Viscosity Approximation Research Articles

On the Monotone Mappings in CAT(0) Spaces

In this paper, we first introduce a monotone mapping and its resolvent in general metric spaces.Then, we give two new iterative methods by combining the resolvent method with Halpern's iterative method and viscosity approximation method for finding a fixed point of monotone mappings and a solution of variational inequalities. We prove convergence theorems of the proposed iterations in CAT(0) metric spaces.

Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces

We propose a parallel iterative scheme with viscosity approximation method which converges strongly to a solution of the multiple-set split equality common fixed point problem for quasi-pseudocontractive mappings in real Hilbert spaces. We also give an application of our result to approximation of minimization problem from intensity-modulated radiation therapy. Finally, we present numerical examples to demonstrate the behaviour of our algorithm. This result improves and generalizes many existing results in literature in this direction.

Inertial Subgradient Extragradient Methods for Solving Variational Inequality Problems and Fixed Point Problems

We propose two new iterative algorithms for solving K-pseudomonotone variational inequality problems in the framework of real Hilbert spaces. These newly proposed methods are obtained by combining the viscosity approximation algorithm, the Picard Mann algorithm and the inertial subgradient extragradient method. We establish some strong convergence theorems for our newly developed methods under certain restriction. Our results extend and improve several recently announced results. Furthermore, we give several numerical experiments to show that our proposed algorithms performs better in comparison with several existing methods.

On the existence of L2-valued thermodynamic entropy solutions for a hyperbolic system with boundary conditions

We prove existence of L2-weak solutions of a quasilinear wave equation with boundary conditions. This describes the isothermal evolution of a one dimensional non-linear elastic material, attached to a fixed point on one side and subject to a force (tension) applied to the other side. The L2-valued solutions appear naturally when studying the hydrodynamic limit from a microscopic dynamics of a chain of anharmonic springs connected to a thermal bath. The proof of the existence is done using a vanishing viscosity approximation with extra Neumann boundary conditions added. In this setting we obtain a uniform a priori estimate in L2, allowing us to use L2 Young measures, together with the classical tools of compensated compactness. We then prove that the viscous solutions converge to weak solutions of the quasilinear wave equation strongly in Lp, for any that satisfy, in a weak sense, the boundary conditions. Furthermore, these solutions satisfy, beside the local Lax entropy condition, the Clausius inequality: the change of the free energy is bounded by the work done by the boundary tension. In this sense they are the correct thermodynamic solutions, and we conjecture their uniqueness.

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

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.

Effect of the Curing Properties and Viscosities of Nonconductive Films on the Solder Joint Morphology and Reliability of Chip-On-Board Packages Using Cu-Pillar/Sn–Ag Bumps

In this article, flip chip assembly using nonconductive films (NCFs) was evaluated for chip-on-board (COB) packages using Cu/Sn–Ag hybrid bumps. In general, significant thermomechanical stress can be induced at the Cu-pillar/Sn–Ag solder joint because of a large coefficient of thermal expansion (CTE) mismatch between the Si chip and printed circuit board (PCB). If a concave-shaped solder joint morphology is generated at the solder joint, thermomechanical stress could be concentrated at the solder joint, resulting in an early failure of T/C reliability. To obtain the solder joint morphology without a concave-shaped solder joint, NCFs’ curing properties and viscosity should be controlled. In particular, the NCFs’ viscosity at the solder melting temperature, at which metallurgical bonding between molten solder and Cu pads occurs, will significantly affect the solder joint morphology. For this reason, solder joint formation was investigated at each viscosity of NCFs and bonding temperature during the thermo compression (TC) flip chip bonding process. In addition, specific viscosities at certain bonding temperatures during the flip chip bonding process were obtained from measured degrees of cure and viscosity approximation. Finally, the material properties of NCFs were optimized without NCF traps at the solder joint and concave-shaped solder joint, and T/C reliability was evaluated for each silica powder content. As a result, it was confirmed that the COB packages using Cu/Sn–Ag hybrid bumps and optimized NCFs show stable resistance changes within 10% of the initial resistance after the T/C reliability test.

