Strong Convergence Research Articles

An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces

Abstract In this article, we study the split variational inclusion and fixed point problems using Bregman weak relatively nonexpansive mappings in the p p -uniformly convex smooth Banach spaces. We introduce an inertial shrinking projection self-adaptive iterative scheme for the problem and prove a strong convergence theorem for the sequences generated by our iterative scheme under some mild conditions in real p p -uniformly convex smooth Banach spaces. The algorithm is designed to select its step size self-adaptively and does not require the prior estimate of the norm of the bounded linear operator. Finally, we provide some numerical examples to illustrate the performance of our proposed scheme and compare it with other methods in the literature.

Modified Tseng Method for Solving Pseudomonotone Variational Inequality Problem in Banach Spaces

This article examines the process for solving the fixed-point problem of Bregman strongly nonexpansive mapping as well as the variational inequality problem of the pseudomonotone operator. Within the context of p-uniformly convex real Banach spaces that are also uniformly smooth, we introduce a modified Halpern iterative technique combined with an inertial approach and Tseng methods for finding a common solution of the fixed-point problem of Bregman strongly nonexpansive mapping and the pseudomonotone variational inequality problem. Using our iterative approach, we develop a strong convergence result for approximating the solution of the aforementioned problems. We also discuss some consequences of our major finding. The results presented in this paper complement and build upon many relevant discoveries in the literature.

Digital twin method for the stress field of a deep-diving spherical shell based on a simulation database

In this paper, a Digital Twin (DT) method is proposed for quickly predicting the stress fields of deep-diving spherical shells based on a simulation database. The DT method can completely cover the stress field distribution of the entire pressure-resistant shell and provide stress predictions for different sizes of deep-diving protection shells. The DT model has a three-level structure. The Level-1 DT maps the Finite Element Model to the digital model, which is verified by experiments. The difference between the experimental and numerical results is less than 9.4%. Then, the Level-2 DT deduces the digital model and expands the simulation database samples. The uncalculated stress field results in the simulation database could be obtained by the local Lagrangian interpolation method. Furthermore, the Level-3 DT can rapidly predict the stress field distribution in deep-diving spherical shell digital models. By comparison with different optimization algorithms, the Particle Swarm Optimization algorithm demonstrates strong optimization capabilities and fast convergence, and the error compared with the simulation results is less than 1%. Finally, combined with actual sensor measurements and historical experience data, this method can be used to carry out real-time mapping and synchronous deduction of the service state of the dynamic stress digital twin.

A strong convergence algorithm for approximating a common solution of variational inequality and fixed point problems in real Hilbert space

In this paper, we propose an iterative algorithm for approximating a common solution of a variational inequality and fixed-point problem. The algorithm combines the subgradient extragradient technique, inertial method and a modified viscosity approach. Using this algorithm, we state and prove a strong convergence algorithm for obtaining a common solution of a pseudomonotone variational inequality problem and fixed-point of an η-demimetric mapping in a real Hilbert space. We give an application of this result to some theoretical optimization problems. Furthermore, we report some numerical examples to show the efficiency of our method by comparing it with previous methods in the literature. Our result extends, improves and unifies many other results in this direction in the literature. Mathematics Subject Classification (2010): 47H09, 49J35, 90C47. Received 21 May 2021; Accepted 14 July 2021

Forward–Reflected–Backward Splitting Algorithms with Momentum: Weak, Linear and Strong Convergence Results

Forward–Reflected–Backward Splitting Algorithms with Momentum: Weak, Linear and Strong Convergence Results

On a Faster Iterative Method for Solving Fractional Delay Differential Equations in Banach Spaces

In this paper, we consider a faster iterative method for approximating the fixed points of generalized α-nonexpansive mappings. We prove several weak and strong convergence theorems of the considered method in mild conditions within the control parameters. In order to validate our findings, we present some nontrivial examples of the considered mappings. Furthermore, we show that the class of mappings considered is more general than some nonexpansive-type mappings. Also, we show numerically that the method studied in our article is more efficient than several existing methods. Lastly, we use our main results to approximate the solution of a delay fractional differential equation in the Caputo sense. Our results generalize and improve many well-known existing results.

