scholarly journals A New Accelerated Viscosity Iterative Method for an Infinite Family of Nonexpansive Mappings with Applications to Image Restoration Problems

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 615
Author(s):  
Jenwit Puangpee ◽  
Suthep Suantai
Keyword(s):  
Image Restoration ◽  
Medical Image ◽  
Numerical Experiments ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Convergence Result ◽  
Approximation Technique ◽  
Viscosity Approximation ◽  
Viscosity Iterative Method ◽  
Restoration Problem

The image restoration problem is one of the popular topics in image processing which is extensively studied by many authors because of its applications in various areas of science, engineering and medical image. The main aim of this paper is to introduce a new accelerated fixed algorithm using viscosity approximation technique with inertial effect for finding a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space and prove a strong convergence result of the proposed method under some suitable control conditions. As an application, we apply our algorithm to solving image restoration problem and compare the efficiency of our algorithm with FISTA method which is a popular algorithm for image restoration. By numerical experiments, it is shown that our algorithm has more efficiency than that of FISTA.

scholarly journals Two-Step Viscosity Approximation Scheme for Variational Inequality in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Liping Yang ◽  
Weiming Kong
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Algorithm ◽  
Approximation Scheme ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Viscosity Approximation ◽  
Convergence Conditions ◽  
Uniformly Smooth ◽  
Necessary And Sufficient

This paper introduces and analyzes a viscosity iterative algorithm for an infinite family of nonexpansive mappings{Ti}i=1∞in the framework of a strictly convex and uniformly smooth Banach space. It is shown that the proposed iterative method converges strongly to a common fixed point of{Ti}i=1∞, which solves specific variational inequalities. Necessary and sufficient convergence conditions of the iterative algorithm for an infinite family of nonexpansive mappings are given. Results shown in this paper represent an extension and refinement of the previously known results in this area.

An iterative method for solving minimization, variational inequality and fixed point problems in reflexive Banach spaces

2018 ◽  
Vol 9 (3) ◽  
pp. 167-184 ◽  
Author(s):  
Lateef Olakunle Jolaoso ◽  
Ferdinard Udochukwu Ogbuisi ◽  
Oluwatosin Temitope Mewomo
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Monotone Operators ◽  
Convergence Result ◽  
Reflexive Banach Spaces ◽  
Strongly Monotone Operators ◽  
System Of Equilibrium Problems

Abstract In this paper, we propose an iterative algorithm for approximating a common fixed point of an infinite family of quasi-Bregman nonexpansive mappings which is also a solution to finite systems of convex minimization problems and variational inequality problems in real reflexive Banach spaces. We obtain a strong convergence result and give applications of our result to finding zeroes of an infinite family of Bregman inverse strongly monotone operators and a finite system of equilibrium problems in real reflexive Banach spaces. Our result extends many recent corresponding results in literature.

scholarly journals A Fast Fixed-Point Algorithm for Convex Minimization Problems and Its Application in Image Restoration Problems

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2619
Author(s):  
Panadda Thongpaen ◽  
Rattanakorn Wattanataweekul
Keyword(s):  
Fixed Point ◽  
Image Restoration ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Convex Minimization ◽  
Fixed Point Algorithm ◽  
Minimization Problems ◽  
Convex Minimization Problems ◽  
Weak Convergence Theorem ◽  
Inertial Technique

In this paper, we introduce a new iterative method using an inertial technique for approximating a common fixed point of an infinite family of nonexpansive mappings in a Hilbert space. The proposed method’s weak convergence theorem was established under some suitable conditions. Furthermore, we applied our main results to solve convex minimization problems and image restoration problems.

