Fixed-point Iteration Technique Research Articles

A fixed point evolution algorithm based on expanded Aitken rapid iteration method for global numeric optimization

Evolution algorithms based on mathematical models whose reproduction operators are derived from mathematical models are a promising branch of metaheuristic algorithms. Aitken rapid iteration method, as a fixed point iteration technique for solving nonlinear equations, performs a procedure of progressive display of a root and generates an iterative sequence that exhibits a convergent trend. Inspired by the idea that an iterative sequence gradually converges to the optimal point during the progressive display procedure of a fixed point of an equation, a fixed point evolution algorithm based on the expanded Aitken rapid iteration method (FPEea) is proposed. To develop FPEea, an expanded Aitken rapid model is first constructed. Then, three polynomials which are derived from the expanded Aitken rapid model are used as the reproduction operator of FPEea to produce offspring. The performance of FPEea is investigated on CEC2019 and CEC2020 benchmark function sets, as well as four engineering design problems. Experimental results show that FPEea is an effective and competitive algorithm compared with several classical evolution algorithms and state-of-the-art algorithms.

Maximum weighted correntropy filters for nonlinear non‐Gaussian systems

AbstractIn this paper, we focus on the filtering issue for nonlinear non‐Gaussian systems. Considering the limitations of the Gaussian function with a single kernel bandwidth, we design a novel extended version called weighted Gaussian function, which consists of a weighting coefficient and two distinct kernel bandwidths. The additional coefficient can directly balance two kernel bandwidths, which improves the flexibility and performance of the correntropy. We develop a cost function utilizing the suggested weighted Gaussian function and statistical linearization method, followed by deriving a maximum weighted correntropy filter based on the maximum correntropy criterion. The proposed algorithm is demonstrated to converge to a Gaussian filter as the kernel bandwidths approach infinity. In addition, we derive the corresponding information filter, which takes the form of information matrix and information vector. It is computationally efficient and easier to generalize to multisensor systems. The performance of the proposed algorithms is compared with other filters based on the third‐order spherical cubature rule and the fixed point iteration technique in a target tracking system. Simulation results confirm the effectiveness of the new approaches.

Memory-friendly fixed-point iteration method for nonlinear surface mode oscillations of acoustically driven bubbles: from the perspective of high-performance GPU programming

A fixed-point iteration technique is presented to handle the implicit nature of the governing equations of nonlinear surface mode oscillations of acoustically excited microbubbles. The model is adopted from the theoretical work of Shaw [1], where the dynamics of the mean bubble radius and the surface modes are bi-directionally coupled via nonlinear terms. The model comprises a set of second-order ordinary differential equations. It extends the classic Keller–Miksis equation and the linearized dynamical equations for each surface mode. Only the implicit parts (containing the second derivatives) are reevaluated during the iteration process. The performance of the technique is tested at various parameter combinations. The majority of the test cases needs only a single reevaluation to achieve 10-9 error. Although the arithmetic operation count is higher than the Gauss elimination, due to its memory-friendly matrix-free nature, it is a viable alternative for high-performance GPU computations of massive parameter studies.

Supply Chain Management of E-Waste for End-of-Life Electronic Products with Reverse Logistics

Sustainable development and environmental pollution have become valuable stimulating factors for the resource recovery of end-of-life products through reverse logistics. E-waste is considered in reverse logistics. Electronic waste is solely responsible for environmental hazards and contains valuable raw materials that can be recycled/repaired, so reverse logistics is essential to minimizing their inappropriate disposal. This paper presents the mathematical model for multi-electronic products, considering multi-manufacturers and multi-retailers. After the end-of-life product, the reverse logistics network collects the e-waste in return processors where testing, sorting, and disassembling are carried out and then sent to the repair and recycling units. Components that are not repaired/recycled are shipped to the secondary manufacturer as raw materials. An electronic product’s reverse supply chain is employed to incorporate the idea of e-waste nullification. The fixed point iteration technique is used to solve the proposed model. A numerical example is analyzed to demonstrate the model’s efficacy where the total cost is minimized. The model’s validity and usefulness in reducing e-waste are validated through managerial insights into the model and sensitivity analysis of the key factors. The proposed policy suggests that the e-waste nullification strategy might be a useful apparatus for managers in ensuring long-term sustainability.

