Meromorphic iterative roots of linear fractional functions

Iterative root problem can be regarded as a weak version of the problem of embedding a homeomorphism into a flow. There are many results on iterative roots of monotone functions. However, this problem gets more diffcult in non-monotone cases. Therefore, it is interesting to find iterative roots of linear fractional functions (abbreviated as LFFs), a class of non-monotone functions on ℝ. In this paper, iterative roots of LFFs are studied on ℂ. An equivalence between the iterative functional equation for non-constant LFFs and the matrix equation is given. By means of a method of finding matrix roots, general formulae of all meromorphic iterative roots of LFFs are obtained and the precise number of roots is also determined in various cases. As applications, we present all meromorphic iterative roots for functions z and 1/z. This work was supported by the Youth Fund of Sichuan Provincial Education Department of China (Grant No. 07ZB042)     

Banach空间中m-d增生映射零点的强弱收敛定理

本文研究了m-d增生映射的零点以及有限个m-d增生映射公共零点的迭代设计问题.利用Lyapunov泛函与广义f投影映射等技巧,在Banach空间中,证明了迭代序列强收敛或弱收敛到m-d增生映射的零点或有限个m-d增生映射的公共零点.与以往的相关研究工作相比,迭代设计中考虑了误差项、迭代格式被简化、限定条件被削弱.     

A numerical method of nondifferentiable convex optimization with constraints

An iterative process is examined for minimizing a convex nondifferentiable functional on a convex closed set in a real Hilbert space. Convergence of the proposed process is proved. A two-sided bound on the optimal functional value is given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 57, pp. 124–131, 1985.     

On set-valued solution of an <Emphasis Type="Italic">n</Emphasis>-order iterative functional equation

A n-order iterative functional equation is considered for set-value function in a nonempty convex compact subset D in Banach space. A result on the existence and uniqueness of its solution is presented without monotonicity. Besides, continuous dependence of the solution upon the given function is obtained.     

Algorithm for solving a new class of general mixed variational inequalities in Banach spaces

In this paper, a new concept of η-proximal mapping for a proper subdifferentiable functional (which may not be convex) on a Banach space is introduced. An existence and Lipschitz continuity of the η-proximal mapping are proved. By using properties of the η-proximal mapping, a new class of general mixed variational inequalities is introduced and studied in Banach spaces. An existence theorem of solutions is established and a new iterative algorithm for solving the general mixed variational inequality is suggested. A convergence criteria of the iterative sequence generated by the new algorithm is also given.     

Numerical solution of the zero-head seepage problem by discretization of the variational inequality

The problem of zero-head seepage through a cutoff is reduced to solving a variational inequality which is discretized by the finite element method. The discrete variational inequality is solved by a two-layer iterative process. A rate of convergence bound is obtained for the approximate solution and the optimal parameters of the two-layer iterative process are determined. A numerical experiment supports the theoretical results.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 45–56, 1986.     

An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.      

A numerical method for solving nonlinear ill-posed problems

A two-step iterative process for the numerical solution of nonlinear problems is suggested. In order to avoid the ill-posed inversion of the Fréchet derivative operator, some regularization parameter is introduced. A convergence theorem is proved. The proposed method is illustrated by a numerical example in which a nonlinear inverse problem of gravimetry is considered. Based on the results of the numerical experiments practical recommendations for the choice of the regularization parameter are given. Some other iterative schemes are considered.     

On Backward p(x)-Parabolic Equations for Image Enhancement

In this study, we investigate the backward p(x)-parabolic equation as a new methodology to enhance images. We propose a novel iterative regularization procedure for the backward p(x)-parabolic equation based on the nonlinear Landweber method for inverse problems. The proposed scheme can also be extended to the family of iterative regularization methods involving the nonlinear Landweber method. We also investigate the connection between the variable exponent p(x) in the proposed energy functional and the diffusivity function in the corresponding Euler-Lagrange equation. It is well known that the forward problems converges to a constant solution destroying the image. The purpose of the approach of the backward problems is twofold. First, solving the backward problem by a sequence of forward problems, we obtain a smooth image which is denoised. Second, by choosing the initial data properly, we try to reduce the blurriness of the image. The numerical results for denoising appear to give improvement over standard methods as shown by preliminary results.     

Extragradient-type method for optimal control problem with linear constraints and convex objective function

The paper presents a method for solving optimal control problem with free right end and linear differential equations constraints. The proposed iterative process of extragradient-type is formulated in the functional subspace of piecewise continuous controls of L ₂. The convergence of the method is proved.     

Solutions of linear iterative functional equations behaving like the forcing term in the equation

Summary. We are dealing with those continuous solutions j \varphi of the functional equation¶¶j°f=g·j+h \varphi\circ f=g\cdot \varphi+h ¶that are asymptotically comparable at the origin (the fixed point of f) with the function h. Connections with a linear iterative functional inequality of second order are also mentioned.     

Factorization of moving-average spectral densities by state-space representations and stacking

To factorize a spectral density matrix of a vector moving average process, we propose a state space representation. Although this state space is not necessarily of minimal dimension, its associated system matrices are simple and most matrix multiplications involved are nothing but index shifting. This greatly reduces the complexity of computation. Moreover, in this article we stack every q consecutive observations of the original process MA(q) and generate a vector MA(1) process. We consider a similar state space representation for the stacked process. Consequently, the solution hinges on a surprisingly compact discrete algebraic Riccati equation (DARE), which involves only one Toeplitz and one Hankel block matrix composed of autocovariance functions. One solution to this equation is given by the so-called iterative projection algorithm. Each iteration of the stacked version is equivalent to q iterations of the unstacked one. We show that the convergence behavior of the iterative projection algorithm is characterized by the decreasing rate of the partial correlation coefficients for the stacked process. In fact, the calculation of the partial correlation coefficients via the Whittle algorithm, which takes a very simple form in this case, offers another solution to the problem. To achieve computational efficiency, we apply the general Newton procedure given by Lancaster and Rodman to the DARE and obtain an algorithm of quadratic convergence rate. One immediate application of the new algorithms is polynomial stabilization. We also discuss various issues such as check of positivity and numerical implementation.     

