Fixed-point Iteration Method Research Articles

Preconditioned inexact fixed point iteration method for solving tensor absolute value equation

Preconditioned inexact fixed point iteration method for solving tensor absolute value equation

Distributed adaptive moving horizon estimation for multi-sensor networks subject to quantization effects

This paper investigates the distributed state estimation problem for multi-sensor networks with quantized measurements. Within the Bayesian framework, a distributed adaptive moving horizon estimation algorithm is developed. Unlike the existing methods regarding quantized errors roughly as bounded uncertainties, the posterior distributions of the errors are demanded to be derived. To overcome the difficulty of evaluating the posterior distributions for series of the states and quantized errors jointly, the variational Bayesian methodology is adopted to approximate the true distributions. Based on the fixed-point iteration method, the update rules are analytically derived, with the convergence criterion provided. Furthermore, by incorporating the average consensus algorithm into the prediction process, all sensors can achieve consensus on their estimates in a distributed manner. Finally, a numerical example of target tracking under logarithmic and uniform quantization effects is given to illustrate the validity of the proposed algorithm.

Computing the Network’s Equilibrium Point at the Fault Clearing Instant in Transient Stability Studies

This paper proposes an approach for computing the network’s equilibrium point related to the fault clearing time in transient stability studies. The computation of this point is not a trivial task, particularly when the algebraic network’s equations are expressed in the power balance form. A natural attempt to solve this problem is using Newton’s method. However, convergence issues are found because of the lack of a general strategy for initializing nodal voltages at the clearing time. This problem has not been widely discussed in the existing literature and, therefore, is comprehensively analyzed in this paper. Furthermore, the paper proposes the use of a network’s model based on current injections and an extended admittance matrix to overcome the problem. This model is efficiently solved via the fixed-point iteration method, which involves factorization of the extended admittance matrix into the product of a lower triangular matrix [L] and an upper triangular matrix [U]. This solution executes a just once and only forward–backward substitution during the iterative solution process. Case studies clearly demonstrate the proposal’s effectiveness in computing the equilibrium point in operating conditions where Newton’s method fails to converge.

A binary-medium-based constitutive model for rocks under cyclic loading

The present study extends the existing three-scale constitutive model to investigate the mechanical response of mudstone under cyclic loading conditions. The research framework encompasses macro, meso, and micro scales, with the mudstone sample treated as a macro Representative Volume Element (RVE) at the macroscale, bonded and frictional elements with distinct mesostructures at the mesoscale, and three-phase composites with varying micro-component contents at the microscale. The transitions at micro, meso, and macro scales are realized through various homogenization methods and the refined binary medium framework.To enhance the model’s predictive capabilities under cyclic loading, isotropic and kinematic hardening effects are introduced to capture the elastoplastic behavior of the frictional elements. To further enhance the model’s numerical stability and convergence, the fixed-point iteration method is introduced to leverage the relationships among macroscopic strain, and the strains of the bonded and frictional elements within the binary-medium framework. Model verification against experimental data enables the prediction of stress and deformation patterns in mudstone samples subjected to cyclic loading, with a notable reduction in model parameters. The model’s enhanced capabilities in capturing material behavior and numerical stability make it a valuable tool for analyzing the response of mudstone structures to cyclic loading scenarios.

Broad learning system based on maximum multi-kernel correntropy criterion

The broad learning system (BLS) is an effective machine learning model that exhibits excellent feature extraction ability and fast training speed. However, the traditional BLS is derived from the minimum mean square error (MMSE) criterion, which is highly sensitive to non-Gaussian noise. In order to enhance the robustness of BLS, this paper reconstructs the objective function of BLS based on the maximum multi-kernel correntropy criterion (MMKCC), and obtains a new robust variant of BLS (MKC-BLS). For the multitude of parameters involved in MMKCC, an effective parameter optimization method is presented. The fixed-point iteration method is employed to further optimize the model, and a reliable convergence proof is provided. In comparison to the existing robust variants of BLS, MKC-BLS exhibits superior performance in the non-Gaussian noise environment, particularly in the multi-modal noise environment. Experiments on multiple public datasets and real application validate the efficacy of the proposed method.

Alcohol-alcohol-based draw solute minimizes the reverse solute flux in forward osmosis desalination.