Strong convergence of inertial shrinking projection method for split best proximity point problem and mixed equilibrium problem

Strong convergence of inertial shrinking projection method for split best proximity point problem and mixed equilibrium problem

A new double inertial subgradient extragradient algorithm for solving split pseudomonotone equilibrium problems and fixed point problems

The purpose of this article is to suggest a modified subgradient extragradient method that includes double inertial extrapolations and viscosity approach for finding the common solution of split equilibrium problem and fixed point problem. The strong convergence result of the suggested method is obtained under some standard assumptions on the control parameters. Our method does not require solving two strongly convex optimization problems in the feasible sets per iteration, and the step-sizes do not depend on bifunctional Lipschitz-type constants. Furthermore, unlike several methods in the literature, our method does not depend on the prior knowledge of the operator norm of the bounded linear operator. Instead, the step-sizes are self adaptively updated. We apply our method to solve split variational inequality problem. Lastly, we conduct some numerical test to compare our method with some well known methods in the literature.

Stability and Data Dependence Results for Jungck-type Iteration Scheme

We propose and study a new Jungck-type iteration scheme to approximate coincidence points of contractive mappings. The strong convergence, stability, and data dependency results have been discussed. Numerical experiments demonstrate that the newly introduced Jungck-type iteration scheme yields a higher convergence rate in comparison with other Jungck-type iteration schemes available in the literature.

A new perspective on Saturn’s polar vortices: Coupling the thermosphere and stratosphere

We argue that the observed temperature and altitude structure at the base of Saturn’s polar thermosphere implies the presence of a deep, warm-cored, anticyclonic vortex that extends deep into the mesosphere, and possibly into the stratosphere. This would be a unique structure in the solar system: a permanent, large-scale middle atmospheric feature that is a direct consequence of magnetosphere-atmosphere coupling. We investigate the plausibility of the suggestion that this deep vortex is linked to the observed non-seasonal polar hotspots in Saturn’s stratosphere. We develop a simple model of the dynamics and thermal structure of the middle atmosphere polar vortex, assuming vertical viscous transport is dominant and that the zonal winds are close to geostrophic balance. The model requires the existence of a small poleward ageostrophic wind which causes compressional heating, producing the required pressure gradients. This scenario is supported by a re-analysis of previously published thermospheric modelling results. Using this model we estimate the extent to which the thermospheric vortex penetrates into the deeper atmosphere. The results are very sensitive to the distribution of eddy viscosity. The models which best match the thermospheric temperatures produce a polar hotspot in the stratosphere that is hotter and broader than observed, indicating that our model, while a useful starting point, is incomplete. We also point out a puzzle concerning the half-widths of the polar hotspot and the polar altitude bulge at the base of the thermosphere. We are able to explain the greater width of the hotspot provided that there is enhanced downwards vertical transport of angular momentum closer to the pole, a situation that is expected given the strong convergence and downwelling predicted in this region. Finally, we speculate that the tropospheric polar cyclone could be connected to the anticyclonic vortex in the thermosphere.

Quantile Optimization via Multiple-Timescale Local Search for Black-Box Functions

Black-Box Quantile Optimization via Finite-Difference-Based Gradient Approximation Risk management necessitates consideration of metrics such as quantiles to supplement conventional mean performance measures. In “Quantile Optimization via Multiple-Timescale Local Search for Black-Box Functions,” J. Hu, M. Song, and M. C. Fu consider the problem where the goal is to optimize the quantile of a black-box output. They introduce two new iterative multitimescale stochastic approximation algorithms utilizing finite-difference-based gradient estimators. The first algorithm requires 2d + 1 samples of the black-box function per iteration, where d is the number of decision variables. The second employs a simultaneous-perturbation-based gradient estimator that uses only three samples per iteration, irrespective of the number of decision variables. The authors prove strong local convergence for both algorithms and analyze their finite-time convergence rates through a novel fixed-point argument. These algorithms perform well across a varied set of benchmark problems.

