Fixed-point Iteration Scheme Research Articles

On “A new three-step fixed point iteration scheme with strong convergence and applications”

<p>This note delves into the convergence analysis of several iterative methods and elucidates their behaviors. Furthermore, we demonstrate that the findings presented in “A new three-step fixed point iteration scheme with strong convergence and applications” are not entirely novel. In particular, some of the results either overlap with or restate previously established methods without introducing significant innovations.</p>

Computational insights into tumor invasion dynamics: A finite element approach

The finite element scheme is proposed and analyzed for the solution of an acid-mediated tumor invasion model. The reaction–diffusion equation shows the evolution in the tumor cell density, H+ ions concentration, and healthy tissue density over time. The coupled non-linear partial differential equations are discretized in time with the implicit Euler method and in space with standard Galerkin finite element. To solve the non-linear and coupled terms of the system a fixed point iteration scheme is presented. Moreover, a mass-lumped scheme is adopted to reduce the computation cost. The cut-off method is used to compute the bounded solutions of the PDEs. Finally, The effects of proliferation rate and healthy tissue degradation rate are investigated.

A coupled steady thermo-electromagnetic problem in axisymmetric geometries. Mathematical and numerical analysis

Abstract This paper focuses on the analysis of a steady thermo-electromagnetic problem related to the modeling of induction heating processes. Taking advantage of the cylindrical symmetry, the original three-dimensional problem can be reduced to a two-dimensional one on a meridional section, provided that the current density has only the azimuthal component. A variational formulation is presented in appropriately weighted Sobolev spaces, and the existence of a solution is established by employing a fixed-point argument. Furthermore, uniqueness and additional regularity results are proved under reasonable assumptions on the physical coefficients. A finite element approximation combined with a fixed-point iteration scheme is proposed for the numerical solution of the problem. A priori error estimates are obtained to quantify the accuracy of the approximation. Finally, numerical results are reported to validate the theoretical estimates and assess the performance of the method in a physical application.

Development of multi-physics calculation method in lead cooled fast reactor code system MOSASAUR

A full-core three-dimensional multi-physics calculation method was researched based on the Lead cooled fast reactor (LFR) code system MOSASAUR, which is focused on both core steady-state and transient analyses of LFR. In the previous version of MOSASAUR, cross-sections generation module and core steady-state simulation module have been developed. In this research, the stiffness confinement method is employed to solve the time-dependent multi-group neutron transport equation to expand the capabilities of core transient analyses. And the thermal hydraulic calculation method is developed as thermal feedback module for core steady-state and transient analyses. Based on the fixed-point iteration scheme, the neutron module and thermal hydraulic module are coupled at each time step. The LMW benchmark and the problem based on ORAL 19-rod bundle are employed to verify the accuracy of transient calculation and thermal hydraulic module respectively. Finally, the multi-physics calculation method is utilized to simulate the transient process of the MicroURANUS core. The simulation results demonstrate the capability of the constructed multi-physics calculation framework to accurately simulate the transient behaviors of LMRs. Numerical results showed the good accuracy of the newly-developed multi-physics calculation module with MOSASAUR.

Multidisciplinary analysis and structural optimization for the aeroelastic sizing of a UAV wing using open-source code integration

