteration method which uses Kellogg’s method. The new algorithm has some advantages over the traditional SSOR algorithm. A Cyclic Reduction algorithm is introduced via a decoupling in Kellogg’s method.

Low-complexity signal detection for massive MIMO systems via trace iterative method

Linear minimum mean square error(MMSE)detection has been shown to achieve near-optimal performance for massive multiple-input multiple-output(MIMO)systems but inevitably involves complicated matrix inversion,which entails high complexity.To avoid the exact matrix inversion,a considerable number of implicit and explicit approximate matrix inversion based detection methods is proposed.By combining the advantages of both the explicit and the implicit matrix inversion,this paper introduces a new low-complexity signal detection algorithm.Firstly,the relationship between implicit and explicit techniques is analyzed.Then,an enhanced Newton iteration method is introduced to realize an approximate MMSE detection for massive MIMO uplink systems.The proposed improved Newton iteration significantly reduces the complexity of conventional Newton iteration.However,its complexity is still high for higher iterations.Thus,it is applied only for first two iterations.For subsequent iterations,we propose a novel trace iterative method(TIM)based low-complexity algorithm,which has significantly lower complexity than higher Newton iterations.Convergence guarantees of the proposed detector are also provided.Numerical simulations verify that the proposed detector exhibits significant performance enhancement over recently reported iterative detectors and achieves close-to-MMSE performance while retaining the low-complexity advantage for systems with hundreds of antennas.

Variational Iteration Method for Solving Time Fractional Burgers Equation Using Maple

The Time Fractional Burger equation was solved in this study using the Mabel software and the Variational Iteration approach. where a number of instances of the Time Fractional Burger Equation were handled using this technique. Tables and images were used to present the collected numerical results. The difference between the exact and numerical solutions demonstrates the effectiveness of the Mabel program’s solution, as well as the accuracy and closeness of the results this method produced. It also demonstrates the Mabel program’s ability to quickly and effectively produce the numerical solution.

The Alternating Group Explicit Iterative Method for the Regularized Long-Wave Equation

An Alternating Group Explicit (AGE) iterative method with intrinsic parallelism is constructed based on an implicit scheme for the Regularized Long-Wave (RLW) equation. The method can be used for the iteration solution of a general tridiagonal system of equations with diagonal dominance. It is not only easy to implement, but also can directly carry out parallel computation. Convergence results are obtained by analysing the linear system. Numerical experiments show that the theory is accurate and the scheme is valid and reliable.

Simulation of Steel Reinforcement on the Nonlinear Behaviour of Slender Glulam Beam Columns by Using the Newton-Raphson Method

The current theory in NF EN 1995-1-1/NA of Eurocode 5, which is based on maximum deflection, has been investigated on softwoods. Therefore, this theory is not adapted for slender glulam beam columns made of tropical hardwood species from the Congo Basin. This maximum deflection is caused by a set of loads applied to the structure. However, Eurocode 5 doesn’t provide how to predict this deflection in case of long-term load for such structures. This can be done by studying load-displacement (P-Δ) behaviour of these structures while taking into account second order effects. To reach this goal, a nonlinear analysis has been performed on a three-dimensional beam column embedded on both ends. Since conducting experimental investigations on large span structural products is time-consuming and expensive especially in developing countries, a numerical model has been implemented using the Newton-Raphson method to predict load-displacement (P-Δ) curve on a slender glulam beam column made of tropical hardwood species. On one hand, the beam has been analyzed without wood connection. On the other hand, the beam has been analyzed with a bolted wood connection and a slotted-in steel plate. The load cases considered include self-weight and a uniformly applied long-term load. Combinations of serviceability limit states (SLS) and ultimate limit states (ULS) have also been considered, among other factors. A finite-element software RFEM 5 has been used to implement the model. The results showed that the use of steel can reduce displacement by 20.96%. Additionally, compared to the maximum deflection provided by Eurocode 5 for softwoods, hardwoods can exhibit an increasing rate of 85.63%. By harnessing the plastic resistance of steel, the bending resistance of wood can be increased by 32.94%.