Strong convergence theorem for split null point, split variational inequality and fixed point problems for a non-expansive semigroup

The primary goal of this paper is to find a common solution of a split null point problem, split variational inequality problem and fixed point problem for a non-expansive semigroup with the help of new type of iterative algorithm. We construct that the sequences induced by the proposed iterative method to solve the above stated problems in the setting of real Hilbert spaces and obtain a strong convergence theorem. We also deduce certain consequences from the main convergence result. At the end, we provide a numerical experiment to demonstrate the convergence analysis of proposed iterative method. The methodology and conclusion described in this work extend and unify previously published findings in this field.

Strong Convergence Theorems for Split Common Fixed Point Problem of Bregman Generalized Asymptotically Nonexpansive Mappings in Banach Spaces

In this paper, a new iterative scheme is introduced and also strong convergence theorems for solving split common fixed point problem for uniformly continuous Bregman generalized asymptotically nonexpansive mappings in uniformly convex and uniformly smooth Banach spaces are presented. The results are proved without the assumption of semicompactness property and or Opial condition

Socio‐techno‐economic‐environmental sizing of hybrid renewable energy system using metaheuristic optimization approaches

AbstractElectricity supply reliability in an electricity distribution network is majorly affected due to unexpected power failures and power cuts. A hybrid renewable energy system (HRES) with the optimal size of renewable energy sources can substantially improve power reliability. Therefore, this article develops an objective function incorporating socio‐techno‐economic‐environmental (STEE) factors for HRES optimal sizing to supply reliable power to a rural village. The factors considered in the objective function are namely social (employment generation factor, human progress index, and land cost), technical (excess energy factor, renewable energy portion, and loss of power supply probability), economical (total net present cost, cost of energy, and annualized cost of system), and environmental (emission cost). In this article, for the first time, marine predators algorithm (MPA) based metaheuristic optimizer is devised to address the sizing optimization problem of HRES. Three HRES configurations, having different arrangements of diesel generator (DG), biogas generator (BG), battery (BAT), wind turbine (WT), and photovoltaic (PV), are examined utilizing MPA, particle swarm optimization (PSO), salp swarm algorithm (SSA), and genetic algorithm (GA) for optimal configuration. Due to the lowest value of economical and environmental factors and the highest value of the social factor, the PV‐WT‐BAT‐BG‐DG configuration is optimal compared to other investigated configurations with MPA. Comparing the four optimizers, MPA has the best STEE factor values, as well as stronger convergence, greater ability to escape from local minima, and higher ability to approach the global optimum. Additionally, by contrasting it with the PSO result for the three HRES configurations, the MPA result quality is confirmed. Furthermore, the cost of energy (0.1799 $/kWh) of the optimal configuration is less than the latest addressed in the literature.

Convergence and stability of the balanced Euler method for stochastic pantograph differential equations with Markovian switching

In this paper, we concern the strong convergence and stability of the balanced Euler method for stochastic pantograph differential equations with Markovian switching (SPDEs-MS). We present the balanced Euler method of SPDEs-MS and consider its moment boundedness under polynomial growth condition plus Khasminskii-type condition. We also study its strong convergence order and its mean-square stability. Two numerical examples are given to illustrate the theoretical results.

Random invariant densities for markov operator cocycles and random mean ergodic theorem

In this paper, we consider random invariant densities and the mean ergodic theorem for Markov operator cocycles which are applicable to quenched type random dynamical systems. We give necessary and sufficient conditions for the existence of random invariant densities for Markov operator cocycles and establish the mean ergodic theorem for generalized linear operator cocycles over a weakly sequentially complete Banach space. The advantage of the result is that we show the implication of weak precompactness for almost every environment to strong convergence in the global sense.

