Multilevel Iteration Methods for Solving Linear Operator Equations of the First Kind

In this paper we develop two multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for linear ill-posed problems. The two algorithms and their convergence analyses are presented in an abstract framework.

Integration of the Coupled Orbit-Attitude Dynamics Using Modified Chebyshev-Picard Iteration Methods

This paper presents Modified Chebyshev-Picard Iteration(MCPI)methods for long-term integration of the coupled orbit and attitude dynamics.Although most orbit predictions for operational satellites have assumed that the attitude dynamics is decoupled from the orbit dynamics,the fully coupled dynamics is required for the solutions of uncontrolled space debris and space objects with high area-to-mass ratio,for which cross sectional area is constantly changing leading to significant change on the solar radiation pressure and atmospheric drag.MCPI is a set of methods for solution of initial value problems and boundary value problems.The methods refine an orthogonal function approximation of long-time-interval segments of state trajectories iteratively by fusing Chebyshev polynomials with the classical Picard iteration and have been applied to multiple challenging aerospace problems.Through the studies on integrating a torque-free rigid body rotation and a long-term integration of the coupled orbit-attitude dynamics through the effect of solar radiation pressure,MCPI methods are shown to achieve several times speedup over the Runge-Kutta 7(8)methods with several orders of magnitudes of better accuracy.MCPI methods are further optimized by integrating the decoupled dynamics at the beginning of the iteration and coupling the full dynamics when the attitude solutions and orbit solutions are converging during the iteration.The approach of decoupling and then coupling during iterations provides a unique and promising perspective on the way to warm start the solution process for the longterm integration of the coupled orbit-attitude dynamics.Furthermore,an attractive feature of MCPI in maintaining the unity constraint for the integration of quaternions within machine accuracy is illustrated to be very appealing.

ON HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR CONTINUOUS SYLVESTER EQUATIONS

We present a Hermitian and skew-Herrnitian splitting （HSS） iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semi- definite matrices. The unconditional convergence of the HSS iteration method is proved and an upper bound on the convergence rate is derived. Moreover, to reduce the computing cost, we establish an inexact variant of the HSS iteration method and analyze its convergence property in detail. Numerical results show that the HSS iteration method and its inexact variant are efficient and robust solvers for this class of continuous Sylvester equations.

TWO ITERATION METHODS FOR SOLVING LINEAR ALGEBRAIC SYSTEMS WITH LOW ORDER MATRIX A AND HIGH ORDER MATRIX B:Y=(AB)Y+Ф

This paper presents optimum an one-parameter iteration (OOPI) method and a multi-parameter iteration direct (MPID) method for efficiently solving linear algebraic systems with low order matrix A and high order matrix B: Y = (A B)Y + Φ. On parallel computers (also on serial computer) the former will be efficient, even very efficient under certain conditions, the latter will be universally very efficient.

THE GENERALIZED LOCAL HERMITIAN AND SKEW-HERMITIAN SPLITTING ITERATION METHODS FOR THE NON-HERMITIAN GENERALIZED SADDLE POINT PROBLEMS

For large and sparse saddle point problems, Zhu studied a class of generalized local Hermitian and skew-Hermitian splitting iteration methods for non-Hermitian saddle point problem [M.-Z. Zhu, Appl. Math. Comput. 218 （2012） 8816-8824 ]. In this paper, we further investigate the generalized local Hermitian and skew-Hermitian splitting （GLHSS） iteration methods for solving non-Hermitian generalized saddle point problems. With different choices of the parameter matrices, we derive conditions for guaranteeing the con- vergence of these iterative methods. Numerical experiments are presented to illustrate the effectiveness of our GLHSS iteration methods as well as the preconditioners.

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.

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.

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.

Existence and iteration of monotone positive solutions for a third-order two-point boundary value problem

The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.

A Family of Fifth-order Iterative Methods for Solving Nonlinear Equations

In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.

New Optimal Newton-Householder Methods for Solving Nonlinear Equations and Their Dynamics

The classical iterative methods for finding roots of nonlinear equations,like the Newton method,Halley method,and Chebyshev method,have been modified previously to achieve optimal convergence order.However,the Householder method has so far not been modified to become optimal.In this study,we shall develop two new optimal Newton-Householder methods without memory.The key idea in the development of the new methods is the avoidance of the need to evaluate the second derivative.The methods fulfill the Kung-Traub conjecture by achieving optimal convergence order four with three functional evaluations and order eight with four functional evaluations.The efficiency indices of the methods show that methods perform better than the classical Householder’s method.With the aid of convergence analysis and numerical analysis,the efficiency of the schemes formulated in this paper has been demonstrated.The dynamical analysis exhibits the stability of the schemes in solving nonlinear equations.Some comparisons with other optimal methods have been conducted to verify the effectiveness,convergence speed,and capability of the suggested methods.

