Fixed-point Iterative Procedure Research Articles

STABILIZATION OF FIXED POINTS IN CHAOTIC MAPS USING NOOR ORBIT WITH APPLICATIONS IN CARDIAC ARRHYTHMIA

Controlling chaos through stability in fixed and periodic states is used in various engineering problems such as heat convection, reduction control, spine-wave instability, traffic control models, cardiac arrhythmia, chemical chaos, etc. Traditionally, this process is done in the coordination of chaos and stability in fixed and periodic points by using fixed point iterative procedures. Therefore, this article deals with a novel alliance between stabilization in one-dimensional discrete maps and Noor fixed point iterative procedure. The procedure contains $\alpha$, $\beta$, $\gamma$ and $r$, as its four new control parameters due to which the stability rate increases more rapidly than the other existing procedures. The stability theorems and a few time varying examples for fixed and periodic points are studied using Noor control system. Further, the Lyapunov exponent property is also described and the maximum Lyapunov value is determined to examine the stability in fixed and periodic points. Moreover, an improved control-based cardiac arrhythmia model is discussed in the Noor control system. Surprisingly, it is noted that the added new parameters $\alpha$, $\beta$, and $\gamma$ may increase the stability in chaotic arrhythmia rapidly.

Bus-Privacy-Preserving Distributed Economic Dispatch Method for Power Systems Using Iteration Functions

The sufficiency of electricity market competition and the economy of power system operation require that the economic dispatch should not only preserve the bus privacy but consider network losses. The existing economic dispatch methods, however, can’t preserve the privacy of each bus from leakage and even complicate the solution when they reasonably consider network losses, due to the coupling complexity of network losses. To address these issues, this paper presents a novel distributed economic dispatch (DED) method based on KKT optimality conditions and iteration functions. It focuses on an economic dispatch model that contains bus active power balance equations and generator power limit inequalities. First, the model’s KKT optimality conditions with inequalities are simplified to equivalent pure equality-type optimality conditions. Then a set of iteration functions is created using the Taylor series, which overcomes the difficulty caused by the fact that unknown bus voltage phases are completely contained in the sine and cosine functions of optimality conditions. Finally, a fast fixed-point iteration procedure is proposed for bus agents to solve the economic dispatch model. The proposed method reasonably considers network losses by using the set of bus active power balance equations. It not only preserves the privacy of each bus from leakage but is straightforward in solution (does not complicate the solution). In addition, it is fully bus-level distributed. Numerical simulation results of systems with different sizes verified the effectiveness and advantages of the proposed method.

Analytical prediction of nonlinear behaviors of beams post-tensioned by unbonded tendons considering shear deformation and tension stiffening effect

ABSTRACT This is a proposal for a nonlinear analytical model that was developed to predict the structural response of post-tensioned beam with unbonded tendon. The structural response includes deflections under the service load before or after cracking, as well as the ultimate load capacity. A comprehensive procedure for the deflection calculation was implemented, considering the additional shear deformation and tension stiffening effect. The shear deformation calculation is based on the Timoshenko beam theory, considering the shear modulus reduction upon cracking. Furthermore, a modification of the tension stiffening model from CEB–FIP Model Code 1990 is proposed to account for the significant strain concentration in the post-yielding region. A fixed-point iterative procedure is introduced in the prediction model to compute the incremental strain in the unbonded tendon. Each iteration brings the estimated strain increase in the tendon closer to the actual value until it is reasonably accurate. Predictive accuracy of the proposed model in both load–deflection relationships and strain rates, as a function of deflections, is successfully validated against experimental beam tests and a nonlinear finite element analysis considering concrete plasticity, respectively.

A new fixed point iteration method for nonlinear third-order BVPs

In this article, we shall present a novel approach based on embedding Green's function into Ishikawa fixed point iterative procedure for the numerical solution of a broad class of boundary value problems of third order. A linear integral operator expressed in terms of Green's function is constructed, then the well-known Ishikawa fixed point iterative scheme is applied to obtain a new iterative scheme. The aim of our alternative strategy is to overcome the major deficiency of other iterative schemes that usually result in the deterioration of the error as the domain increases. Furthermore the proposed strategy will improve the rate of convergence of other existing methods that are based on Picard's and Mann's iterative schemes. Convergence results of the iterative algorithm have been proved. A number of numerical examples shall be solved to illustrate the method and demonstrate its reliability and accuracy. Moreover, we shall compare our results with both the analytical and the numerical solutions obtained by other methods that exist in the literature.