Halpern-Type Inertial Iteration Methods with Self-Adaptive Step Size for Split Common Null Point Problem

In this paper, two Halpern-type inertial iteration methods with self-adaptive step size are proposed for estimating the solution of split common null point problems (SpCNPP) in such a way that the Halpern iteration and inertial extrapolation are computed simultaneously in the beginning of each iteration. We prove the strong convergence of sequences driven by the suggested methods without estimating the norm of bounded linear operator when certain appropriate assumptions are made. We demonstrate the efficiency of our iterative methods and compare them with some related and well-known results using relevant numerical examples.

Global Finite-Energy Solutions of the Compressible Euler–Poisson Equations for General Pressure Laws with Large Initial Data of Spherical Symmetry

We are concerned with global finite-energy solutions of the three-dimensional compressible Euler–Poisson equations with gravitational potential and general pressure law, especially including the constitutive equation of white dwarf stars. In this paper, we construct global finite-energy solutions of the Cauchy problem for the Euler–Poisson equations with large initial data of spherical symmetry as the inviscid limit of the solutions of the corresponding Cauchy problem for the compressible Navier–Stokes–Poisson equations. The strong convergence of the vanishing viscosity solutions is achieved through entropy analysis, uniform estimates in Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}, and a more general compensated compactness framework via several new ingredients. A key estimate is first established for the integrability of the density over unbounded domains independent of the vanishing viscosity coefficient. Then a special entropy pair is carefully designed via solving a Goursat problem for the entropy equation such that a higher integrability of the velocity is established, which is a crucial step. Moreover, the weak entropy kernel for the general pressure law and its fractional derivatives of the required order near vacuum (ρ=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =0$$\\end{document}) and far-field (ρ=∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho =\\infty $$\\end{document}) are carefully analyzed. Owing to the generality of the pressure law, only the Wloc-1,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{-1,p}_{\ extrm{loc}}$$\\end{document}-compactness of weak entropy dissipation measures with p∈[1,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in [1,2)$$\\end{document} can be obtained; this is rescued by the equi-integrability of weak entropy pairs which can be established by the estimates obtained above, so that the div-curl lemma still applies. Finally, based on the above analysis of weak entropy pairs, the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} compensated compactness framework for the compressible Euler equations with general pressure law is established. This new compensated compactness framework and the techniques developed in this paper should be useful for solving further nonlinear problems with similar features.

Iterative Stability Analysis for Generalized α-Nonexpensive Mappings with Fixed Points

This article introduces a novel iterative process, denoted as F★, designed for the class of generalized α-Nonexpensive mappings. The study establishes strong and weak convergence theorems within the context of Banach spaces, supported by carefully chosen assumptions. The convergence results contribute to the theoretical foundation of iterative processes in functional analysis. The presented framework is applied to address nonlinear integral equations, showcasing the versatility and applicability of the proposed F* for the class of generalized iteration process. Additionally, the article includes numerical examples that not only validate the theoretical findings but also provide insights into the practical utility of the developed methodology.

Strongly Convergent Homogeneous Approximations to Inhomogeneous Markov Jump Processes and Applications

The study of time-inhomogeneous Markov jump processes is a traditional topic within probability theory that has recently attracted substantial attention in various applications. However, their flexibility also incurs a substantial mathematical burden which is usually circumvented by using well-known generic distributional approximations or simulations. This article provides a novel approximation method that tailors the dynamics of a time-homogeneous Markov jump process to meet those of its time-inhomogeneous counterpart on an increasingly fine Poisson grid. Strong convergence of the processes in terms of the Skorokhod J1 metric is established, and convergence rates are provided. Under traditional regularity assumptions, distributional convergence is established for unconditional proxies, to the same limit. Special attention is devoted to the case where the target process has one absorbing state and the remaining ones transient, for which the absorption times also converge. Some applications are outlined, such as univariate hazard-rate density estimation, ruin probabilities, and multivariate phase-type density evaluation. Funding: M. Bladt and O. Peralta would like to acknowledge financial support from the Swiss National Science Foundation Project 200021_191984. O. Peralta acknowledges financial support from NSF Award #1653354 and AXA Research Fund Award on “Mitigating risk in the wake of the pandemic”.