Numerical simulation for nonlinear space-fractional reaction convection-diffusion equation with its application

In this research, we adopt a fully implicit approach with a weighted shifted Grunwald–Letnikov difference operator to find the numerical solution of the one and two-dimensional nonlinear space fractional convection–diffusion-reaction equation over a bounded domain. We employ the Peacemann-Rachford alternative direction implicit technique for 2D problems, which is computationally efficient and is used to reduce a 2D problem to a 1D problem by alternately applying x and y directions. This type of problem allows us to successfully predict and investigate pollutant distribution in groundwater, which is based on two processes: convection and diffusion. Because computing the numerical solution of the nonlinear system of equations at each time step using the fixed-point iteration technique is computationally expensive, we have linearized our problem. The numerical scheme’s unconditional stability and second-order convergence in both space and time are theoretically investigated. Finally, numerical experiments with several variations of the nonlinear reaction term are performed, confirming the theoretical analysis and demonstrating the effectiveness of the suggested strategy.

Computation of invariant sets via immersion for discrete-time nonlinear systems

In this paper, we propose an approach for computing invariant sets of discrete-time nonlinear systems by lifting the nonlinear dynamics into a higher dimensional linear model. In particular, we focus on the maximal admissible invariant set contained in some given constraint set. For special types of nonlinear systems, which can be exactly immersed into higher dimensional linear systems with state transformations, invariant sets of the original nonlinear system can be characterized using the higher dimensional linear representation. For general nonlinear systems without the immersibility property, approximate immersions are defined in a local region within some tolerance and linear approximations are computed by leveraging the fixed-point iteration technique for invariant sets. Given the bound on the mismatch between the linear approximation and the original system, we provide an invariant inner approximation of the maximal admissible invariant set by a tightening procedure.

Generalized Second-Order Value Iteration in Markov Decision Processes

Value iteration is a fixed point iteration technique utilized to obtain the optimal value function and policy in a discounted reward Markov decision process (MDP). Here, a contraction operator is constructed and applied repeatedly to arrive at the optimal solution. Value iteration is a first-order method and, therefore, it may take a large number of iterations to converge to the optimal solution. Successive relaxation is a popular technique that can be applied to solve a fixed point equation. It has been shown in the literature that under a special structure of the MDP, successive overrelaxation technique computes the optimal value function faster than standard value iteration. In this article, we propose a second-order value iteration procedure that is obtained by applying the Newton–Raphson method to the successive relaxation value iteration scheme. We prove the global convergence of our algorithm to the optimal solution asymptotically and show the second-order convergence. Through experiments, we demonstrate the effectiveness of our proposed approach.

Analysis of fabrication and crack-induced porosity migration in mixed oxide fuels for sodium fast reactors by the finite element method

We present an engineering-scale model for the migration of porosity in a fuel pellet experiencing a temperature gradient. The system of coupled pore advection and heat diffusion equations governing the problem is solved by a fixed-point iteration technique. The coupling between porosity and temperature fields is considered via the dependency of pore advection velocity on the local temperature and temperature gradient, and via the dependency of fuel thermal conductivity and of the volumetric power source on the local porosity. We employ the finite element method to discretize the resulting equations. As pure advection solutions obtained by this method are well-known to present spurious spatial oscillations, we introduce stabilization techniques in the pore advection equation. The proposed model is first tested against a benchmark problem representative for the conditions of an uranium-plutonium oxide fuel pellet irradiated in a sodium fast reactor. The results are compared to the those obtained by a model implemented in the BISON fuel performance code. The analysis shows how the results of the newly developed model are in line with those obtained by the reference model, and underlines a superior stability of the solution. The model is then applied to analyze the contribution of as-fabricated and crack-induced porosities in determining the fuel restructuring and in particular the central hole formation. A comparison to experimental data shows the impact of considering crack-induced porosity to predict the extent of the central void.