Modified two-grid method for solving coupled Navier-Stokes/Darcy model based on Newton iteration

A new decoupled two-gird algorithm with the Newton iteration is proposed for solving the coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. Moreover the error estimate is given, which shows that the same order of accuracy can be achieved as solving the system directly in the fine mesh when h = H2. Both theoretical analysis and numerical experiments illustrate the efficiency of the algorithm for solving the coupled problem.

A Novel Method of Newton Iteration in Complex Field and Lattice Search for Locating Partial Discharges in Transformers

在变压器局部放电特高频定位的多传感器时间差测量中不可避免地存在误差,这往往导致时间差方程组在实数域内无解。为了得到时间差方程组的最优近似解,提出了复数域牛顿迭代网格搜索方法。该方法在复数域内进行迭代;当迭代结果为实数时就将其作为定位结果;当迭代结果为复数时,以其实部为中心点坐标,在该点周围局部区域内采用网格搜索法计算最优近似解。通过在变压器上开展试验检验了该方法的可行性和准确性。研究表明,在一定时间误差下,在实数域采用牛顿迭代法计算时不收敛,而在复数域可以收敛,平均定位误差为0.20 m。复数域牛顿迭代网格搜索方法能够在一定时间误差下求解时间差方程组,给出最优近似解,实现局部放电定位。

ANCIENT CHINESE ALGORITHM: THE YING BUZU SHU (METHOD OF SURPLUS AND DEFICIENCY)VS NEWTON ITERATION METHOD

Air exploratory discussion of an ancient Chinese algorithm, the Ying Buzu Shu, in about 2nd century BC, known as the rule of double false position in the West is given. In addition to pointing out that the rule of double false position is actually a translation version of the ancient Chinese algorithm, a comparison with well-known Newton iteration method is also made. If derivative is introduced, the ancient Chinese algorithm reduces to the Newton method. A modification of the ancient Chinese algorithm is also proposed, and some of applications to nonlinear oscillators are illustrated.

基于两步正则化Gauss-Newton迭代算法的ECT图像重建

电容层析成像(ECT)技术求解图像重建问题属于非线性问题,并且存在严重的不适定性。为提高图像重建精度,提出了一种基于两步正则化Gauss-Newton迭代算法的ECT图像重建方法。针对标准正则化Gauss-Newton迭代算法在图像重建中存在的不收敛问题,引入了两步迭代方法;改进了正则化矩阵,提高了解估计的精确度;考虑到Gauss-Newton算法对迭代初值的依赖性,加入了同伦算法。最后,进行仿真和静态实验,并与线性反投影(LBP)算法、Landweber算法、Tikhonov正则化算法进行对比。结果表明,该方法可有效提高图像重建精度。

Variational iteration method for solving the mechanism of the Equatorial Eastern Pacific El Nino-Southern Oscillation

A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered.

Variational iteration solving method for El Nio phenomenon atmospheric physics of nonlinear model

A class of E1 Niйo atmospheric physics oscillation model is considered. The E1 Niйo atmospheric physics oscillation is an abnormal phenomenon involved in the tropical Pacific ocean-atmosphere interactions. The conceptual oscillator model should consider the variations of both the eastern and western Pacific anomaly patterns. An E1 Niйo atmospheric physics model is proposed using a method for the variational iteration theory. Using the variational iteration method, the approximate expansions of the solution of corresponding problem are constructed. That is, firstly, introducing a set of functional and accounting their variationals, the Lagrange multiplicators are counted, and then the variational iteration is defined, finally, the approximate solution is obtained. From approximate expansions of the solution, the zonal sea surface temperature anomaly in the equatorial eastern Pacific and the thermocline depth anomaly of the sea-air oscillation for E1 Niйo atmospheric physics model can be analyzed. E1 Niйo is a very complicated natural phenomenon. Hence basic models need to be reduced for the sea-air oscillator and are solved. The variational iteration is a simple and valid approximate method.

ON THE MONOTONE CONVERGENCE OF THE PROJECTED ITERATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate of this method is investigated.

A two-grid algorithm based on Newton iteration for the stream function form of the Navier-Stokes equations

In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems （with same stiffness matrix but different right-hand side） on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.

