Fixed-point Iteration 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

THE FIXED POINT ITERATIONS METHOD FOR THE CONTRACTION MAPPING

This paper studies the fixed point iterations method that considers the contraction mapping and one of its applications for the parameter re-estimation of the normal (Gaussian) distribution, where the presence of outliers is considered. In order to study the observed method Banach’s fixed point theorem is presented, where it is shown that the contraction property is directly related to the Jacobian of the observed mapping. Furthermore, convergence analysis is conducted in order to estimate the convergence order of the observed model.

Use of molybdenum as a pressure calibrant in a cubic press and sound velocity measurement of tungsten to 5 GPa

ABSTRACT The pressure limit of the conventional cubic press is normally below 6 GPa, and the suitable pressure calibrant of the fixed-point method is scarce. In this study, we developed a new method for pressure determination using an acoustic travel time approach. Based on the reported ultrasonic travel times of polycrystalline molybdenum under high pressure in literature, an acoustic gauge of polycrystalline molybdenum was derived. The shear wave travel times of molybdenum were measured at high pressures and utilized for the pressure calibration in a cubic press. The pressures derived from this new travel time method are in excellent agreement with those from the fixed-point method. Subsequent compressional and shear wave velocities of polycrystalline tungsten were measured up to about 5 GPa at room temperature to further verify this method. This travel time method for pressure calibration is valuable for the sound velocity measurement of other materials in an offline laboratory.

Computing the Noncommutative Inner Rank by Means of Operator-Valued Free Probability Theory

AbstractWe address the noncommutative version of the Edmonds’ problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm.

Tensor power flow formulations for multidimensional analyses in distribution systems

In this paper, we present two multidimensional power flow formulations based on a fixed-point iteration (FPI) algorithm to efficiently solve hundreds of thousands of Power flows (PFs) in distribution systems. The presented algorithms are the base for a new TensorPowerFlow (TPF) tool and shine for their simplicity, benefiting from multicore Central processing unit (CPU) and Graphics processing unit (GPU) parallelization. We also focus on the mathematical convergence properties of the algorithm, showing that its unique solution is at the practical operational point. The proof is validated using numerical simulations showing the robustness of the FPI algorithm compared to the classical Newton–Raphson (NR) approach. In the case study, a benchmark with different PF solution methods is performed, showing that for applications requiring a yearly simulation at 1-minute resolution, the computation time is decreased by a factor of 164, compared to the NR in its sparse formulation. Finally, a set of applications is described, highlighting the potential of the proposed formulations over a wide range of analyses in distribution systems.

Generalized Linear Differential Equation using Hyers - Ulam Stability Approach

In this paper, We demonstrate the Hyers - Ulam stability of linear differential equation of fourth order. We interact with the differential equation\begin{align*}\gamma^{iv} (\omega) + \rho_1 \gamma{'''} (\omega)+ \rho_2 \gamma{''} (\omega) + \rho_3 \gamma' (\omega) + \rho_4 \gamma(\omega) = \chi(\omega),\end{align*}where $\gamma \in c^4 [\alpha,\beta], \chi \in [\alpha,\beta]$. Hyers-Ulam stability concerns the robustness of solutions of functional equations under small perturbations, ensuring that a solution approximately satisfying the equation is close to an exact solution. We extend this concept to fourth-order linear differential equations and continuous functions. Using fixed-point methods and various norms, we establish conditions under which such equations exhibit Hyers-Ulam stability. Several illustrative examples are provided to demonstrate the application of these results in specific cases, contributing to the growing understanding of stability in higher-order differential equations. Our findings have implications in both theoretical research and practical applications in physics and engineering.

Efficient Analysis of Overdispersed Data Using an Accurate Computation of the Dirichlet Multinomial Distribution.