The Method for Solving Fixed Point Problem of G-Nonexpansive Mapping in Hilbert Spaces Endowed with Graphs and Numerical Example

The main aim of this paper is to study a strong convergence theorem of viscosity approximation method for G-nonexpansive mapping defined on a Hilbert space endowed with a directed graph. By using our main result, we give a numerical expample to approximate the value of π.

The macroscopic viscosity approximation: A first-principle relationship between molecular diffusion and viscosity

In this article, a new relationship between viscosity and molecular diffusion at infinite dilution is proposed for better rationalization and prediction of these properties, based on a “macroscopic viscosity approximation” (MVA), i.e., by assuming viscosity around a solute as equal to the macroscopic, measurable viscosity of the solvent. This implies that activation energies of the viscous flow and diffusion process are equal. The hypothesis is validated by our correlation analysis (mean difference of 0.10 kcal/mol, R2 = 0.96). The new approach, named “Modified Stokes–Einstein” (MSE), achieves better performance than the widely used Wilke–Chang (WC) correlation both in organic solvents [mean relative error (MRE) of 15% vs 24%, respectively] and in water (MRE of 13% vs 21%, respectively). Contrary to the popular WC correlation as well as all other available approaches in the literature, the MSE approach can be used consistently for water, without requiring any ad hoc association parameter, and is not fitted on diffusion and/or viscosity data, making all of its underlying hypotheses explicit. Based on the MVA and the MSE, a simple atomic count estimation method for the activation energy of the flow allows us to simultaneously predict viscosity and diffusion coefficients with an MRE of 21%–22%, again slightly better than the WC correlation, but not requiring any experimental data as the input. This work provides rationalized and efficient means for prediction of diffusion coefficients at infinite dilution and pure liquid viscosities wherever such properties are required, for example, as inputs for mixing rules to predict flow and transport behavior of complex systems.

Viscosity approximation methods for generalized asymptotically nonexpansive mappings and monotone inclusions in Hadamard spaces

Viscosity approximation methods for generalized asymptotically nonexpansive mappings and monotone inclusions in Hadamard spaces

Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space

ABSTRACT In this paper, we propose a new viscosity type inertial extragradient method with Armijo line-search technique for approximating a common solution of equilibrium problem with pseudo-monotone bifunction and fixed points of relatively nonexpansive mapping in a real Hilbert space. Two advantages of our algorithm are that its convergence does not require the bifunction to satisfy any Lipschitz-type condition and only one strongly convex program and one projection onto the feasible set are perform at each iteration. Under some mild conditions on the control sequences, we state and prove a strong convergence theorem and also present two numerical examples to illustrate the performance of our algorithm. The results in this paper improve and generalize many recent results in this direction in the literature.

Viscosity Approximation Methods for 
 
 
 
 *
 
 
 
 −Nonexpansive Multi-Valued Mappings in Convex Metric Spaces

In this paper, we prove convergence theorems for viscosity approximation processes involving * −nonexpansive multi-valued mappings in complete convex metric spaces. We also consider finite and infinite families of such mappings and prove convergence of the proposed iteration schemes to common fixed points of them. Our results improve and extend some corresponding results.

Viscosity approximation methods for split common fixed point problems without prior knowledge of the operator norm

In this work, we study the split common fixed point problem which was first introduced by Censor and Segal [14]. We introduce an algorithm based on the viscosity approximation method without prior knowledge of the operator norm by selecting the stepsizes in the same adaptive way as L?opez et al. [22] for solving the problem for two attracting quasi-nonexpansive operators in real Hilbert spaces. A strong convergence result of the proposed algorithm is established under some suitable conditions. We also modify our algorithm to extend to the class of demicontractive operators and the class of hemicontractive operators, and obtain strong convergence results. Moreover, we apply our main result to other split problems, that is, the split feasibility problem and the split variational inequality problem. Finally, a numerical result is also given to illustrate the convergence behavior of our algorithm.

