Dynamical Comparison of Several Third-Order Iterative Methods for Nonlinear Equations

There are several ways that can be used to classify or compare iterative methods for nonlinear equations, for instance; order of convergence, informational efficiency, and efficiency index. In this work, we use another way, namely the basins of attraction of the method. The purpose of this study is to compare several iterative schemes for nonlinear equations. All the selected schemes are of the third-order of convergence and most of them have the same efficiency index. The comparison depends on the basins of attraction of the iterative techniques when applied on several polynomials of different degrees. As a comparison, we determine the CPU time (in seconds) needed by each scheme to obtain the basins of attraction, besides, we illustrate the area of convergence of these schemes by finding the number of convergent and divergent points in a selected range for all methods. Comparisons confirm the fact that basins of attraction differ for iterative methods of different orders, furthermore, they vary for iterative methods of the same order even if they have the same efficiency index. Consequently, this leads to the need for a new index that reflects the real efficiency of the iterative scheme instead of the commonly used efficiency index.     

一类混合型Lyapunov矩阵方程的对称正定解

本文研究了一类混合型Lyapunov矩阵方程的对称正定解问题。首先将此方程转化为等价的含参矩阵方程,然后运用矩阵分解和紧凸集上不动点定理,给出了方程具有对称正定解的一些必要和充分条件:其次建立两种求方程对称正定解的参数迭代算法,分析了迭代的收敛性及参数的选取方法,并指出这两种算法的适应性和特点;数值算例表明上述算法的可行性和有效性,并对比出两种迭代的敛速。     

Tibetan Sentiment Classification Method Based on Semi-Supervised Recursive Autoencoders

We apply the semi-supervised recursive autoencoders (RAE) model for the sentiment classification task of Tibetan short text, and we obtain a better classification effect. The input of the semi-supervised RAE model is the word vector. We crawled a large amount of Tibetan text from the Internet, got Tibetan word vectors by using Word2vec, and verified its validity through simple experiments. The values of parameter α and word vector dimension are important to the model effect. The experiment results indicate that when α is 0.3 and the word vector dimension is 60, the model works best. Our experiment also shows the effectiveness of the semi-supervised RAE model for Tibetan sentiment classification task and suggests the validity of the Tibetan word vectors we trained.     

Using a Lie-Group Adaptive Method for the Identification of a Nonhomogeneous Conductivity Function and Unknown Boundary Data

Only the left-boundary data of temperature and heat flux are used to estimate an unknown parameter function α(x) in T_t(x,t) = ∂(α(x)T_x)/∂x + h(x,t), as well as to recover the right-boundary data. When α(x) is given the above problem is a well-known inverse heat conduction problem (IHCP). This paper solves a mixed-type inverse problem as a combination of the IHCP and the problem of parameter identification, without needing to assume a function form of α(x) a priori, and without measuring extra data as those used by other methods. We use the one-step Lie-Group Adaptive Method (LGAM) for the semi-discretizations of heat conduction equation, respectively, in time domain and spatial domain to derive algebraic equations, which are used to solve α(x) through a few iterations. To test the stability of the present LGAM we also add a random noise in the initial data. When α(x) is identified, a sideways approach is employed to recover the unknown boundary data. The convergence speed and accuracy are examined by numerical examples.     

Hydrothermal processings to produce magnetic particulates

α-Fe₂O₃, γ-Fe₂O₃ , and α-FeOOH powders were used as starting materials to prepare barium hexaferrite hydrothermally. Since strontium hexaferrite and lanthanum-doped calcium hexaferrite have coercivities similar to that of barium ferrite, the hydrothermal synthesis of strontium hexaferrite and lanthanum-doped calcium hexaferrite were also explored. The reaction products obtained with the various starting materials are described. Electron micrographs showed that α-Fe₂O₃, γ-Fe₂O₃, or α-FeOOH dissolved in the solution first, and then barium hexaferrite nucleated and grew from the solution     

Convergence acceleration of iterative algorithms. Applications to thin shell analysis and Navier–Stokes equations

This work deals with the convergence acceleration of iterative nonlinear methods. Two convergence accelerating techniques are evaluated: the Modified Mininal Polynomial Extrapolation Method (MMPE) and the Padé approximants. The algorithms studied in this work are iterative correctors: Newton’s modified method, a high-order iterative corrector presented in Damil et al. (Commun Numer Methods Eng 15:701–708, 1999) and an original algorithm for vibration of viscoelastic structures. We first describe the iterative algorithms for the considered nonlinear problems. Secondly, the two accelerating techniques are presented. Finally, through several numerical tests from the thin shell theory, Navier–Stokes equations and vibration of viscoelastic shells, the advantages and drawbacks of each accelerating technique is discussed.     