Modeling count data using suitable statistical distributions has been instrumental for analyzing the patterns it conveys. However, failing to address critical aspects, like overdispersion, jeopardizes the effectiveness of such an analysis. In this paper, overdispersed count data is modeled using the Dirichlet Multinomial (DM) distribution by maximizing its likelihood using a fixed-point iteration algorithm. This is achieved by estimating the DM distribution parameters while comparing the recent Languasco-Migliardi (LM), and the Yu-Shaw (YS) procedures, which address the well-known computational difficulties of evaluating its log-likelihood. Experiments were conducted using multiple datasets from different domains spanning polls, images, and IoT network traffic. They all showed the superiority of the LM procedure as it succeeded at estimating the DM parameters at the designated level of accuracy in all experiments, while the YS procedure failed to produce sufficiently accurate results (or any results at all) in several experiments. Moreover, the LM procedure achieved a speedup that ranged from 2-fold to 20-fold over YS.

Application of CR Iteration Scheme in the Generation of Mandelbrot Sets of zp+logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + \\log c^t$$\\end{document} Function

Mandelbrot set is one of the most fascinating objects in mathematics, so in the literature, one can find many studies related to this set. Moreover, there are many different extensions and generalizations of the classical Mandelbrot set. The two most popular extensions are the use of various kinds of functions and iterations. In this paper, firstly, we replace the constant c in the classical zp+c\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$z^p + c$$\\end{document} function used in the definition of Mandelbrot set with logct\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\log c^t$$\\end{document}, where t∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\in \\mathbb {R}$$\\end{document} and t≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\ge 1$$\\end{document}. Secondly, we replace the Picard iteration with the CR iteration scheme. For this combination of function and iteration, we prove the escape criterion for the escape-time algorithm used to generate some images of the proposed sets. Moreover, we study the dependency between the iteration’s parameters and two numerical measures (average number of iterations and generation time). We show that this dependency is complex and non-linear.

Adaptive time step selection for spectral deferred correction

AbstractSpectral Deferred Correction (SDC) is an iterative method for the numerical solution of ordinary differential equations. It works by refining the numerical solution for an initial value problem by approximately solving differential equations for the error, and can be interpreted as a preconditioned fixed-point iteration for solving the fully implicit collocation problem. We adopt techniques from embedded Runge-Kutta Methods (RKM) to SDC in order to provide a mechanism for adaptive time step size selection and thus increase computational efficiency of SDC. We propose two SDC-specific estimates of the local error that are generic and do not rely on problem specific quantities. We demonstrate a gain in efficiency over standard SDC with fixed step size and compare efficiency favorably against state-of-the-art adaptive RKM.

Control-Oriented Free-Boundary Equilibrium Solver for Tokamaks

A free-boundary equilibrium solver for an axisymmetric tokamak geometry was developed based on the finite difference method and Picard iteration in a rectangular computational area. The solver can run either in forward mode, where external coil currents are prescribed until the converged magnetic flux function ψ(R,Z) map is achieved, or in inverse mode, where the desired plasma boundary, with or without an X-point, is prescribed to determine the required coil currents. The equilibrium solutions are made consistent with prescribed plasma parameters, such as the total plasma current, poloidal beta, or safety factor at a specified flux surface. To verify the mathematical correctness and accuracy of the solver, the solution obtained using this numerical solver was compared with that from an analytic fixed-boundary equilibrium solver based on the EAST geometry. Additionally, the proposed solver was benchmarked against another numerical solver based on the finite-element and Newton-iteration methods in a triangular-based mesh. Finally, the proposed solver was compared with equilibrium reconstruction results in DIII-D experiments.

Stability of uncertain delay differential equations via Lyapunov direct and fixed point methods

Stability of uncertain delay differential equations via Lyapunov direct and fixed point methods

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 novel approach to the convergence analysis of chaotic dynamics in fractional order Chua’s attractor model employing fixed points

This article uses a fixed point method to connect with Chua’s attractor model, incorporating the Atangana–Baleanu derivative using a two-step Lagrange polynomial. We introduce novel fixed point theorems using some special contractions followed by graphical representations of the convergence behavior of the iterative process. The uniqueness and existence of fixed points are demonstrated through theoretical proofs and numerical simulations. This approach of its first kind substantiates the existence and uniqueness of the model, enriching the understanding of its chaotic dynamics. Furthermore, numerical investigations explore distinct scenarios of Chua’s model, encompassing the classic Chua’s attractor, the double scroll attractor, chaos in Chua’s circuit, and behaviors such as cycle behavior and higher parametric values. Each case is analyzed graphically, providing insights into the system’s complex dynamics and validating theoretical predictions.