PurposeThis paper aims to propose a structural sizing approach of an unmanned aerial vehicle (UAV) wing that takes into account the aeroelasticity effects through a fluid–structure interaction analysis.Design/methodology/approachThe sizing approach proposed in this study is an iterative process, each iteration of which consists of two sub-loops, a multidisciplinary analysis (MDA) loop followed by a structural optimization loop. The MDA loop seeks the aeroelastic equilibrium between aerodynamic forces and structural displacements using a fixed-point iteration scheme. Once the equilibrium is reached, the converged pressure loads are used for the structural optimization, which aims to find the structural thicknesses that minimize the wing weight under failure criteria. The two sub-loops are run sequentially in an iterative process until the mass is converged. The analysis models are implemented in open-source software, namely, PANUKL for aerodynamics and MYSTRAN for structures, while the whole process is automated with Python and integrated in the open-source optimization framework OpenMDAO.FindingsThe approach was applied to the design of the Predator MQ-1 wing. The results of the MDAs show the convergence of the wing deformations to the flight shape after few iterations. At the end of the aeroelastic sizing loop, the result is a structurally sized wing with minimal weight considering the aeroelasticity effects.Originality/valueThe approach proposed takes into account the wing aero-structural coupling effects while sizing its structure instead of a fixed load distribution. In addition, the approach is fully based on open-source codes, which are freely available for public use and can be fully reproducible.

A New Robust Iterative Scheme Applied in Solving a Fractional Diffusion Model for Oxygen Delivery via a Capillary of Tissues

In this paper, we constructed a new and robust fixed point iterative scheme called the UO iterative scheme for the approximation of a contraction mapping. The scheme converges strongly to the fixed point of a contraction mapping. A rate of convergence result is shown with an example, and our scheme, when compared, converges faster than some existing iterative schemes in the literature. Furthermore, the stability and data dependence results are shown. Our new scheme is applied in the approximation of the solution to the oxygen diffusion model. Finally, our results are applied in the approximation of the solution to the boundary value problems using Green’s functions with an example.

A modified fixed point iterative scheme based on Green’s function with application to BVPs

This paper presents a new faster iteration approach that is based upon the well-known three-step fixed point iteration scheme called SP-iterative scheme for a broad and larger class of non-linear BVPs. The scheme is constructed using Green’s function based integral operator and the new modified scheme in the paper will be called SP–Green’s iterative scheme. We present a fixed point convergence theorem using some assumption on the sequences of scalars in our scheme. We also prove the weak [Formula: see text]-stability for our scheme and present a numerical example to support our scheme. We provide eventually applications of our scheme to a second-order BVP and a singular BVP of isothermal gas sphere.

A Novel Meshfree Method for Nonlinear Equations in Flow through Porous Media and Electrohydrodynamic Flows

In this study, an efficient meshfree numerical method is introduced for solving the nonlinear boundary value problems. The method of fundamental solutions (MFS) is one of the most popular among meshfree methods. While traditionally limited to linear and homogeneous problems, this study extends the applicability of the MFS to include nonhomogeneous and nonlinear equations. To achieve this, an extended MFS is combined with a fixed-point iteration scheme. This developed framework is benchmarked to address two different flow problems. The first involves fluid flow through porous media in a channel governed by the nonlinear Brinkman-Forchheimer equation. The second problem pertains to electrohydrodynamic (EHD) flow in a circular conduit. The obtained solutions are compared with the finite element method and the solutions available in the existing literature.

Recent Developments in Iterative Algorithms for Digital Metrics

This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, we extend the idea of Iteration process Mann, Ishikawa, Agarwal, and Thakur based on the ϝ-Stable Iterative Scheme in digital metric space. We also design some fractal images, which frame the compression of Fixed-Point Iterative Schemes and contractive mappings. Furthermore, we present a concrete example that exemplifies the motivation behind our investigations. Moreover, we provide an application of the proposed Fractal image and Sierpinski triangle that compress the works by storing images as a collection of digital contractions, which addresses the issue of storing images with less storage memory in this paper.

An efficient non-negative least mean squares algorithm based on q-gradient for system identification

The system identification under non-negative constraints problem is a common and important one in the real-life problems. A non-negative least mean squares algorithm was proposed to address such problems. However, it suffers from slow and unbalanced convergence. Motivated by the introduction of non-Newtonian gradient in least mean squares algorithm can accelerate the convergence. In this paper, an efficient non-negative least mean squares algorithm based on q-gradient is developed. At first, the q-gradient of the mean square error cost function is derived from the definition of Jackson's derivative. Then, the modified algorithm is developed by replacing the conventional gradient with the q-gradient on a fixed-point iteration scheme using Karush-Kuhn-Tucker condition. Simulations for stationary and non-stationary system identification are conducted to illustrate the effectiveness of the developed algorithm for constrain the system parameters to be non-negative and accelerate the convergence speed. Besides, it can balance the convergence rate for widely distributed parameters, especially for the convergence of small parameters. For practical application, the accuracy of system identification when the order of the adaptive filter is larger than the order of the system is also verified by the simulation.