Continuous analogue to iterative optimization for PDE-constrained inverse problems

The parameters of many physical processes are unknown and have to be inferred from experimental data. The corresponding parameter estimation problem is often solved using iterative methods such as steepest descent methods combined with trust regions. For a few problem classes also continuous analogues of iterative methods are available. In this work, we expand the application of continuous analogues to function spaces and consider PDE (partial differential equation)-constrained optimization problems. We derive a class of continuous analogues, here coupled ODE (ordinary differential equation)–PDE models, and prove their convergence to the optimum under mild assumptions. We establish sufficient bounds for local stability and convergence for the tuning parameter of this class of continuous analogues, the retraction parameter. To evaluate the continuous analogues, we study the parameter estimation for a model of gradient formation in biological tissues. We observe good convergence properties, indicating that the continuous analogues are an interesting alternative to state-of-the-art iterative optimization methods.     

An iterative method for an inverse source problem of time-fractional diffusion equation

We propose an iteration method to recover a space-dependent source for a time-fractional diffusion equation from the final measurement. Based on the conditional stability of the inverse problem, we prove the convergence of the iterative regularization method under the a priori parameter choice rule and the a posteriori parameter choice rule, respectively. Numerical examples in one dimension and two dimension are given to validate the effectiveness of the presented method.     

A Quasi-Boundary Semi-Analytical Method for Backward in Time Advection-Dispersion Equation

In this paper, we take the advantage of an analytical method to solve the advection-dispersion equation (ADE) for identifying the contamination problems. First, the Fourier series expansion technique is employed to calculate the concentration field C(x, t) at any time t< T. Then, we consider a direct regularization by adding an extra term αC(x,0) on the final condition to carry off a second kind Fredholm integral equation. The termwise separable property of the kernel function permits us to transform itinto a two-point boundary value problem. The uniform convergence and error estimate of the regularized solution C^α(x,t) are provided and a strategy to select the regularized parameter is suggested. The solver used in this work can recover the spatial distribution of the groundwater contaminant concentration. Several numerical examples are examined to show that the new approach can retrieve all past data very well and is good enough to cope with heterogeneous parameters’ problems, even though the final data are noised seriously.     

The development of <100> texture in Fe-Cr-Co-Mo permanentmagnet alloys

The cold-rolled and recrystallization textures of Fe-Cr-Co-Mo permanent magnet alloys are described. The studied composition is Fe-30%Cr-15%Co-3%Mo (in wt.%). The cold-rolled texture can be considered to be {111}<110>, {111}<112>, {100}<110>, and {211}<110>, while the recrystallization texture can be considered to be {111}<100>, {110}<112>, {211}<110>, and {110}<110>. The secondary recrystallization is caused by heat-treating the alloys in the sequence of α, α+γ, α+γ+σ, α phase region. This results in a favorable texture of {110}<110> and <100> direction, aligning along the transverse direction (TD) of the strips. The best magnetic properties obtained in this study were 1.2 T (12.0 kG), iH _c=82.0 kAm^-1 (1025 Oe), and (BH)max= 60.8 kJm^-3 (7.6 MGOe) with TD alloys     

Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory

This paper investigates the nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory and Timoshenko beam theory. The piezoelectric nanobeam is subjected to an applied voltage and a uniform temperature change. The nonlinear governing equations and boundary conditions are derived by using the Hamilton principle and discretized by using the differential quadrature (DQ) method. A direct iterative method is employed to determine the nonlinear frequencies and mode shapes of the piezoelectric nanobeams. A detailed parametric study is conducted to study the influences of the nonlocal parameter, temperature change and external electric voltage on the size-dependent nonlinear vibration characteristics of the piezoelectric nanobeams.     