Hybrid viscosity approximation methods for systems of variational inequalities and hierarchical fixed point problems

In this paper, we propose an implicit iterative method and an explicit iterative method for solving a general system of variational inequalities with a hierarchical fixed point problem constraint for an infinite family of nonexpansive mappings. We show that the proposed algorithms converge strongly to a solution of the general system of variational inequalities, which is a unique solution of the hierarchical fixed point problem.

Strong convergence of viscosity forward-backward algorithm to the sum of two accretive operators in Banach space

ABSTRACT In recent years, the Forward-Backward algorithm (FBA) received much attention due to its various applications in image recovery, signal processing, and machine learning. In this paper, we consider the FBA in the setting of Banach spaces that are uniformly convex and q-uniformly smooth. We introduce two viscosity FBA, and one of them with weakly contractive mapping, which generalizes many previous results on the viscosity approximation method with fixed contraction. Moreover, we establish their strong convergences under more general conditions.

Convergence Theorems of Variational Inequality for Asymptotically Nonexpansive Nonself Mapping in CAT(0) Spaces

The aim of this manuscript is to get the strong convergence theorems of the Moudafi’s viscosity approximation methods for an asymptotically nonexpansive nonself mapping in C A T ( 0 ) spaces.

Tseng Type Methods for Inclusion and Fixed Point Problems with Applications

An algorithm is introduced to find an answer to both inclusion problems and fixed point problems. This algorithm is a modification of Tseng type methods inspired by Mann’s type iteration and viscosity approximation methods. On certain conditions, the iteration obtained from the algorithm converges strongly. Moreover, applications to the convex feasibility problem and the signal recovery in compressed sensing are considered. Especially, some numerical experiments of the algorithm are demonstrated. These results are compared to those of the previous algorithm.

Viscosity approximations considering boundary point method for fixed-point and variational inequalities of quasi-nonexpansive mappings

Existing viscosity approximation schemes have been extensively investigated to solve equilibrium problems, variational inequalities, and fixed-point problems, and most of which contain that contraction is a self-mapping defined on certain bounded closed convex subset C of Hilbert spaces H for standard viscosity approximation. In particular, if the zero point does not belong to C, the standard viscosity approximation cannot be applied to solve the minimum norm fixed point of some nonlinear operators. In this paper, we introduce three generalized viscosity approximation algorithms with boundary point method for quasi-nonexpansive mappings, which overcome this deficiency above. These three proposed algorithms have simple expressions; moreover, they are easy to implement in the actual computation process. Especially, we can find the minimum norm fixed point of quasi-nonexpansive mappings under contraction is a zero operator.

Bounded perturbation resilience and superiorization techniques for a modified proximal gradient method

ABSTRACTIn this article, we combine the viscosity approximation method and proximal operator to propose modified proximal gradient methods for solving the unconstrained convex optimization problems. The bounded perturbation resilience of these methods is investigated in the general Hilbert space H. Under reasonable parameter conditions, we prove that our algorithms strongly converge to the unique solution of a variational inequality problem. Our results improve and extend the corresponding results reported by many authors recently.

Composite inertial subgradient extragradient methods for variational inequalities and fixed point problems

In this paper, we introduce and investigate composite inertial gradient-based algorithms with a line-search process for solving a variational inequality problem (VIP) with a pseudomonotone and Lipschitz continuous mapping and a common fixed-point problem (CFPP) of finitely many nonexpansive mappings and a strictly pseudocontractive mapping in the framework infinite-dimensional Hilbert spaces. The proposed algorithms are based on an inertial subgradient–extragradient method with the line-search process, hybrid steepest-descent methods, viscosity approximation methods and Mann iteration methods. Under weak conditions, we prove strong convergence of the proposed algorithms to the element in the common solution set of the VIP and CFPP, which solves a certain hierarchical VIP defined on this common solution set.

Viscosity method for generalized equilibrium problems with perturbation and nonexpansive mappings in Hilbert spaces

In this paper, we introduce a new iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem with perturbation and the fixed points for a new kind of combination of two nonexpansive mapping in Hilber space, we prove that the sequence converges strongly to a common element of two sets. Our results generalize and improve some results in the litretures.