Convergence of nested classical iterative methods for linear systems

Summary Classical iterative methods for the solution of algebraic linear systems of equations proceed by solving at each step a simpler system of equations. When this system is itself solved by an (inner) iterative method, the global method is called a two-stage iterative method. If this process is repeated, then the resulting method is called a nested iterative method. We study the convergence of such methods and present conditions on the splittings corresponding to the iterative methods to guarantee convergence forany number of inner iterations. We also show that under the conditions presented, the spectral radii of the global iteration matrices decrease when the number of inner iterations increases. The proof uses a new comparison theorem for weak regular splittings. We extend our results to larger classes of iterative methods, which include iterative block Gauss-Seidel. We develop a theory for the concatenation of such iterative methods. This concatenation appears when different numbers of inner interations are performed at each outer step. We also analyze block methods, where different numbers of inner iterations are performed for different diagonal blocks.Dedicated to Richard S. Varga on the occasion of his sixtieth birthdayP.J. Lanzkron was supported by Exxon Foundation Educational grant 12663 and the UNISYS Corporation; D.J. Rose was supported by AT&T Bell Laboratories, the Microelectronic Center of North Carolina and the Office of Naval Research under contract number N00014-85-K-0487; D.B. Szyld was supported by the National Science Foundation grant DMS-8807338.     

An iterative algorithm based on -proximal mappings for a system of generalized implicit variational inclusions in Banach spaces

In this paper, we give the notion of M-proximal mapping, an extension of P-proximal mapping given in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383], for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a system of generalized implicit variational inclusions in Banach spaces and show its equivalence with a system of implicit Wiener–Hopf equations using the concept of M-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational inclusions and discuss the convergence and stability analysis of the iterative algorithm.     

One-step jackknife for M-estimators computed using Newton's method

To estimate the dispersion of an M-estimator computed using Newton's iterative method, the jackknife method usually requires to repeat the iterative process n times, where n is the sample size. To simplify the computation, one-step jackknife estimators, which require no iteration, are proposed in this paper. Asymptotic properties of the one-step jackknife estimators are obtained under some regularity conditions in the i.i.d. case and in a linear or nonlinear model. All the one-step jackknife estimators are shown to be asymptotically equivalent and they are also asymptotically equivalent to the original jackknife estimator. Hence one may use a dispersion estimator whose computation is the simplest. Finite sample properties of several one-step jackknife estimators are examined in a simulation study.The research was supported by Natural Sciences and Engineering Research Council of Canada.     

On the optimization of linear,time-variant,multivariable control systems using the contraction mapping principle

The problem of optimizing a linear, time-variant, multivariable control system with quadratic cost functional is first phrased in the context of functional analysis. It is shown that the optimum control is given implicitly as the solution of a matrix integral equation. After the fixed-point contraction mapping theorem is invoked, an iterative method for solving this equation is developed and conditions for its convergence to a unique optimum are derived. Techniques for transforming the system operator are discussed so that, when convergence of the original sequence of iterations cannot be assured, that of the transformed system can. However, it is shown specifically that there is always a finite optimization interval for which the procedure may be used. Bounds are also given for the errors, in the sense of norms, between the control aftern iterations and its ultimate value and between the cost functional aftern iterations and its ultimate value. These bounds are used to decide when to terminate the sequence. Solutions of the iterative scheme using a hybrid computer in parallel and serial modes are discussed and the delays inherent in both methods calculated. It is concluded that the method can be used to track an optimum control system, which drifts from optimum because of parameter variations, with little delay and particularly when the optimization interval is extrapolated only a little into the future. Comparison of the proposed scheme with the steepest-descent approach developed by Balakrishnan shows that the present scheme requires one-third of the computations per step and, therefore, may converge more quickly.The author is indebted to the Principal and the Governors of Sunderland Polytechnic for the facilities placed at his disposal and permission to publish this work.     

Stability of functional equations in single variable

This paper discusses Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equation, nonlinear functional equation and iterative equation. Surveying many known and related results, we clarify the relations between Hyers-Ulam stability and other senses of stability such as iterative stability, continuous dependence and robust stability, which are used for functional equations. Applying results of nonlinear functional equations we give the Hyers-Ulam stability of Böttcher's equation. We also prove a general result of Hyers-Ulam stability for iterative equations.     

An approach to the Gummel map by vector extrapolation methods

The numerical approximation of nonlinear partial differential equations requires the computation of large nonlinear systems, that are typically solved by iterative schemes. At each step of the iterative process, a large and sparse linear system has to be solved, and the amount of time elapsed per step grows with the dimensions of the problem. As a consequence, the convergence rate may become very slow, requiring massive cpu-time to compute the solution. In all such cases, it is important to improve the rate of convergence of the iterative scheme. This can be achieved, for instance, by vector extrapolation methods. In this work, we apply some vector extrapolation methods to the electronic device simulation to improve the rate of convergence of the family of Gummel decoupling algorithms. Furthermore, a different approach to the topological ε-algorithm is proposed and preliminary results are presented.     

Fast interior point solvers for H1-regularized PDE-constrained optimization problems

We consider Newton systems arising from the interior point solution of PDE-constrained optimization problems. In particular, we examine problems where the control variable is regularized by an H¹-norm within the cost functional. We present preconditioned iterative methods for the resulting matrix systems, and justify the potency of our approach through numerical experiments. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.