一种超松弛原始对偶不动点算法及其应用

近年来,关于两个凸函数和的优化问题受到极大关注,其中一凸函数可微且其梯度满足 Lipschitz 连续性,另一凸函数包含有界线性算子。提出一种超松弛原始对偶不动点算法求解这一类问题,相比于原始对偶不动点算法,所提算法扩展了松弛参数的选择范围。通过定义合适的范数,运用非扩张算子不动点理论,证明所提迭代算法的收敛性,并证明算法的遍历收敛率。在对目标函数一些强的条件下,证明算法具有全局线性收敛率。最后,为验证算法的有效性和优越性,将所提算法运用于求解全变分图像复原模型,数值结果表明,选择松弛参数大于 $1$ (即超松弛) 的原始对偶不动点算法比松弛参数小于 $1$ 时算法收敛更快。     

The Global Nonlinear Galerkin Method for the Solution of von Karman Nonlinear Plate Equations: An Optimal & Faster Iterative Method for the Direct Solution of Nonlinear Algebraic Equations F(x) = 0, using x<sup style='margin-left:-6.5px'>·</sup> = λ[αF + (1 - α)B^TF]

The application of the Galerkin method, using global trial functions which satisfy the boundary conditions, to nonlinear partial differential equations such as those in the von Karman nonlinear plate theory, is well-known. Such an approach using trial function expansions involving multiple basis functions, leads to a highly coupled system of nonlinear algebraic equations (NAEs). The derivation of such a system of NAEs and their direct solutions have hitherto been considered to be formidable tasks. Thus, research in the last 40 years has been focused mainly on the use of local trial functions and the Galerkin method, applied to the piecewise linear system of partial differential equations in the updated or total Lagrangean reference frames. This leads to the so-called tangent-stiffness finite element method. The piecewise linear tangent-stiffness finite element equations are usually solved by an iterative Newton-Raphson method, which involves the inversion of the tangent-stiffness matrix during each iteration. However, the advent of symbolic computation has made it now much easier to directly derive the coupled system of NAEs using the global Galerkin method. Also, methods to directly solve the NAEs, without inverting the tangent-stiffness matrix during each iteration, and which are faster and better than the Newton method are slowly emerging. In a previous paper [Dai, Paik and Atluri (2011a)], we have presented an exponentially convergent scalar homotopy algorithm to directly solve a large set of NAEs arising out of the application of the global Galerkin method to von Karman plate equations. While the results were highly encouraging, the computation time increases with the increase in the number of NAEs-the number of coupled NAEs solved by Dai, Paik and Atluri (2011a) was of the order of 40. In this paper we present a much improved method of solving a larger system of NAEs, much faster. If F(x) = 0 [F_i(x_j) = 0] is the system of NAEs governing the modal amplitudes x_j [j = 1, 2...N], for large N, we recast the NAEs into a system of nonlinear ODEs: x^· = λ[αF + (1 - α)B^TF], where λ and α are scalars, and B_ij = ∂F_i / ∂x_j. We derive a purely iterative algorithm from this, with optimum value for λ and α being determined by keeping x on a newly defined invariant manifold [Liu and Atluri (2011b)]. Several numerical examples of nonlinear von Karman plates, including the post-buckling behavior of plates with initial imperfections are presented to show that the present algorithms for directly solving the NAEs are several orders of magnitude faster than those in Dai, Paik and Atluri (2011a). This makes the resurgence of simple global Galerkin methods, as alternatives to the finite element method, to directly solve nonlinear structural mechanics problems without piecewise linear formulations, entirely feasible.     

基于线激光传感器的工件尺寸测量系统的误差补偿方法

提出了一种基于线激光传感器的工件尺寸测量系统的误差补偿方法,利用坐标系投影和图像处理技术进行误差补偿。设定传感器坐标系OM-XMYMZM和设备坐标系O-XYZ,分析坐标轴夹角φ、δ、γ对工件尺寸坐标值X、Y、Z的误差,建立了基于φ、δ、γ在XOY、YOZ、XOZ平面上的投影角α、β、θ的误差补偿模型。利用图像处理技术测得α、β、θ,计算经过误差补偿的工件尺寸坐标值X′、Y′、Z′。对尺寸100mm×100mm×10mm的长方体工件进行测量实验,分别测量了长度、宽度、圆心距、圆直径、圆线距、台阶高度。测量结果表明:经误差补偿后的工件尺寸测量误差在40μm以内,优于未补偿前的520μm;均方根误差低于40μm,优于未补偿前的580μm。其中,圆心距误差补偿效果最显著,测量误差减小了560μm;圆直径误差补偿效果最不明显,测量误差减小了10μm。     

An accelerated iterative method for the dynamics of constrained multibody systems

