Fixed-point Iteration Research Articles

Viscosity-type inertial iterative methods for variational inclusion and fixed point problems

<abstract><p>In this paper, we have introduced some viscosity-type inertial iterative methods for solving fixed point and variational inclusion problems in Hilbert spaces. Our methods calculated the viscosity approximation, fixed point iteration, and inertial extrapolation jointly in the starting of every iteration. Assuming some suitable assumptions, we demonstrated the strong convergence theorems without computing the resolvent of the associated monotone operators. We used some numerical examples to illustrate the efficiency of our iterative approaches and compared them with the related work.</p></abstract>

Data transfers for full core heterogeneous reactor high-fidelity multiphysics studies

Multiphysics simulations for nuclear reactor analysis are usually performed by resorting to operator splitting and fixed point iterations between single-physics solvers. This enables the separate solution of each physics, such as neutronics, fuel performance, and thermal hydraulics, on meshes tailored to the requirements of the respective numerical discretizations of the equations. As the equations are coupled, several fields must be transferred between singlephysics solves. Projecting fields between meshes while preserving order of accuracy, conservation properties, and mapping non-overlapping geometries is a complex endeavour. This conference paper will present the transfers as implemented in MOOSE, which can handle arbitrary meshes, arbitrary mappings, conservation of integral quantities, and are made to scale with distributed simulations on both ends of the transfers. Their adequacy for advanced nuclear reactor multiphysics coupling is shown through examples and numerical studies.

Stability analysis on the post-quantum structure of a boundary value problem: application on the new fractional $ (p, q) $-thermostat system

<abstract><p>In this paper, we discussed some qualitative properties of solutions to a thermostat system in the framework of a novel mathematical model designed by the new $ (p, q) $-derivatives in fractional post-quantum calculus. We transformed the existing standard model into a new control thermostat system with the help of the Caputo-like $ (p, q) $-derivatives. By the properties of the $ (p, q) $-gamma function and applying the fractional Riemann-Liouville-like $ (p, q) $-integral, we obtained the equivalent $ (p, q) $-integral equation corresponding to the given Caputo-like post-quantum boundary value problem ($ (p, q) $-BOVP) of the thermostat system. To conduct an analysis on the existence of solutions to this $ (p, q) $-system, some theorems were proved based on the fixed point methods and the stability analysis was done from the Ulam-Hyers point of view. In the applied examples, we used numerical data to simulate solutions of the Caputo-like $ (p, q) $-BOVPs of the thermostat system with respect to different parameters. The effects of given parameters in the model will show the performance of the thermostat system.</p></abstract>

The efficacy of a new fixed point method in nonlinear equation solutions

The key objective of this research is to establish a new approach to the classical fixed point method for solving nonlinear equations by using the new theorem of a-fixed point established in [1] and the conformable fractional derivative defined in [6]. As an application, we establish a new a-fixed point algorithm as a numerical result.

Fuzzy stability of mixed type functional equations in Modular spaces

In this article, our aim is to find some stability results for mixed type cubic and quartic functional equations in fuzzy modular spaces using the fundamental results of fixed-point theory. The fixed point method provides one of the effective techniques that can be used to investigate the fuzzy stability of a mixed type cubic and quartic functional equations.

An extrapolated fixed-point optimization method for strongly convex smooth optimizations

<abstract><p>In this work, we focused on minimizing a strongly convex smooth function over the common fixed-point constraints. We proposed an extrapolated fixed-point optimization method, which is a modified version of the extrapolated sequential constraint method with conjugate gradient direction. We proved the convergence of the generated sequence to the unique solution to the considered problem without boundedness assumption. We also investigated some numerical experiments to underline the effectiveness and performance of the proposed method.</p></abstract>

Fractional tempered differential equations depending on arbitrary kernels

<abstract><p>In this paper, we expanded the concept of tempered fractional derivatives within both the Riemann-Liouville and Caputo frameworks, introducing a novel class of fractional operators. These operators are characterized by their dependence on a specific arbitrary smooth function. We then investigated the existence and uniqueness of solutions for a particular class of fractional differential equations, subject to specified initial conditions. To aid our analysis, we introduced and demonstrated the application of Picard's iteration method. Additionally, we utilized the Gronwall inequality to explore the stability of the system under examination. Finally, we studied the attractivity of the solutions, establishing the existence of at least one attractive solution for the system. Throughout the paper, we provide examples and remarks to support and reinforce our findings.</p></abstract>

Robust Federated Learning: Maximum Correntropy Aggregation Against Byzantine Attacks.