Recently, alcohol-based draw solute (DS), i.e., alcohol with water, is one of the trending research topics in forward osmosis (FO) because of its performance and ease of regeneration. Nevertheless, the higher reverse solute flux (RSF) of the alcohol-based DS hinders its commercialization in water and wastewater treatment applications. This research aims to minimize the RSF of the alcohol-based DS in FO by investigating the possibility of using alcohol-alcohol-based draw solutes for the first time. Three alcohol-alcohol-based draw solutions, namely, (1) E70 + IPA30 (ethanol: 70% + isopropanol: 30%), (2) E40 + IPA60 (ethanol: 40% + isopropanol: 60%), and (3) E10 + IPA90 (ethanol: 10% + isopropanol: 90%), were prepared and the properties (including osmolality, shear stress, and viscosity) of the DS were first investigated followed by the parametric investigation (concerning temperature and concentration). The results were further analyzed with the fixed-point iterative method in MATLAB to obtain the performance parameters. Results reveal that the E10 + IPA90 mixture yields a lower RSF of 40.62g/m2/h and specific reverse solute flux of3.78g/L with a considerably good water flux and recovery percentage of 11.47 LMH and 26.29%, respectively, as compared to other DS E70 + IPA30 and E40 + IPA60 at 25°C. Thus, E10 + IPA90 is recommended as a potential candidate to be used as a DS in FO.

A Novel Fixed-Point Iteration Approach for Solving Troesch’s Problem

This paper introduces a novel F fixed-point iteration method that leverages Green’s function for solving the nonlinear Troesch problem in Banach spaces, which are symmetric spaces. The Troesch problem, characterized by its challenging boundary conditions and nonlinear nature, is significant in various physical and engineering applications. The proposed method integrates fixed-point theory with Green’s function techniques to develop an iteration process that ensures convergence, stability, and accuracy. The numerical experiments demonstrate the method’s efficiency and robustness, highlighting its potential for broader applications in solving nonlinear differential equations in Banach spaces.

On the efficient non-linear solver for hydraulic fracturing and well cementing simulations based on Anderson acceleration

The aim of this study is to create a fast and stable iterative technique for numerical solution of a quasi-linear elliptic pressure equation. We developed a modified version of the Anderson acceleration (AA) algorithm to fixed-point (FP) iteration method. It computes the approximation to the solutions at each iteration based on the history of vectors in extended space, which includes the vector of unknowns, the discrete form of the operator, and the equation's right-hand side. Several constraints are applied to AA algorithm, including a limitation of the time step variation during the iteration process, which allows switching to the base FP iterations to maintain convergence. Compared to the base FP algorithm, the improved version of the AA algorithm enables a reliable and rapid convergence of the iterative solution for the quasi-linear elliptic pressure equation describing the flow of particle-laden yield-stress fluids in a narrow channel during hydraulic fracturing, a key technology for stimulating hydrocarbon-bearing reservoirs. In particular, the proposed AA algorithm allows for faster computations and resolution of unyielding zones in hydraulic fractures that cannot be calculated using the FP algorithm. The quasi-linear elliptic pressure equation under consideration describes various physical processes, such as the displacement of fluids with viscoplastic rheology in a narrow cylindrical annulus during well cementing, the displacement of cross-linked gel in a proppant pack filling hydraulic fractures during the early stage of well production (fracture flowback), and multiphase filtration in a rock formation. We estimate computational complexity of the developed algorithm as compared to Jacobian-based algorithms and show that the performance of the former one is higher in modelling of flows of viscoplastic fluids. We believe that the developed algorithm is a useful numerical tool that can be implemented in commercial simulators to obtain fast and converged solutions to the non-linear problems described above.

A Two-steps Fixed-point Method for the Simplicial Cone Constrained Convex Quadratic Optimization

In this paper, we deal with the resolution of the simplicial cone constrained convex quadratic optimization (abbreviated SCQO). It is known that the optimality conditions of SCQO is only a standard linear complementarity problem (LCP). Under a suitable condition, the solution of LCP is equivalent to find the solution of an absolute value equations AVE. For its numerical solution, we propose an efficient two-steps fixed point iterative method for solving the AVE. Moreover, we show that this method converges globally linear to the unique solution of the AVE and which is in turn an optimal solution of SCQO. Some numerical results are reported to demonstrate the efficiency of the proposed algorithm. Keywords: Quadratic programming, simplicial cones, absolute value equations, linear complementarity problem, Picard’s fixed point iterative method.

Improved mixture correntropy cubature Kalman filter for attitude estimation

Abstract This work proposes an algorithm based on improved mixture correntropy cubature Kalman filtering (IMC-CKF) to address the issues of low accuracy and susceptibility to complex noise and outlier interference in inertial navigation attitude estimation. First, a combination of Gaussian kernel and Cauchy kernel is suggested to construct mixture correntropy to tackle the problem of single kernel-based correntropy being insufficient to confronted with complex noise. Second, the model fitting loss based on mean square error and measurement fitting loss based on mixture correntropy are used to establish the objective function. The maximum correntropy criterion (MCC) replaces the minimum mean square error (MMSE) criterion, and the fixed-point iteration method is used to solve the objective function, deriving the mixture correntropy matrix to adjust the measurement noise variance. Finally, the semi-trapezoidal membership function is used to determine the mixture correntropy coefficient. Accordingly, the algorithm can adaptively select the proportions of each kernel function according to the respective noise interference situation. Simulations and dynamic–static experiments have been conducted, and the algorithm has been compared with single kernel-based correntropy algorithms and other robust algorithms to confirm its superior precision and stability under complex noise and outlier interference.