An iterative factored topography-dependent eikonal solver for anisotropic media

Accurate and efficient eikonal solvers for heterogeneous media play an important role in many areas of seismology, such as seismic tomography, migration, and earthquake localization. Incorporating seismic anisotropy and complex topography remain a computational challenge for finite-difference eikonal solvers. In recent years, the topography-dependent eikonal equation (TDEE) has been proposed as an effective way to calculate seismic traveltimes for isotropic and anisotropic media with irregular topography. However, the Lax-Friedrichs sweeping method used in previous studies to approximate the viscosity solution of TDEE for anisotropic media is more dissipative and needs a much higher number of iterations to converge. In addition, the TDEE solution for the initial point source has an upwind source singularity, which makes all TDEE solvers, even the high-order ones, exhibit polluted convergence and relatively large errors that propagate from the point source to the entire computational domain. To solve these problems, we have formulated the factored topography-dependent anisotropic eikonal (FTDAE) equation in tilted transversely isotropic (TTI) media using the factorization principle. Then, the resulting quartic equation can be numerically solved by using a fixed-point iteration technique based on the simpler elliptical anisotropic eikonal (EAE) equation with a high-order source term due to anelliptical anisotropy introduced by TTI media. At each iteration, the unknown traveltime in the EAE equation is factored into two functions: One of the functions is specified analytically to capture the source singularity, such that the unknown factor is differentiable in the source neighborhood and could be solved by the fast sweeping method. Numerical examples indicate that our FTDAE solver can treat source singularity successfully and achieve high accuracy after just a few iterations, independently of the mesh size, which could provide a more efficient and robust tool for traveltime calculation in the presence of seismic anisotropy and complex surfaces.

Online Construction of Variable Span Linear Filters Using a Fixed-Point Approach

The variable span linear filters (VSLFs) constitute a unified framework of conventional subspace and linear filtering techniques for noise reduction. The construction of VSLFs, however, relies on the generalized eigendecomposition (GEVD) methods, which are computationally expensive. This in turn stymies the employment of such filters in practical online processing problems. To address this issue, we first propose in this paper a fixed-point iteration technique to extract the generalized eigenvectors. It is based on maximizing the pre-whitened generalized Rayleigh quotient (GRQ). We then integrate this technique with online statistic estimation to construct VSLFs. Our proposed method is computationally efficient and can also harness parallel architectures. To show its effectiveness, we consider a speech enhancement application and compare the results with those of several existing methods.

Linear Iterative Power Flow Approach Based on the Current Injection Model of Load and Generator

Power flow (PF) is a fundamental tool for operation, automation and optimization of the power systems. Due to the nonlinearity of the PF system equations, the classical PF solutions are computationally very demanding. As a common approach in solving the nonlinear equations, linearization is a potential technique which can simplify and accelerate the PF calculations. In this context, this paper proposes a linear fast iterative method based on the fixed-point iteration technique in which a linearized model of generator along with a ZI load model are integrated in a simplified system of linear equations (SLE) of $Yv=i$ . The relaxation method is used during the deriving process of generator equivalent current in this approach. However, the already developed ZI load model based on the curve-fitting technique has been exploited in this work. The accuracy of the proposed PF method has been compared with calculated results from DIgSILENT PowerFactory on the benchmark IEEE 33-bus test system and on a large medium voltage network in Germany.

Modeling and performance analysis of smart map application in the Multi-access Edge Computing paradigm

Multi-access Edge Computing (MEC) recognized as an emerging technology that provides cloud computing services in the proximity of Radio Access Network (RAN). Mobile subscribers could offload their computation-intensive or memory-intensive tasks to the nearest Base Stations (BS) and deliver the cloud services from MEC servers. This technology can meet the requirements of low latency and location awareness. Smart map applications are an indispensable part of our digital life in large cities. In this paper, we introduce how the map application could gain from MEC technology. We concentrate on performance modeling and analysis of direction-finding in the map application. We study the impact of variation of workload, the number of servers, queue length, and connection failure on the task rejection probability and mean sojourn time. We model each phase of direction-finding service by the Markov Reward Model (MRM). The numerical results presented by applying models in the SHARPE software package. Furthermore, inter-dependencies between sub-models resolved by the fixed-point iteration technique. Moreover, the analytical results verified by the Discrete-Event Simulation (DES), which conducted in MATLAB. The result could help the MEC providers to assess their platform performance.