A Distributed Newton Method for Processing Signals Defined on the Large-Scale Networks

In the graph signal processing(GSP)framework,distributed algorithms are highly desirable in processing signals defined on large-scale networks.However,in most existing distributed algorithms,all nodes homogeneously perform the local computation,which calls for heavy computational and communication costs.Moreover,in many real-world networks,such as those with straggling nodes,the homogeneous manner may result in serious delay or even failure.To this end,we propose active network decomposition algorithms to select non-straggling nodes(normal nodes)that perform the main computation and communication across the network.To accommodate the decomposition in different kinds of networks,two different approaches are developed,one is centralized decomposition that leverages the adjacency of the network and the other is distributed decomposition that employs the indicator message transmission between neighboring nodes,which constitutes the main contribution of this paper.By incorporating the active decomposition scheme,a distributed Newton method is employed to solve the least squares problem in GSP,where the Hessian inverse is approximately evaluated by patching a series of inverses of local Hessian matrices each of which is governed by one normal node.The proposed algorithm inherits the fast convergence of the second-order algorithms while maintains low computational and communication cost.Numerical examples demonstrate the effectiveness of the proposed algorithm.

Doubly Periodic Wave Solutions of Jaulent-Miodek Equations Using Variational Iteration Method Combined with Jacobian-function Method

One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.

A Unification of the Concepts of the Variational Iteration, Adomian Decomposition and Picard Iteration Methods;and a Local Variational Iteration Method

This paper compares the variational iteration method(VIM),the Adomian decomposition method(ADM)and the Picard iteration method(PIM)for solving a system of first o rder n onlinear o rdinary d ifferential e quations(ODEs).A unification of the concepts underlying these three methods is attempted by considering a very general iterative algorithm for VIM.It is found that all the three methods can be regarded as special cases of using a very general matrix of Lagrange multipliers in the iterative algorithm of VIM.The global variational iteration method is briefly reviewed,and further recast into a Local VIM,which is much more convenient and capable of predicting long term complex dynamic responses of nonlinear systems even if they are chaotic.

Iteration dependent interval based open‐closed‐loop iterative learning control for time varying systems with vector relative degree

For linear time varying(LTV)multiple input multiple output(MIMO)systems with vector relative degree,an open‐closed‐loop iterative learning control(ILC)strategy is developed in this article,where the time interval of operation is iteration dependent.To compensate the missing tracking signal caused by iteration dependent interval,the feedback control is introduced in ILC design.As the tracking signal of many continuous iterations is lost in a certain interval,the feedback control part can employ the tracking signal of current iteration for compensation.Under the assumption that the initial state vibrates around the desired initial state uniformly in mathematical expectation sense,the expectation of ILC tracking error can converge to zero as the number of iteration tends to infinity.Under the circumstance that the initial state varies around the desired initial state with a bound,as the number of iteration tends to infinity,the expectation of ILC tracking error can be driven to a bounded range,whose upper bound is proportional to the fluctuation.It is revealed that the convergence condition is dependent on the feed-forward control gains,while the feedback control can accelerate convergence speed by selecting appropriate feedback control gains.As a special case,the controlled system with integrated high relative degree is also addressed by proposing a simplified iteration dependent interval based open‐closed‐loop ILC method.Finally,the effectiveness of the developed iteration dependent interval based open‐closed‐loop ILC is illustrated by a simulation example with two cases on initial state.

Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method

How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations(NAEs).This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear terms.We decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear system.Through the maximal orthogonal projection concept,to minimize a merit function within a selected interval of splitting parameters,the optimal parameters can be quickly determined.In each step,a linear system is solved by the Gaussian elimination method,and the whole iteration procedure is convergent very fast.Several numerical tests show the high performance of the optimal split-linearization iterative method(OSLIM).

Solutions of the Duffin-Kemmer-Petiau equation in the presence of Hulthn potential in(1+2) dimensions for unity spin particles using the asymptotic iteration method

The relativistic Duffin-Kemmer-Petiau equation in the presence of Hulthen potential in （1 ＋2） dimensions for spin-one particles is studied. Hence, the asymptotic iteration method is used for obtaining energy eigenvalues and eigenfunctions.