As an emerging decentralized machine learning technique, federated learning organizes collaborative training and preserves the privacy and security of participants. However, untrustworthy devices, typically Byzantine attackers, pose a significant challenge to federated learning since they can upload malicious parameters to corrupt the global model. To defend against such attacks, we propose a novel robust aggregation method-maximum correntropy aggregation (MCA), which applies the maximum correntropy criterion (MCC) to derive a central value from parameters. Different from the previous use of MCC for denoising, we utilize it as a similarity metric to measure parameter distribution and aggregate a robust center. Correntropy in MCC, with all even-order moments of the parameter, contains high-order statistical properties, which allows for a comprehensive capture of parameter characteristics, thus helping to prevent interference from attackers. Meanwhile, correntropy extracts information from the parameters themselves, without requiring the proportion of malicious attackers. Through the fixed-point iteration, we solve the optimization objective, demonstrating the linear convergence of the iteration formula. Theoretical analysis reveals the robustness aggregation property of MCA and the error bound between MCA and the global optimal solution, with linear convergence to the optimal solution neighborhood. By performing independent identically distribution (IID) and non-IID experiments on three different datasets, we show that MCA exhibits significant robustness under mainstream attacks, whereas other methods cannot withstand all of them.

Numerical analysis of the frequency-dependent Jiles-Atherton hysteresis model using the example of Terfenol-D material

<p>The Jiles-Atherton model has been widely used in describing the hysteretic property of a magnetic material or device. However, the calculation errors are not so easily discovered. With a complex expression, the frequency-dependent Jiles-Atherton model should be solved numerically with appropriate settings. This paper proposes an effective solving method for this model and describes some necessary analysis built on the numerical results. In the numerical method proposed in this manuscript, the anhysteretic magnetization was calculated by the secant method, and the trapezoidal rule was utilized to form the implicit function, which can be calculated by the fixed-point iteration. Compared to the other common methods, the proposed one has a friendly expression and fast computation speed. The Terfenol-D material was taken as an example for the numerical analysis. The feasible region was determined and the commonly used approximation that neglects the term of the magnetic field when calculating the magnetic induction intensity was tested. At last, the required number of sampling points per period was reached to guarantee high precision from analyzing its influence on the computation precision. The proposed numerical method is helpful for high-precision solutions of the frequency-dependent Jiles-Atherton model. The results from the numerical analysis can also help users avoid some incorrect calculations when employing this hysteresis model.</p>

Nontrivial solutions for a Hadamard fractional integral boundary value problem

<abstract><p>In this paper, we studied a Hadamard-type fractional Riemann-Stieltjes integral boundary value problem. The existence of nontrivial solutions was obtained by using the fixed-point method when the nonlinearities can be superlinear, suberlinear, and have asymptotic linear growth. Our results improved and generalized some results of the existing literature.</p></abstract>

Fixed points of multiplicative closed graph operators on b-multiplicative metric spaces

This article discusses fixed point iterations of some multiplicative contraction mappings with variations in b-multiplicative metric space domains. A map that satisfies the multiplicative contraction condition has been proposed for an increasing sequence of subsets of a b-multiplicative metric space, where one element of the sequence is mapped into the following member of the sequence. Additionally, several fixed point conclusions are established for various multiplicative contraction mappings with multiplicative closed graphs.

A faster fixed point iterative algorithm and its application to optimization problems

<p>In this paper, we studied the AA-iterative algorithm for finding fixed points of the class of nonlinear generalized $ (\alpha, \beta) $-nonexpansive mappings. First, we proved weak convergence and then proved several strong convergence results of the scheme in a ground setting of uniformly convex Banach spaces. We gave a few numerical examples of generalized $ (\alpha, \beta) $-nonexpansive mappings to illustrate the major outcomes. One example was constructed over a subset of a real line while the other one was on the two dimensional space with a taxicab norm. We considered both these examples in our numerical computations to show that our iterative algorithm was more effective in the rate of convergence corresponding to other fixed point algorithms of the literature. Some 2D and 3D graphs were obtained that supported graphically our results and claims. As applications of our major results, we solved a class of fractional differential equations, 2D Voltera differential equation, and a convex minimization problem. Our findings improved and extended the corresponding results of the current literature.</p>

