Stability and convergence of F iterative scheme with an application to the fractional differential equation

In this paper, we prove that F iterative scheme is almost stable for weak contractions. Further, we prove convergence results for weak contractions as well as for generalized non-expansive mappings due to Hardy and Rogers via F iterative scheme. We also prove that F iterative scheme converges faster than the some known iterative schemes for weak contractions. An illuminative numerical example is formulated to support our assertion. Finally, utilizing our main result the solution of nonlinear fractional differential equation is approximated.

     

Convergence Properties of Iterative Learning Control Processes in the Sense of the Lebesgue‐P Norm

This paper addresses the convergence issue of first‐order and second‐order PD‐type iterative learning control schemes for a type of partially known linear time‐invariant systems. By taking advantage of the generalized Young inequality of convolution integral, the convergence is analyzed in the sense of the Lebesgue‐p norm and the convergence speed is also discussed in terms of Q_p factors. Specifically, we find that: (1) the sufficient condition on convergence is dominated not only by the derivative learning gains, along with the system input and output matrices, but also by the proportional learning gains and the system state matrix; (2) the strictly monotone convergence is guaranteed for the first‐order rule while, in the case of the second‐order scheme, the monotonicity is maintained after some finite number of iterations; and (3) the iterative learning process performed by the second‐order learning scheme can be Q_p‐faster, Q_p‐equivalent, or Q_p‐slower than the iterative learning process manipulated by the first‐order rule if the learning gains are appropriately chosen. To manifest the validity and effectiveness of the results, several numerical simulations are conducted. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A fast higher degree total variation minimization method for image restoration

Based on the spectral decomposition theory, this paper presents a unified analysis of higher degree total variation (HDTV) model for image restoration. Under this framework, HDTV is reinterpreted as a family of weighted L₁–L₂ mixed norms of image derivatives. Due to the equivalent formulation of HDTV, we construct a modified functional for HDTV-based image restoration. Then, the minimization of the modified functional can be decoupled into two separate sub-problems, which correspond to the deblurring and denoising. Thus, we design a fast and efficient image restoration algorithm using an iterative Wiener deconvolution with fast projected gradient denoising (IWD-FPGD) scheme. Moreover, we show the convergence of the proposed IWD-FPGD algorithm for the special case of second-degree total variation. Finally, the systematic performance comparisons of the proposed IWD-FPGD algorithm demonstrate the effectiveness in terms of peak signal-to-noise ratio, structural similarity and convergence rate.     

Convergence of P-regular splitting iterative methods for non-Hermitian positive semidefinite linear systems

In this paper, we study the convergence of P-regular splitting iterative methods for singular and non-singular non-Hermitian positive semidefinite linear systems, which generalize the known results. Some numerical experiments are performed to illustrate the convergence results.     

Multipoint efficient iterative methods and the dynamics of Ostrowski's method

ABSTRACT

In this work, we introduce a modification into the technique, presented in A. Cordero, J.L. Hueso, E. Martínez, and J.R. Torregrosa [Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett. 25 (2012), pp. 2369–2374], that increases by two units the convergence order of an iterative method. The main idea is to compose a given iterative method of order p with a modification of Newton's method that introduces just one evaluation of the function, obtaining a new method of order p+2, avoiding the need to compute more than one derivative, so we improve the efficiency index in the scalar case. This procedure can be repeated n times, with the same approximation to the derivative, obtaining new iterative methods of order p+2n. We perform different numerical tests that confirm the theoretical results. By applying this procedure to Newton's method one obtains the well known fourth order Ostrowski's method. We finally analyse its dynamical behaviour on second and third degree real polynomials.     

Dynamics of a fifth-order iterative method

In this paper, we study the dynamical behaviour of a two-point iterative method with order of convergence five to solve nonlinear equations in the complex plane. In fact, we complement the dynamical study started in previous works with a more systematic analysis for polynomials with at most two different roots and different multiplicities. In addition, we characterize some polynomials of degree greater or equal to 4, such that the related methods are not generally convergent. We also analyse the degrees of the rational functions associated with two-point methods when they are applied to polynomials of degree n, showing their dependence on n ² and how this fact considerably complicates the dynamical study.     