Numerical solutions of boundary value problems via fixed point iteration

Numerical solutions of boundary value problems via fixed point iteration

A recent fixed point method based on two inertial terms

A recent fixed point method based on two inertial terms

A Meshless Radial Point Interpolation Method for Solving Fractional Navier–Stokes Equations

This paper aims to develop a meshless radial point interpolation (RPI) method for obtaining the numerical solution of fractional Navier–Stokes equations. The proposed RPI method discretizes differential equations into highly nonlinear algebraic equations, which are subsequently solved using a fixed-point method. Furthermore, a comprehensive analysis regarding the effects of spatial and temporal discretization, polynomial order, and fractional order is conducted. These factors’ impacts on the accuracy and efficiency of the solutions are discussed in detail. It can be shown that the meshless RPI method works quite well for solving some benchmark problems accurately.

Numerical simulation of a fractional Davey–Stewartson model via a conservative scheme: preservation of mass, energy and momenta

In the present manuscript, we depart from a two-dimensional form of the nonlinear Davey–Stewartson system from fluid dynamics, consisting of two partial differential equations with two unknown functions, one of them is complex-valued and the other real-valued. This model possesses four conserved quantities, namely, the mass, the energy and two momenta. In a first stage, the mathematical model is generalized to the fractional scenario considering Riesz fractional operators and three different orders. We prove mathematically that this system also possesses four quantities that are preserved in time, and which are fractional extensions of the classical mass, energy and momentum operators. Motivated by this fact, we introduce a numerical scheme for the mathematical model by making use of fractional-order centred differences. By using similar arguments as in the analysis of the continuous system, we propose some discretizations of the mass, the energy and the momenta, and we proved theoretically that these quantities are also conserved in the discrete case. In our theoretical analysis, we establish mathematically the properties of consistency for the mathematical model as well as for the discrete conserved quantities. Computationally, the finite-difference scheme is implemented using a fixed-point algorithm. Various computer simulations are performed to illustrate the conservation of the discrete quantities. Finally, we provide some interesting perspectives of research at the end of this work. This report is the first work in which a conservative finite-difference scheme for the Davey–Stewartson system is designed and rigorously analysed for the conservation properties, for both the integer-order and fractional cases.

Improved order in Hilfer fractional differential systems: Solvability and optimal control problem for hemivariational inequalities

The main objective of this study is to analyze the optimal control results and solvability concerning Hilfer fractional differential hemivariational inequalities within the range of (1<ϱ<2). Initially, we explore the solvability of a mild solutions for the Hilfer fractional hemivariational inequality. Subsequently, we investigate the optimal control strategies for the given problems by employing cost functionals, cosine operators, mild solutions, the fixed point method for multivalued functions, and a generalized Clarke subdifferential approach. To illustrate our findings, we provide a practical example and a filter diagram.

Coordinate-update algorithms can efficiently detect infeasible optimization problems

Coordinate update/descent algorithms are widely used in large-scale optimization due to their low per-iteration cost and scalability, but their behavior on infeasible or misspecified problems has not been much studied compared to the algorithms that use full updates. For coordinate-update methods to be as widely adopted to the extent so that they can be used as engines of general-purpose solvers, it is necessary to also understand their behavior under pathological problem instances. In this work, we show that the normalized iterates of randomized coordinate-update fixed-point iterations (RC-FPI) converge to the infimal displacement vector and use this result to design an efficient infeasibility detection method. We then extend the analysis to the setup where the coordinates are defined by non-orthonormal basis using the Friedrichs angle and then apply the machinery to decentralized optimization problems.