Primal-Dual Fixed Point Algorithms Based on Adapted Metric for Distributed Optimization with Bounded Communication Delays

Primal-Dual Fixed Point Algorithms Based on Adapted Metric for Distributed Optimization with Bounded Communication Delays

Some fixed point and stability results in $ b $-metric-like spaces with an application to integral equations on time scales

<abstract><p>This paper presents the stability theorem for the $ T $-Picard iteration scheme and establishes the existence and uniqueness theorem for fixed points concerning $ T $-mean nonexpansive mappings within $ b $-metric-like spaces. The outcome of our fixed point theorem substantiated the existence and uniqueness of solutions to the Fredholm-Hammerstein integral equations defined on time scales. Additionally, we provided two numerical examples from distinct time scales to support our findings empirically.</p></abstract>

On the Usage of Orthogonal Polynomials with Picard Iteration Method to Find Numerical Solutions of Neutrosophic Non-Linear Ordinary and Partial Differential Equations

This research aims to modify the Picard iteration method by hybridizing it with some orthogonal polynomials and then applying the hybrid method in solving neutrosophic nonlinear elementary value problems. This method is based on modifying the Picard iteration method by approximating the right-hand side of the neutrosophic differential equation of the studied problem either by Legendre polynomials or by Chebyshev polynomials of the first kind to obtain two different hybrids of the Picard iteration method. Also, we apply this modification to neutrosophic elementary value problems represented by neutrosophic nonlinear and right-handed nonlinear differential equations to demonstrate the reliability and efficiency of the proposed modified method. For this goal, we prove how effective this method is, we calculate the neutrosophic absolute error of approximate solutions resulting from the application of the proposed modification of the Picard iteration method and with the exact solution.

Fast and Accurate Predictor-Corrector Methods Using Feedback-Accelerated Picard Iteration for Strongly Nonlinear Problems

Fast and Accurate Predictor-Corrector Methods Using Feedback-Accelerated Picard Iteration for Strongly Nonlinear Problems

A Unified Model for Large-Scale Inexact Fixed-Point Iteration: A Stochastic Optimization Perspective

A Unified Model for Large-Scale Inexact Fixed-Point Iteration: A Stochastic Optimization Perspective

A Modified Fixed-Point Iteration Algorithm for Magnetic Field Computation With Hysteresis Models

A Modified Fixed-Point Iteration Algorithm for Magnetic Field Computation With Hysteresis Models

Analysis of fractal-fractional Alzheimer's disease mathematical model in sense of Caputo derivative

<abstract> <p>Alzheimer's disease stands as one of the most widespread neurodegenerative conditions associated with aging, giving rise to dementia and posing significant public health challenges. Mathematical models are considered as valuable tools to gain insights into the mechanisms underlying the onset, progression, and potential therapeutic approaches for AD. In this paper, we introduce a mathematical model for AD that employs the fractal fractional operator in the Caputo sense to characterize the temporal dynamics of key cell populations. This model encompasses essential elements, including amyloid-<italic>β</italic> ($\mathbb{ A_\beta }$), neurons, astroglia and microglia. Using the fractal fractional operator, we have established the existence and uniqueness of solutions for the model under consideration, employing Leray-Schaefer's theorem and the Banach fixed-point methods. Utilizing functional techniques, we have analyzed the proposed model stability under the Ulam-Hyers condition. The suggested model has been numerically simulated by using a fractional Adams-Bashforth approach, which involves a two-step Lagrange polynomial. For numerical simulations, different ranges of fractional order values and fractal dimensions are considered. This new fractal fractional operator in the form of the Caputo derivative was determined to yield better results than an ordinary integer order. Various outcomes are shown graphically by for different fractal dimensions and arbitrary orders.</p> </abstract>

Solving delay integro-differential inclusions with applications

<abstract><p>This work primarily delves into three key areas: the presence of mild solutions, exploration of the topological and geometrical makeup of solution sets, and the continuous dependency of solutions on a second-order semilinear integro-differential inclusion. The Bohnenblust-Karlin fixed-point method has been integrated with Grimmer's theory of resolvent operators. Ultimately, the study delves into a mild solution for a partial integro-differential inclusion to showcase the achieved outcomes.</p></abstract>