A novel Picard–Ishikawa–Green's iterative scheme for solving third‐order boundary value problems

The purpose of this paper is to introduce a novel fixed point iterative scheme based on Green's function, called the Picard–Ishikawa–Green's iterative scheme and use it in approximating the solution of boundary value problems (BVPs). It is proved that Picard–Ishikawa–Green's scheme converges strongly for an integral operator which represents the solution of BVP and the scheme is stable. Moreover, we prove that the integral operator is a contraction. Furthermore, it is shown that the novel scheme converges faster than all of Ishikawa–Green's, Khan–Green's, and Mann–Green's schemes. Finally, numerical examples are given to substantiate the validity of our results for third‐order BVPs. Our results extend and generalize several other results in literature.

Higher-Order Finite Element Methods for the Nonlinear Helmholtz Equation

In this work, we analyze the finite element method with arbitrary but fixed polynomial degree for the nonlinear Helmholtz equation with impedance boundary conditions. We show well-posedness and error estimates of the finite element solution under a resolution condition between the wave number k, the mesh size h and the polynomial degree p of the form “k(kh)p+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k(kh)^{p+1}$$\\end{document} sufficiently small” and a so-called smallness of the data assumption. For the latter, we prove that the logarithmic dependence in h from the case p=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p=1$$\\end{document} in Wu and Zou (SIAM J Numer Anal 56(3):1338–1359, 2018) can be removed for p≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge 2$$\\end{document}. We show convergence of two different fixed-point iteration schemes. Numerical experiments illustrate our theoretical results and compare the robustness of the iteration schemes with respect to the size of the nonlinearity and the right-hand side data.

A novel iterative scheme for solving delay differential equations and third order boundary value problems via Green's functions

<abstract><p>In this paper, we constructed a novel fixed point iterative scheme called the Modified-JK iterative scheme. This iteration process is a modification of the JK iterative scheme. Our scheme converged weakly to the fixed point of a nonexpansive mapping and strongly to the fixed point of a mapping satisfying condition (E). We provided some examples to show that the new scheme converges faster than some existing iterations. Stability and data dependence results were proved for this iteration process. To substantiate our results, we applied our results to solving delay differential equations. Furthermore, the newly introduced scheme was applied in approximating the solution of a class of third order boundary value problems (BVPs) by embedding Green's functions. Moreover, some numerical examples were presented to support the application of our results to BVPs via Green's function. Our results extended and generalized other existing results in literature.</p></abstract>

A Fixed-Point Iteration Scheme for Sensitivity-Based Distributed Optimal Control

A Fixed-Point Iteration Scheme for Sensitivity-Based Distributed Optimal Control

Joint Beamforming and Phase Shifts Design for RIS-Aided Multi-User Full-Duplex Systems in Smart Cities.

Full-duplex (FD) and reconfigurable intelligent surface (RIS) are potential technologies for achieving wireless communication effectively. Therefore, in theory, the RIS-aided FD system is supposed to enhance spectral efficiency significantly for the ubiquitous Internet of Things devices in smart cities. However, this technology additionally induces the loop-interference (LI) of RIS on the residual self-interference (SI) of the FD base station, especially in complicated urban outdoor environments, which will somewhat counterbalance the performance benefit. Inspired by this, we first establish an objective and constraints considering the residual SI and LI in two typical urban outdoor scenarios. Then, we decompose the original problem into two subproblems according to the variable types and jointly design the beamforming matrices and phase shifts vector methods. Specifically, we propose a successive convex approximation algorithm and a soft actor-critic deep reinforcement learning-related scheme to solve the subproblems alternately. To prove the effectiveness of our proposal, we introduce benchmarks of RIS phase shifts design for comparison. The simulation results show that the performance of the low-complexity proposed algorithm is only slightly lower than the exhaustive search method and outperforms the fixed-point iteration scheme. Moreover, the proposal in scenario two is more outstanding, demonstrating the application predominance in urban outdoor environments.