Unconditionally positivity and boundedness preserving schemes for a FitzHugh–Nagumo equation

In this paper, a semi-explicit scheme is constructed for the space-independent FitzHugh–Nagumo equation. Qualitative stability analysis shows that the semi-explicit scheme is dynamically consistent with the space independent equation. Then, the semi-explicit scheme is extended to construct a new finite difference scheme for the full FitzHugh–Nagumo equation in one- and two-space dimensions, respectively. According to the theory of M-matrices, it is proved that these new schemes are able to preserve the positivity and boundedness of solutions of the corresponding equations for arbitrary step sizes. The consistency and numerical stability of these schemes is also analysed. Combined with the property of the strictly diagonally dominant matrix, the convergence of these schemes is established. Numerical experiments illustrate our results and display the advantages of our schemes in comparison to some other schemes.     

A Non-monotone Line Search Algorithm for?Unconstrained Optimization

The monotone line search schemes have been extensively used in the iterative methods for solving various optimization problems. It is well known that the non-monotone line search technique can improve the likelihood of finding a global optimal solution and the numerical performance of the methods, especially for some difficult nonlinear problems. The traditional non-monotone line search approach requires that a maximum of recent function values decreases. In this paper, we propose a new line search scheme which requires that a convex combination of recent function values decreases. We apply the new line search technique to solve unconstrained optimization problems, and show the proposed algorithm possesses global convergence and R-linear convergence under suitable assumptions. We also report the numerical results of the proposed algorithm for solving almost all the unconstrained testing problems given in CUTEr, and give numerical comparisons of the proposed algorithm with two famous non-monotone methods.     

P-剖分方法的收敛性

p-剖分方法是一种高效的stationary subdivision方法。本文利用一种特殊的控制多边形δ－控制多边形给出了p-剖分方法一致收敛的一个充分条件。     

Nonlinear iterative methods for solving the split common null point problem in Banach spaces

In this work, we study the split common null point problem in the framework of Banach spaces. We propose an iterative scheme for solving the problem and then prove strong convergence theorem of the sequences generated by our iterative scheme under suitable conditions. We finally provide some numerical examples to support the main theorem.     

Analysis of three stabilized finite volume iterative methods for the steady Navier–Stokes equations

In this work, three stabilized finite volume iterative schemes for the stationary Navier–Stokes equations are considered. Under the finite volume discretization at each iterative step, the iterative scheme I consists in solving the steady Stokes problem, iterative scheme II consists in solving the stationary linearized Navier–Stokes equations and iterative scheme III consists in solving the steady Oseen equations, respectively. We discuss the stabilities and convergence of three iterative methods. The iterative schemes I and II are stable and convergent under some strong uniqueness conditions, while iterative scheme III is unconditionally stable and convergent under the uniqueness condition. Finally, some numerical results are presented to verify the performance of these iterative schemes.     

Iterative learning control for nonlinear differential inclusion systems

In this paper, we propose an iterative learning control strategy to track a desired trajectory for a class of uncertain systems governed by nonlinear differential inclusions. By imposing Lipschitz continuous condition on a set‐valued mapping described by a closure of the convex hull of a set and using D‐type and PD‐type updating laws with initial iterative learning, we establish the iterative learning process and give a new convergence analysis with the help of Steiner‐type selector. Finally, numerical examples are provided to verify the effectiveness of the proposed method with suitable selection of set‐valued mappings. An application to the speed control of robotic fish is also given.     

数据丢失对迭代学习控制的影响分析

针对一类线性系统,分析数据丢失对迭代学习控制算法的影响.首先基于lifting方法给出跟踪误差渐近收敛和单调收敛的条件,并分析收敛速度与数据丢失率的关系,结果表明收敛速度随着数据丢失程度的增加而变慢.其次,为抑制迭代变化扰动的影响,给出一种存在数据丢失时的鲁棒迭代学习控制器设计方法,并将控制器设计问题转化为求取线性矩阵不等式的可行解.仿真示例验证了理论分析的结果以及鲁棒迭代学习控制算法的有效性.

     

Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints

The efficiency of the classic alternating direction method of multipliers has been exhibited by various applications for large-scale separable optimization problems, both for convex objective functions and for nonconvex objective functions. While there are a lot of convergence analysis for the convex case, the nonconvex case is still an open problem and the research for this case is in its infancy. In this paper, we give a partial answer on this problem. Specially, under the assumption that the associated function satisfies the Kurdyka–?ojasiewicz inequality, we prove that the iterative sequence generated by the alternating direction method converges to a critical point of the problem, provided that the penalty parameter is greater than 2L, where L is the Lipschitz constant of the gradient of one of the involved functions. Under some further conditions on the problem's data, we also analyse the convergence rate of the algorithm.     