A Hybrid Compensator Configuration for VAR Control and Harmonic Suppression in All-Electric Shipboard Power Systems

This paper proposes a cost-effective compensator to suppress harmonics and compensate the power factor of all-electric shipboard power systems (SPSs). This compensator, which is based on a fixed capacitor-thyristor controlled reactor (FC-TCR), behaves as a low-pass filter and therefore can reject both low and high-order harmonics. Moreover, the FC-TCR compensator is featured by the low switching losses; hence, it can effectively be implemented for low and medium voltage applications. The design of the filter is detailed via equivalent lossy circuits, which exhibit the mechanism of the harmonic mitigation. Furthermore, theoretical analyses and mathematical developments are suggested to enhance the filtering performance. Besides, details of the Fixed-Point iteration technique which is applied to extract the firing angle of the TCR are conducted. A practical SPS is selected to ensure and demonstrate the performance of proposed methodology via thorough simulations under MATLAB/Simulink environment. The obtained results verify the theoretical analyses and confirm the effectiveness of proposed solution.

Vibration analysis of functionally graded thermoelastic nonlocal sphere with dual-phase-lag effect

The vibration of a functionally graded axisymmetric nonlocal thermoelastic hollow sphere with dual-phase-lag effect is addressed in this paper. Surfaces of the sphere are assumed to be thermally insulated or isothermal and stress free. According to a simple power law, the material is assumed to be graded in the radial direction. The linear theory of modified thermoelasticity with a dual phase lag based on Eringen’s nonlocal elasticity is employed to model this problem. The Matrix Frobenius method of continued power series is introduced to derive the analytical solutions. The phase velocity relations for the existence of various modes of vibrations in the designed hollow sphere are derived in compact forms. In order to explore the attributes of vibrations, the fixed-point numerical iteration technique is used to solve the secular equations. The numerical computations for the material crust in respect of the natural frequencies, thermoelastic damping and the frequency shifting are presented graphically using MATLAB software tools.

Hierarchical Stochastic Models for Performance, Availability, and Power Consumption Analysis of IaaS Clouds

Infrastructure as a Service (IaaS) is one of the most significant and fastest growing fields in cloud computing. To efficiently use the resources of an IaaS cloud, several important factors such as performance, availability, and power consumption need to be considered and evaluated carefully. Evaluation of these metrics is essential for cost-benefit prediction and quantification of different strategies which can be applied to cloud management. In this paper, analytical models based on Stochastic Reward Nets (SRNs) are proposed to model and evaluate an IaaS cloud system at different levels. To achieve this, an SRN is initially presented to model a group of physical machines which are controlled by a management layer. Afterwards, the SRN models presented for the groups of physical machines in the first stage are combined to capture a monolithic model representing an entire IaaS cloud. Since the monolithic model does not scale well for large cloud systems, two approximate SRN models using folding and fixed-point iteration techniques are proposed to evaluate the performance, availability, and power consumption of the IaaS cloud. The existence of a solution for the fixed-point approximate model is proved using Brouwer's fixed-point theorem. A validation of the proposed monolithic and approximate models against both an ad-hoc discrete-event simulator developed in Java and the CloudSim framework is presented. The analytic-numeric results obtained from applying the proposed models to sample cloud systems show that the errors introduced by approximate models are insignificant while an improvement of several orders of magnitude in the state space reduction of the monolithic model is obtained.