Modal decomposition from partial measurements

A data set over space and time is assumed to have a low-rank representation in separated spatial and temporal modes. The problem of evaluating these modes from a temporal series of partial measurements is considered. Each elementary instantaneous measurement captures only a “window” (in space) of the observed data set, but the position of this window varies in time so as to cover the entire region of interest and would allow for a complete measurement would the scene be static. A novel procedure, alternative to the Gappy Proper Orthogonal Decomposition (GPOD) methodology, is introduced. It is a fixed-point iterative procedure where modes are evaluated sequentially. Tested upon very sparse acquisition (1% of measurements being available) and very noisy synthetic data sets (10% noise), the proposed algorithm is shown to outperform two variants of the GPOD algorithm, with much faster convergence, and better reconstruction of the entire data set.

Numerical solution of nonlinear mixed Volterra-Fredholm integral equations in complex plane via PQWs

This paper presents an efficient numerical method for solving nonlinear mixed Volterra- integral equations in the complex plane. The method is based on the fixed point iteration procedure. In each iteration of this method, periodic quasi-wavelets are used as basis functions to approximate the solution. Also, using the Banach fixed point theorem, some results concerning the error analysis are obtained. Finally, some numerical examples show the implementation and accuracy of this method.

Fluid Structure Interaction of Buoyant Bodies with Free Surface Flows: Computational Modelling and Experimental Validation

In this paper we present a computational model for the fluid structure interaction of a buoyant rigid body immersed in a free surface flow. The presence of a free surface and its interaction with buoyant bodies make the problem very challenging. In fact, with light (compared to the fluid) or very flexible structures, fluid forces generate large displacements or accelerations of the solid and this enhances the artificial added mass effect. Such a problem is relevant in particular in naval and ocean engineering and for wave energy harvesting, where a correct prediction of the hydrodynamic loading exerted by the fluid on buoyant structures is crucial. To this aim, we develop and validate a tightly coupled algorithm that is able to deal with large structural displacement and impulsive acceleration typical, for instance, of water entry problems. The free surface flow is modeled through the volume of fluid model, the finite volume method is utilized is to discretize the flow and solid motion is described by the Newton-Euler equations. Fluid structure interaction is modeled through a Dirichlet-Newmann partitioned approach and tight coupling is achieved by utilizing a fixed-point iterative procedure. As most experimental data available in literature are limited to the first instants after the water impact, for larger hydrodynamic forces, we specifically designed a set of dedicated experiments on the water impact of a buoyant cylinder, to validate the proposed methodology in a more general framework. Finally, to demonstrate that the proposed numerical model could be used for a wide range of engineering problems related to FSI in multiphase flows, we tested the proposed numerical model for the simulation of a floating body.

Probabilistic Identification and Estimation of Noise: Application to MR Images

For proper modelling of signal and noise in MR data requires proper interpretation and analysis of data, the different approaches with this degradation due to random fluctuations in the MR data, probabilistic modeling is power solution, which needs correctness in the computation of noise is challenging task and various stastical approaches can be utilized. After modelling the noise it can be integrated to denoising pipeline, in this research work, the recognition of noise only pixels and the evaluation of standard deviation of noise using median, mean or other optimal sample quantiles are combined in to single frame work for noise assement and uses fixed point iterative procedure to obtain standard deviation of noise. We tested the effectiveness of the algorithm to the MR clinical and synthetic data base.

Choosing the Optimal Multi-Point Iterative Method for the Colebrook Flow Friction Equation

The Colebrook equation is implicitly given in respect to the unknown flow friction factor λ; λ = ζ ( R e , ε * , λ ) which cannot be expressed explicitly in exact way without simplifications and use of approximate calculus. A common approach to solve it is through the Newton–Raphson iterative procedure or through the fixed-point iterative procedure. Both require in some cases, up to seven iterations. On the other hand, numerous more powerful iterative methods such as three- or two-point methods, etc. are available. The purpose is to choose optimal iterative method in order to solve the implicit Colebrook equation for flow friction accurately using the least possible number of iterations. The methods are thoroughly tested and those which require the least possible number of iterations to reach the accurate solution are identified. The most powerful three-point methods require, in the worst case, only two iterations to reach the final solution. The recommended representatives are Sharma–Guha–Gupta, Sharma–Sharma, Sharma–Arora, Džunić–Petković–Petković; Bi–Ren–Wu, Chun–Neta based on Kung–Traub, Neta, and the Jain method based on the Steffensen scheme. The recommended iterative methods can reach the final accurate solution with the least possible number of iterations. The approach is hybrid between the iterative procedure and one-step explicit approximations and can be used in engineering design for initial rough, but also for final fine calculations.