Approximation of fixed point of accretive operators based on a Halpern-Type iterative method

2017 ◽  
Vol 26 (3) ◽  
pp. 263-274
Author(s):  
KADRI DOGAN ◽  
◽  
VATAN KARAKAYA ◽  
Keyword(s):  
Fixed Points ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Convergence Result ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Iterative Sequence ◽  
Uniformly Convex ◽  
Accretive Operators ◽  
The Common

In this study, we introduce a new iterative processes to approximate common fixed points of an infinite family of quasi-nonexpansive mappings and obtain a strongly convergent iterative sequence to the common fixed points of these mappings in a uniformly convex Banach space. Also we prove that this process to approximate zeros of an infinite family of accretive operators and we obtain a strong convergence result for these operators. Our results improve and generalize many known results in the current literature.

scholarly journals Viscosity Approximation Methods for a Family of Nonexpansive Mappings in CAT(0) Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Jinfang Tang
Keyword(s):  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Approximation Methods ◽  
Viscosity Approximation Method ◽  
Viscosity Approximation ◽  
Convergence Problem ◽  
Iterative Schemes ◽  
The Family ◽  
Viscosity Approximation Methods ◽  
Implicit And Explicit

The purpose of this paper is using the viscosity approximation method to study the strong convergence problem for a family of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of the family of nonexpansive mappings are proved which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.

scholarly journals Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Songnian He ◽  
Jun Guo
Keyword(s):  
Fixed Points ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Duality Mapping ◽  
Normalized Duality Mapping ◽  
Uniformly Smooth ◽  
Computational Work

LetCbe a nonempty closed convex subset of a real uniformly smooth Banach spaceX,{Tk}k=1∞:C→Can infinite family of nonexpansive mappings with the nonempty set of common fixed points⋂k=1∞Fix⁡(Tk), andf:C→Ca contraction. We introduce an explicit iterative algorithmxn+1=αnf(xn)+(1-αn)Lnxn, whereLn=∑k=1n(ωk/sn)Tk,Sn=∑k=1nωk,  andwk>0with∑k=1∞ωk=1. Under certain appropriate conditions on{αn}, we prove that{xn}converges strongly to a common fixed pointx*of{Tk}k=1∞, which solves the following variational inequality:〈x*-f(x*),J(x*-p)〉≤0,    p∈⋂k=1∞Fix(Tk), whereJis the (normalized) duality mapping ofX. This algorithm is brief and needs less computational work, since it does not involveW-mapping.

scholarly journals Virtual Element Method for the Laplace-Beltrami equation on surfaces

2018 ◽  
Vol 52 (3) ◽  
pp. 965-993 ◽  
Author(s):  
Massimo Frittelli ◽  
Ivonne Sgura
Keyword(s):  
Numerical Experiments ◽  
Beltrami Equation ◽  
Convergence Result ◽  
Point Of View ◽  
Polygonal Approximation ◽  
Virtual Element Method ◽  
Element Space ◽  
First Order ◽  
Virtual Element ◽  
Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

scholarly journals Iterative Algorithms for Solving the System of Mixed Equilibrium Problems, Fixed-Point Problems, and Variational Inclusions with Application to Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-29 ◽  
Author(s):  
Tanom Chamnarnpan ◽  
Poom Kumam
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Iterative Algorithms ◽  
Infinite Family ◽  
Common Element ◽  
Mixed Equilibrium Problems ◽  
Mixed Equilibrium

We introduce a new iterative algorithm for solving a common solution of the set of solutions of fixed point for an infinite family of nonexpansive mappings, the set of solution of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion for aβ-inverse-strongly monotone mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above three sets under some mild conditions. Furthermore, we give a numerical example which supports our main theorem in the last part.

scholarly journals Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Luo Yi Shi ◽  
Ru Dong Chen
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convex Subset ◽  
Unique Fixed Point ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Approximation Methods ◽  
Viscosity Approximation ◽  
Viscosity Approximation Methods

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mappingTof a closed convex subsetCof a CAT(0) spaceX. Suppose that the set Fix(T)of fixed points ofTis nonempty. For a contractionfonCandt∈(0,1), letxt∈Cbe the unique fixed point of the contractionx↦tf(x)⊕(1-t)Tx. We will show that ifXis a CAT(0) space satisfying some property, then{xt}converge strongly to a fixed point ofTwhich solves some variational inequality. Consider also the iteration process{xn}, wherex0∈Cis arbitrary andxn+1=αnf(xn)⊕(1-αn)Txnforn≥1, where{αn}⊂(0,1). It is shown that under certain appropriate conditions onαn,{xn}converge strongly to a fixed point ofTwhich solves some variational inequality.

Sign in / Sign up

Export Citation Format

Share Document