Fixed-Point Iteration Method for Uncertain Parameters in Dynamic Response of Systems with Viscoelastic Elements

Fixed-Point Iteration Method for Uncertain Parameters in Dynamic Response of Systems with Viscoelastic Elements

Instantaneous Frequency Extraction for Nonstationary Signals via a Squeezing Operator with a Fixed-Point Iteration Method

Instantaneous Frequency Extraction for Nonstationary Signals via a Squeezing Operator with a Fixed-Point Iteration Method

A new inexact fixed point iteration method for solving tensor absolute value equation

The main purpose of this paper is to solve tensor absolute value equation by using preconditioned techniques and the inexact fixed point iteration method. For an existing preconditioner, we present a new inexact fixed point iteration method for solving tensor absolute value equation. Theoretically, we give the convergence of the proposed method. Numerically, we show the efficiency of the proposed method by two numerical examples.

The steady states of antitone electric systems

The steady states of an antitone electric system are described by an antitone function with respect to the componentwise order. When this function is bounded from below by a positive vector, it has only one fixed point. This fixed point is attractive for the fixed point iteration method. In the general case, we find existence and uniqueness results of fixed points using the set of ‐matrices.

Two-step inertial Tseng’s extragradient method for solving quasimonotone variational inequalities

We focus on quasimonotone variational inequality problems with non-Lipschitz operator and combine some operator theory techniques to establish a weak convergent fixed point iterative method to approximate the solution of the presented problems. The convergence analysis assumes very mild conditions and strictly weaker assumption on the cost operator associated with the problems and thus generalizes and extend recent related results in the literature. Finally, several numerical examples illustrate the potential and advantages of the proposed algorithm over some existing methods are presented.

On control intervals of chaos with Fréchet derivative of certain iterations as discrete dynamical system in Banach spaces

Dynamical systems are one of the interesting concepts where iteration algorithms and chaos can be considered together. In iteration algorithms, one of the basic concepts of fixed point theory, it is well known that the behavior of the iteration mechanism is chaotic if the original transformation is taken as chaotic. One of the natural ways to transform a chaotic system into a dynamical system is through control mechanisms. In this paper, we first consider an iteration class defined on Banach spaces, which is prominent in the literature in terms of both speed and convergence rate. Then, we consider the transformation constituting the iteration class as chaotic and obtain the stability and unstability behaviors of the iterations according to the operator norm by using Gâteaux and Fréchet derivatives representing the direction‐dependent derivative. In this way, we aimed to obtain the chaos control intervals obtained with functions in real space with the help of operators in Banach spaces. Using the operator norm obtained with the Fréchet derivative, we derive interesting dynamical system intervals from a chaotic system with parameter variables of iteration algorithms. It is noteworthy that the parameter variables of the studied iteration classes are the same in some transformation classes. In addition, analytical proofs are followed by computer simulations of parameter‐dependent control intervals considering the logistic operator with a chaotic structure. At the same time, the periodic behavior of the iteration algorithms used in our study is illustrated by the Lyapunov exponent with parameter‐dependent control intervals according to operator norm. Finally, as a real‐life problem, it has been shown that the chaos of the logistic population growth model with dynamic features can be controlled by chaos control mechanisms established by fixed point iteration methods.

An inexact fixed point iteration method for solving absolute value equation

An inexact fixed point iteration method for solving absolute value equation

On fixed point iterative methods for solving non-Lipschitz quasi-monotone variational inequalities

On fixed point iterative methods for solving non-Lipschitz quasi-monotone variational inequalities

Analyzing Stability and Data Dependence Notions by a Novel Jungck-Type Iteration Method

Finding the ideal circumstances for a mapping to have a fixed point is the fundamental goal of fixed point theory. These criteria can also be used for the structure under investigation. One of this theory’s most well-known theorems, Banach’s fixed point theorem, has been expanded adopting various methods, making it possible to conduct numerous research studies. Thanks to the Jungck-Contraction Theorem, which has been proven through commutative mappings, many fixed point theorems have been obtained using classical fixed point iteration methods and newly defined methods. This study aims to investigate the convergence, stability, convergence rate, and data dependency of the new four-step fixed-point iteration method. Nontrivial examples are also included to support some of the results herein.

Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge

We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n, where one edge has weight α∈C, with Re(α)<0, and all the others have weights 1. This paper is a sequel of a previous one where we considered Re(α)∈[0,1] (Grudsky et al., 2022 [12]). We prove that for Re(α)<0 and n>Re(α−1)/Re(α), one eigenvalue is negative while the others belong to [0,4] and are distributed as the function x↦4sin2⁡(x/2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4Re(α)2/(2Re(α)−1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [0,4] and the outlier. We also compute the eigenvectors and their norms.