Trajectory Optimization for Loss Minimization in Induction Motor Fed Elevator Systems

This paper presents a novel loss minimization technique for induction motor driven elevator systems (IMDES) to determine an energy optimum velocity pattern. The objective function consists of a differential equation relating total power losses in the induction motor to its rotor speed. The constraints are provided in terms of velocity and acceleration governed by passenger comfort. Since the velocity and acceleration are state variables, Pontryagin's minimum principle is used to determine the most optimum trajectory by forming an appropriate Hamiltonian function. Boundary conditions are specified and used to obtain the coefficients of the rotor speed equation. As the velocity is optimized, the total travel time is not fixed and treated as a variable. A fixed-point iterative procedure is used to estimate the travel time. Additional loss minimization is incorporated by optimizing the machine flux for the constrained portion of the velocity pattern. An algorithm for implementing the optimization scheme is explained in detail. The theoretical claims are supported with analytical and experimental results. Proposed technique is developed for IMDES, but it can be easily adapted to other drive applications as well.

A Bi-Projection Method for Incompressible Bingham Flows with Variable Density, Viscosity, and Yield Stress

A new numerical scheme for solving incompressible Bingham flows with variable density, plastic viscosity and yield stress is proposed. The mathematical and computational difficulties due to the non-differentiable definition of the stress tensor in the plug regions, i.e. where the strain-rate tensor vanishes, is overcome by using a projection formulation as in the Uzawa-like method for viscoplastic flows. This projection definition of the plastic tensor is coupled with a fractional time-stepping scheme designed for Newtonian incompressible flows with variable density. The plastic tensor is treated implicitly in the first sub-step of the fractional time-stepping scheme and a fixed-point iterative procedure is used for its computation. A pseudo-time relaxation term is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed-point algorithm. This is a key feature of the bi-projection scheme which provides a fast and accurate computation of the plastic tensor. Stability and error analyses of the bi-projection scheme are provided. The use of the discrete divergence-free velocity to convect the density in the mass conservation equation allows us to derive lower and upper bounds for the discrete density. The error induced by the pseudo-time relaxation term is controlled by a prescribed numerical parameter so that a first-order estimate of the time error is derived for the velocity field and the density, as well as the dependent parameters that are the plastic viscosity and the yield stress.

Inference with progressively censored k-out-of-n system lifetime data

A system with n independent components which works if and only if a least k of its n components work is called a k-out-of-n system. For exponentially distributed component lifetimes, we obtain point and interval estimators for the scale parameter of the component lifetime distribution of a k-out-of-n system when the system failure time is observed only. In particular, we prove that the maximum likelihood estimator (MLE) of the scale parameter based on progressively Type-II censored system lifetimes is unique. Further, we propose a fixed-point iteration procedure to compute the MLE for k-out-of-n systems data. In addition, we illustrate that the Newton–Raphson method does not converge for any initial value. Finally, exact confidence intervals for the scale parameter are constructed based on progressively Type-II censored system lifetimes.

An efficient method for fiber bridging traction identification based on the R-curve: Formulation and experimental validation

Fiber bridging is one of the main toughening mechanisms in mode I interlaminar and intralaminar fracture of laminated composites. Intact fibers exert closing forces on both faces of the crack, restraining crack propagation. An effective identification procedure is required to characterize bridging tractions and to develop accurate prediction models. The method proposed in this work is based on the resistance (R)-curve and assumes fiber bridging tractions decreasing non-linearly with respect to the crack opening displacement, following a parametric function. The distribution parameters are identified by a fixed-point iterative procedure where the energy release rate computed in a numerical model is matched with the values obtained experimentally in two points of the R-curve. The method is applied and validated through three cases, from low bridging intensity in thin-ply delamination to high bridging intensity in intralaminar fracture.