An accelerated iterative method is suggested for the dynamic analysis of multibody systems consisting of interconnected rigid bodies. The Lagrange multipliers associated with the kinematic constraints are iteratively computed by the monotone reduction of the constraint error vector, and the resulting equations of motion are easily time-integrated by a well established ODE technique. The velocity and acceleration constraints as well as the position constraints are made to be satisfied at the joints at each time step. Exact solution is obtained without the time demanding procedures such as selection of the independent coordinates, decomposition of the constraint Jacobian matrix, and Newton Raphson iterations. An acceleration technique is employed for the faster convergence of the iterative scheme and the convergence analysis of the proposed iterative method is presented. Numerical solutions for the verification problems are presented to demonstrate the efficiency and accuracy of the suggested technique.     

Trajectory optimization of spacecraft high-thrust orbit transfer using a modified evolutionary algorithm

This article introduces a new method to optimize finite-burn orbital manoeuvres based on a modified evolutionary algorithm. Optimization is carried out based on conversion of the orbital manoeuvre into a parameter optimization problem by assigning inverse tangential functions to the changes in direction angles of the thrust vector. The problem is analysed using boundary delimitation in a common optimization algorithm. A method is introduced to achieve acceptable values for optimization variables using nonlinear simulation, which results in an enlarged convergence domain. The presented algorithm benefits from high optimality and fast convergence time. A numerical example of a three-dimensional optimal orbital transfer is presented and the accuracy of the proposed algorithm is shown.     

ON OPTIMAL ARCHGRIDS

Treated in this work is the optimality of flexureless orthogonal archgrids. Various features of fully stressed arches and archgrids are highlighted and an efficient and simple iterative method for obtaining the optimal solution is proposed. Considering the load proportions carried by arches as independent variables, the optimality of the archgrid is ensured by imposing the conditions of equal elevation at nodal intersections and individual optimality of component arches. Computational effort, is reduced by using a simple updating procedure during repetitive calculations. The convergence of the solution is aptly tested by the convergence of elevations and compressive forces in arches. It is shown that the optimal solutions for some skew archgrid systems can be obtained by solving equivalent orthogonal archgrids.     

Iterative algorithm for optimal fiducials under weak perspective projection

In previous work, we designed space fiducials with the aim of making camera pose determination as noise‐insensitive as possible. These fiducials turned out to be sets of points that formed concentric regular polyhedra. Here, we apply an idea of Dementhon and Davis and test and analyze an iterative linear algorithm in conjunction with our optimal fiducials to increase the accuracy of the computed camera pose. We also analyze under what circumstances this iterative algorithm is guaranteed to converge to the correct solution. Comprehensive computer simulations illustrate the behavior of the algorithm and the degree of improvement in pose determination in case of convergence. © 2009 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 19, 27–36, 2009     

Application of a genetic algorithm: near optimal estimation of the rate and equilibrium constants of complex reaction mechanisms

In iterative non-linear least-squares fitting, the reliable estimation of initial parameters that lead to convergence to the global optimum can be difficult. Irrespective of the algorithm used, poor parameter estimates can lead to abortive divergence if initial guesses are far from the true values or in rare cases convergence to a local optimum. For determination of the parameters of complex reaction mechanisms, where often little is known about what value these parameters should take, the task of determining good initial estimates can be time consuming and unreliable. In this contribution, the methodology of applying a genetic algorithm (GA) to the task of determining initial parameter estimates that lie near the global optimum is explained. A generalised genetic algorithm was implemented according to the methodology and the results of its application are also given. The parameter estimates obtained were then used as the starting parameters for a gradient search method, which quickly converged to the global optimum. The genetic algorithm was successfully applied to both simulated kinetic measurements where the reaction mechanism contained one equilibrium constant and two rate constants to be fitted, and to kinetic measurements of the complexation of Cu²⁺ by 1,4,8,11-tetraazacyclotetradecane where two equilibrium and two rate constants were fitted. The implementation of the algorithm is such that it can be generally applied to any reaction mechanism that can be expressed by standard chemistry notation. The control parameters of the algorithm can be varied through a simple user interface to account for parameter range and the number of parameters involved.     

应用最优化方法辨识指数自回归模型

本文研究应用非线性最优化原理辨识指数自回归模型的方法，以Ｎｅｗｔｏｎ一维搜索和线性回归法为基础，提出一种改进的坐标轮换算法，交替地估计模型的线性和非线性参数。实例表明，该算法对指数自回归模型具有良好的辨识效果。