ASYNCHRONOUS RELAXED ITERATIVE METHODS FOR SOLVING LINEAR SYSTEMS OF EQUATIONS

In this paper, the asynchronous versions of classical iterative methods for solving linear systems of equations are considered. Sufficient conditions for convergence of asynchronous relaxed processes are given for H-matrix by which nor only the requirements of [3] on coefficient matrix are lowered, but also a larger region of convergence than that in [3] is obtained.

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.

VISCOSITY ITERATIVE METHODS FOR COMMON FIXED POINTS OF TWO NONEXPANSIVE MAPPINGS WITHOUT COMMUTATIVITY ASSUMPTION IN HILBERT SPACES

In this article, we introduce a new viscosity iterative method for two nonexpansive mappings in Hilbert spaces. We also prove, without commutativity assumption, that the iterates converge to a common fixed point of the mappings which solves some variational inequality. The results presented extend the corresponding results of Shimizu and Takahashi IT. Shimizu, W. Takahashi, Strong convergence to common fixed point of families of nonexpansive mappings, J. Math. Anal. Appl. 211 （1997）, 71-83], and Yao and Chen [Y. Yao, R. Chert, Convergence to common fixed points of average mappings without commutativity assumption in Hilbert spaces, Nonlinear Analysis 67（2007）, 1758-1763].

IMPLICIT ITERATIVE METHODS WITH VARIABLE CONTROL PARAMETERS FOR ILL-POSED OPERATOR EQUATIONS

This paper discusses a kind of implicit iterative methods with some variable parameters, which are called control parameters, for solving ill-posed operator equations. The theoretical results show that the new methods always lead to optimal convergence rates and have some other important features, especially the methods can be implemented parallelly.

Geometric interpretation of several classical iterative methods for linear system of equations and diverse relaxation parameter of the SOR method

Two kinds of iterative methods are designed to solve the linear system of equations, we obtain a new interpretation in terms of a geometric concept. Therefore, we have a better insight into the essence of the iterative methods and provide a reference for further study and design. Finally, a new iterative method is designed named as the diverse relaxation parameter of the SOR method which, in particular, demonstrates the geometric characteristics. Many examples prove that the method is quite effective.

Iterative Methods for Solving the Nonlinear Balance Equation with Optimal Truncation

Two types of existing iterative methods for solving the nonlinear balance equation(NBE)are revisited.In the first type,the NBE is rearranged into a linearized equation for a presumably small correction to the initial guess or the subsequent updated solution.In the second type,the NBE is rearranged into a quadratic form of the absolute vorticity with the positive root of this quadratic form used in the form of a Poisson equation to solve NBE iteratively.The two methods are rederived by expanding the solution asymptotically upon a small Rossby number,and a criterion for optimally truncating the asymptotic expansion is proposed to obtain the super-asymptotic approximation of the solution.For each rederived method,two iterative procedures are designed using the integral-form Poisson solver versus the over-relaxation scheme to solve the boundary value problem in each iteration.Upon testing with analytically formulated wavering jet flows on the synoptic,sub-synoptic and meso-αscales,the iterative procedure designed for the first method with the Poisson solver,named M1a,is found to be the most accurate and efficient.For the synoptic wavering jet flow in which the NBE is entirely elliptic,M1a is extremely accurate.For the sub-synoptic wavering jet flow in which the NBE is mostly elliptic,M1a is sufficiently accurate.For the meso-αwavering jet flow in which the NBE is partially hyperbolic so its boundary value problem becomes seriously ill-posed,M1a can effectively reduce the solution error for the cyclonically curved part of the wavering jet flow,but not for the anti-cyclonically curved part.

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.

Downward continuation of airborne geomagnetic data based on two iterative regularization methods in the frequency domain

Downward continuation is a key step in processing airborne geomagnetic data. However,downward continuation is a typically ill-posed problem because its computation is unstable; thus, regularization methods are needed to realize effective continuation. According to the Poisson integral plane approximate relationship between observation and continuation data, the computation formulae combined with the fast Fourier transform（FFT）algorithm are transformed to a frequency domain for accelerating the computational speed. The iterative Tikhonov regularization method and the iterative Landweber regularization method are used in this paper to overcome instability and improve the precision of the results. The availability of these two iterative regularization methods in the frequency domain is validated by simulated geomagnetic data, and the continuation results show good precision.

IMPROVED HOMOTOPY ITERATION METHOD AND APPLIED TO THE NINE-POINT PATH SYNTHESIS PROBLEM FOR FOUR-BAR LINKAGES

A new algorithm called homotopy iteration method based on the homotopy function is studied and improved. By the improved homotopy iteration method, Polynomial systems with high Order and deficient can be solved fast and efficiently comparing to the original homotopy iteration method. Numerical examples for the ninepoint path synthesis of four-bar linkages show the advantages and efficiency of the improved homotopy iteration method.