Symplectic exponential Runge–Kutta methods for solving nonlinear Hamiltonian systems

Symplecticity is also an important property for exponential Runge–Kutta (ERK) methods in the sense of structure preservation once the underlying problem is a Hamiltonian system, though ERK methods provide a good performance of higher accuracy and better efficiency than classical Runge–Kutta (RK) methods in dealing with stiff problems: y′(t)=My+f(y). On account of this observation, the main theme of this paper is to derive and analyze the symplectic conditions for ERK methods. Using the fundamental analysis of geometric integrators, we first establish one class of sufficient conditions for symplectic ERK methods. It is shown that these conditions will reduce to the conventional ones when M→0, and this means that these conditions of symplecticity are extensions of the conventional ones in the literature. Furthermore, we also present a new class of structure-preserving ERK methods possessing the remarkable property of symplecticity. Meanwhile, the revised stiff order conditions are proposed and investigated in detail. Since the symplectic ERK methods are implicit and iterative solutions are required in practice, we also investigate the convergence of the corresponding fixed-point iterative procedure. Finally, the numerical experiments, including a nonlinear Schrödinger equation, a sine-Gordon equation, a nonlinear Klein–Gordon equation, and the well-known Fermi–Pasta–Ulam problem, are implemented in comparison with the corresponding symplectic RK methods and the prominent numerical results definitely coincide with the theories and conclusions made in this paper.

Likelihood inference for the component lifetime distribution based on progressively censored parallel systems data

ABSTRACTPoint and interval estimators for the scale parameter of the component lifetime distribution of a k-component parallel system are obtained when the component lifetimes are assumed to be independently and identically exponentially distributed. We prove that the maximum likelihood estimator of the scale parameter based on progressively Type-II censored system lifetimes is unique and can be obtained by a fixed-point iteration procedure. In particular, we illustrate that the Newton–Raphson method does not converge for any initial value. Furthermore, exact confidence intervals are constructed by a transformation using normalized spacings and other component lifetime distributions including Weibull distribution are discussed.

An empirical study of the convergence area and convergence speed of Agarwal et al. fixed point iteration procedure

We present an empirical study of the convergence area and speed of Agarwal et al. fixed point iterative procedure in the particular case of the Newton's method associated to the complex polynomials $p_{3}(z)=z^3-1$ and $p_{8}(z)=z^8-1$. In order to obtain an analytical expression for the experimental data related to the mean number of iterations (MNI) and convergence area index (CAI), we use regression analysis and find some linear and nonlinear bi-variable models with good correlation coefficients.

A comparison of some fixed point iteration procedures by using the basins of attraction

Several iterative processes have been defined by researchers to approximate the fixed points of various classes operators. In this paper we present, by using the basins of attraction for the roots of some complex polynomials, an empirical comparison of some iteration procedures for fixed points approximation of Newton’s iteration operator. Some numerical results are presented. The Matlab m-files for generating the basins of attraction are presented, too.

Stability of tripled fixed point iteration procedures for mixed monotone mappings

Recently, Berinde and Borcut [Berinde, V. and Borcut, M., Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces, Nonlinear Anal. 74 (2011), 4889-4897] introduced the concept of tripled fixed point and by now, there are several researches on this subject, in partially metric spaces and in cone metric spaces. In this paper, we introduce the notion of stability definition of tripled fixed point iteration procedures and establish stability results for mixed monotone mappings which satisfy various contractive conditions. Our results extend and complete some existing results in the literature. An illustrative example is also given.

Simple and Efficient Determination of the Tikhonov Regularization Parameter Chosen by the Generalized Discrepancy Principle for Discrete Ill-Posed Problems

Discrete ill-posed problems where both the coefficient matrix and the right hand side are contaminated by noise appear in a variety of engineering applications. In this paper we consider Tikhonov regularized solutions where the regularization parameter is chosen by the generalized discrepancy principle (GDP). In contrast to Newton-based methods often used to compute such parameter, we propose a new algorithm referred to as GDP-FP, where derivatives are not required and where the regularization parameter is calculated efficiently by a fixed-point iteration procedure. The algorithm is globally and monotonically convergent. Additionally, a specialized version of GDP-FP based on a projection method, that is well-suited for large-scale Tikhonov problems, is also proposed and analyzed in detail. Numerical examples are presented to illustrate the effectiveness of the proposed algorithms on test problems from the literature.

Fixed point iterations for a class of nonstandard Sturm–Liouville boundary value problems

The paper examines a particular class of nonlinear integro-differential equations consisting of a Sturm–Liouville boundary value problem on the half-line, where the coefficient of the differential term depends on the unknown function by means of a scalar integral operator. In order to handle the nonlinearity of the problem, we consider a fixed point iteration procedure, which is based on considering a sequence of classical Sturm–Liouville boundary value problems in the weak solution sense. The existence of a solution and the global convergence of the fixed-point iterations are stated without resorting to the Banach fixed point theorem. Moreover, the unique solvability of the problem is discussed and several examples with unique and non-unique solutions are given.