Regularized Normalization Methods for Solving Linear and Nonlinear Eigenvalue Problems

To solve linear and nonlinear eigenvalue problems, we develop a simple method by directly solving a nonhomogeneous system obtained by supplementing a normalization condition on the eigen-equation for the uniqueness of the eigenvector. The novelty of the present paper is that we transform the original homogeneous eigen-equation to a nonhomogeneous eigen-equation by a normalization technique and the introduction of a simple merit function, the minimum of which leads to a precise eigenvalue. For complex eigenvalue problems, two normalization equations are derived utilizing two different normalization conditions. The golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues, and simultaneously, we can obtain precise eigenvectors to satisfy the eigen-equation. Two regularized normalization methods can accelerate the convergence speed for two extensions of the simple method, and a derivative-free fixed-point Newton iterative scheme is developed to compute real eigenvalues, the convergence speed of which is ten times faster than the golden section search algorithm. Newton methods are developed for solving two systems of nonlinear regularized equations, and the efficiency and accuracy are significantly improved. Over ten examples demonstrate the high performance of the proposed methods. Among them, the two regularization methods are better than the simple method.

Parametric Iterative Method for Addressing an Embedded-Steel Constitutive Model with Multiple Roots

In this paper, an iterative procedure to find the solution of a nonlinear constitutive model for embedded steel reinforcement is introduced. The model presents different multiplicities, where parameters are randomly selected within a solvability region. To achieve this, a class of multipoint fixed-point iterative schemes for single roots is modified to find multiple roots, achieving the fourth order of convergence. Complex discrete dynamics techniques are employed to select the members with the most stable performance. The mechanical problem referred to earlier, as well as some academic problems involving multiple roots, are solved numerically to verify the theoretical analysis, robustness, and applicability of the proposed scheme.

Global existence of classical solutions and numerical simulations of a cancer invasion model

In this paper, we study a cancer invasion model both theoretically and numerically. The model is a nonstationary, nonlinear system of three coupled partial differential equations modeling the motion of cancer cells, degradation of the extracellular matrix, and certain enzymes. We first establish existence of global classical solutions in both two- and three-dimensional bounded domains, despite the lack of diffusion of the matrix-degrading enzymes and corresponding regularizing effects in the analytical treatment. Next, we give a weak formulation and apply finite differences in time and a Galerkin finite element scheme for spatial discretization. The overall algorithm is based on a fixed-point iteration scheme. Our theory and numerical developments are accompanied by some simulations in two and three spatial dimensions.

A new three-step fixed point iteration scheme with strong convergence and applications

This work proposes a new three-step iteration scheme to approximate the fixed points of a contractive like mapping. Further, the stability and strong convergence of the proposed scheme are considered, and their applicability to different problems is discussed. Some numerical experiments are presented, showing that the new scheme outperforms all the well-known existing three-step schemes available in the literature.

On Some Fixed Point Iterative Schemes with Strong Convergence and Their Applications

In this paper, a new one-parameter class of fixed point iterative method is proposed to approximate the fixed points of contractive type mappings. The presence of an arbitrary parameter in the proposed family increases its interval of convergence. Further, we also propose new two-step and three-step fixed point iterative schemes. We also discuss the stability, strong convergence and fastness of the proposed methods. Furthermore, numerical experiments are performed to check the applicability of the new methods, and these have been compared with well-known similar existing methods in the literature.