Analysis of Free Vibrations of Axisymmetric Functionally Graded Generalized Viscothermoelastic Cylinder Using Series Solution

The analysis of free vibrations of axisymmetric functionally graded isotropic viscothermoelastic hollow cylinder has been investigated in the radial direction. The material of viscothermoelastic cylinder has been assumed to be graded in the radial direction due to simple exponent law. The governing partial differential equations are transformed into ordinary differential equations with th e help of time harmonic vibrations. The extended power series solution of matrix Frobenius method has been applied to derive the analytical solutions for stresses, displacement and temperature change. The frequency equations of various types of modes of vibrations have been derived in a compact form. To investigate the numerical features of vibrations, the analytical results of frequency equations have been further solved with the help of fixed-point numerical iteration technique using MATLAB software tools. The polymethyl methacrylate material has been used for numerical computations. The numerical results have been presented for frequency shift, natural frequencies, thermoelastic damping, stresses, displacement and variation of temperature. With the increase in the values of grading index, variation of vibrations in field functions go on decreasing. The numerical results show alternate variations in homogenous materials in contrast to inhomogeneous materials. The behaviour of deformation, temperature change, frequency shift and thermoelastic damping have been monitored (increase or decrease) with the help of grading index (in-homogeneity parameter).

Stability of numerical methods under the regime-switching jump-diffusion model with variable coefficients

In this paper we introduce three numerical methods to evaluate the prices of European, American, and barrier options under a regime-switching jump-diffusion model (RSJD model) where the volatility and other parameters are considered as variable coefficients. The prices of the European option, which is one of the financial derivatives, are given by a partial integro-differential equation (PIDE) problem and those of the American option are evaluated by solving a linear complementarity problem (LCP). The proposed methods are constructed to avoid the use of any fixed point iteration techniques at each state of the economy and time step. We analyze the stability of the proposed methods with respect to the discrete l2-norm in the time and spatial variables. A variety of numerical experiments are carried out to show the second-order convergence of the three numerical methods under the regime-switching jump-diffusion model with variable coefficients.

A Novel Fixed Point Feedback Approach Studying the Dynamical Behaviors of Standard Logistic Map

The standard logistic map is one of the oldest and simplest systems which has found a celebrated place in the dynamical systems and in different vital applications of science like image encryption in cryptography, secure communications and traffic control models. Generally, the dynamical systems are characterized by one or more control parameters that determine the dynamical behaviors of the system. Traditionally, the discrete logistic map allows only one parameter [Formula: see text] to determine its complete behavior. This study takes one step forward, using the superior fixed point iterative technique to study the dynamical properties of the discrete logistic map. The proposed technique provides an extra degree of freedom on control parameters that renders superior dynamical properties and may increase the performance of many applications. Analytical analysis as well as numerical simulations are presented to show the effectiveness, flexibility and efficiency of new method.

Joint compressed sensing imaging and phase adjustment via an iterative method for multistatic passive radar

The resolution of the multistatic passive radar imaging system (MPRIS) is poor due to the narrow bandwidth of the signal transmitted by illuminators of opportunity. Moreover, the inaccuracies caused by the inaccurate tracking system or the error position measurement of illuminators or receivers can deteriorate the quality of an image. To improve the performance of an MPRIS, an imaging method based on the tomographic imaging principle is presented. Then the compressed sensing technique is extended to the MPRIS to realize high-resolution imaging. Furthermore, a phase correction technique is developed for compensating for phase errors in an MPRIS. Phase errors can be estimated by iteratively solving an equation that is derived by minimizing the mean recovery error of the reconstructed image based on the principle of fixed-point iteration technique. The technique is nonparametric and can be used to estimate phase errors of any form. The effectiveness and convergence of the technique are confirmed by numerical simulations.

A fixed point iterative method for the solution of two-point boundary value problems for a second order differential equations

Recently Kilicman et al. (2006) propose a variational fixed point iteration technique with the Galerkin method for the determination of the starting function for the solution of second order linear ordinary differential equation with two-point boundary value problem without proving the convergence of the method. In this paper, a fixed point iteration method similar to Mann iteration process is proposed and successfully applied to the solution of two-point boundary value problems. We use an affine function satisfying the boundary conditions as a starting approximate solution. We also show the convergence of the method and design a Maple program for the numerical computations. Examples are given to demonstrate the agreement of the results of the proposed method with that of exact solution and existing methods.