On the P-type and Newton-type ILC schemes for dynamic systems with non-affine-in-input factors

In this paper, P-type learning scheme and Newton-type learning scheme are proposed for quite general nonlinear dynamic systems with non-affine-in-input factors. Using the contraction mapping method, it is shown that both schemes can achieve asymptotic convergence along learning repetition horizon. In order to quantify and evaluate the learning performance, new indices—Q-factor and Q-order—are introduced in particular to evaluate the learning convergence speed. It is shown that the P-type iterative learning scheme has a linear convergence order with limited learning convergence speed under system uncertainties. On the other hand, if more of system information such as the input Jacobian is available, Newton-type iterative learning scheme, which is originated from numerical analysis, can greatly speed up the learning convergence speed. The effectiveness of the two learning control methods are demonstrated through a switched reluctance motor system.     

Some variants of the Chebyshev–Halley family of methods with fifth order of convergence

In this paper we present some techniques for constructing high-order iterative methods in order to approximate the zeros of a non-linear equation f(x)=0, starting from a well-known family of cubic iterative processes. The first technique is based on an additional functional evaluation that allows us to increase the order of convergence from three to five. With the second technique, we make some changes aimed at minimizing the calculus of inverses. Finally, looking for a better efficiency, we eliminate terms that contribute to the error equation from sixth order onwards.

The paper contains a comparative study of the asymptotic error constants of the methods and some theoretical and numerical examples that illustrate the given results. We also analyse the efficiency of the aforementioned methods, by showing some numerical examples with a set of test functions and by using adaptive multi-precision arithmetic in the computation.     

非线性迭代学习控制问题的延拓修正牛顿法

对于非线性迭代学习控制问题,提出基于延拓法和修正Newton法的具有全局收敛性的迭代学习控制新方法.由于一般的Newton型迭代学习控制律都是局部收敛的,在实际应用中有很大局限性.为拓宽收敛范围,该方法将延拓法引入迭代学习控制问题,提出基于同伦延拓的新的Newton型迭代学习控制律,使得初始控制可以较为任意的选择.新的迭代学习控制算法将求解过程分成N个子问题,每个子问题由换列修正Newton法利用简单的递推公式解出.本文给出算法收敛的充分条件,证明了算法的全局收敛性.该算法对于非线性系统迭代学习控制具有全局收敛和计算简单的优点.     

A numerical method for determining monotonicity and convergence rate in iterative learning control

In iterative learning control (ILC), a lifted system representation is often used for design and analysis to determine the convergence rate of the learning algorithm. Computation of the convergence rate in the lifted setting requires construction of large N×N matrices, where N is the number of data points in an iteration. The convergence rate computation is O(N²) and is typically limited to short iteration lengths because of computational memory constraints. As an alternative approach, the implicitly restarted Arnoldi/Lanczos method (IRLM) can be used to calculate the ILC convergence rate with calculations of O(N). In this article, we show that the convergence rate calculation using IRLM can be performed using dynamic simulations rather than matrices, thereby eliminating the need for large matrix construction. In addition to faster computation, IRLM enables the calculation of the ILC convergence rate for long iteration lengths. To illustrate generality, this method is presented for multi-input multi-output, linear time-varying discrete-time systems.     

Modified alternating direction-implicit iteration method for linear systems from the incompressible Navier–Stokes equations

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, Ran and Yuan [On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems, Appl. Math. Comput. 217 (2010), pp. 3050–3068] presented the block symmetric successive over-relaxation (BSSOR) and the modified BSSOR iteration methods based on the special structures of the coefficient matrices. In this study, we present the modified alternating direction-implicit (MADI) iteration method for solving the linear systems. Under suitable conditions, we establish convergence theorems for the MADI iteration method. In addition, the optimal parameter involved in the MADI iteration method is estimated in detail. Numerical experiments show that the MADI iteration method is a feasible and effective iterative solver.     

Some numerical methods in elastoplasticity

In this paper we study some methods for the numerical resolution of the deformation equation in elasto-plastic media, subjected to Hencky's law. We describe an iterative algorithm and show its convergence to the stress field. Then, its performances are given in a typical example with Von Mises plasticity convex set.P ₁-Lagrange Finite Element scheme is implemented by usingModulef's Library for numerical computation.