Iterative Approximation Method Research Articles

The modified general iterative methods for nonexpansive semigroups in banach spaces†

In this paper, we introduce the modified general iterative approximation methods for finding a common fixed point of nonexpansive semigroups which is a unique solution of some variational inequalities. The strong convergence theorems are established in the framework of a reflexive Banach space which admits a weakly continuous duality mapping. The main result extends various results existing in the current literature. Mathematics Subject Classification (2000) 47H05, 47H09, 47J25, 65J15

Rural Electrification in Oyo State of Nigeria: Assessement of Existing 33kv Feeder Network Capacity

The aim of this study was to assess the capacity of the existing primary feeder network (33 kV) and to determine ways of reinforcing it in view of prospective full scale electrification and power demand of the rural areas of Oyo State. This work was done by using data of maximum power demand allocations to fully electrified areas of the State to compute expected range of power demand per capita, and applying normal mode simulations and solutions using iterative approximation and direct matrix algebra methods. The results show that the load per capita range, and potential load demand for the State are 0.011 kW to 0.094 kW and about 534 MW (huge compared to current 103.7 MW supply from the utility) respectively. The capacities of existing networks and the total injection substations’ capacity (225 MVA) are inadequate for meeting the potential demand of Oyo State, including the rural areas. Upgrading the voltage of the existing network to 132 kV will improve its distribution capacity significantly and solve overloading and voltage problems. Also, installation of more 132 kV substations is needed to relieve the existing fully loaded ones. Otherwise, a full scale rural electrification development in Oyo State, intending to feed from the national grid, will be impossible.

Existence theorems and iterative approximation methods for generalized mixed equilibrium problems for a countable family of nonexpansive mappings

In this paper, we introduce the new generalized mixed equilibrium problem basing on hemicontinuous and relaxed monotonic mapping. Using the KKM technique, we obtain the existence of solutions for the generalized mixed equilibrium problem in a Banach space. Furthermore, we also introduce a hybrid projection algorithm for finding a common element in the solution set of a generalized mixed equilibrium problem and the common fixed point set of a countable family of nonexpansive mappings. The strong convergence theorem of the proposed sequence is obtained in a Banach space setting. The main results extend various results existing in the current literature.

A weak convergence theorem for common fixed points of some generalized nonexpansive mappings and nonspreading mappings in a Hilbert space

We study the iterative methods for approximation of fixed points of some generalized nonexpansive mappings and nonspreading mappings in a Hilbert space. Consequently, we obtain the weak convergence theorem for common fixed points of two such mappings.

An Iterative Approximation Method for a Common Fixed Point of Two Pseudocontractive Mappings

We introduce an iterative process for finding an element in the common fixed point sets of two continuous pseudocontractive mappings. As a consequence, we provide an approximation method for a common fixed point of a finite family of pseudocontractive mappings. Furthermore, our convergence theorem is applied to a convex minimization problem. Our theorems extend and unify most of the results that have been proved for this class of nonlinear mappings.

HYBRID VISCOSITY ITERATIVE APPROXIMATION OF ZEROS OF M-ACCRETIVE OPERATORS IN BANACH SPACES

In this paper, let $X$ be a reflexive Banach space which either is uniformly smooth or has a weakly continuous duality map. We prove, under the convergence of no parameter sequences to zero, the strong convergence of their iterative scheme to a zero of $m$-accretive operator $A$ in $X$, which solves a variational inequality on the set $A^{-1}(0)$ of zeros of $A$. Such a result includes their main result as a special case. Furthermore, we also give a weak convergence theorem for hybrid viscosity iterative approximation method involving a maximal monotone operator in a Hilbert space.

A New Composite General Iterative Scheme for Nonexpansive Semigroups in Banach Spaces

We introduce a new general composite iterative scheme for finding a common fixed point of nonexpansive semigroups in the framework of Banach spaces which admit a weakly continuous duality mapping. A strong convergence theorem of the purposed iterative approximation method is established under some certain control conditions. Our results improve and extend announced by many others.

The General Hybrid Approximation Methods for Nonexpansive Mappings in Banach Spaces

We introduce two general hybrid iterative approximation methods (one implicit and one explicit) for finding a fixed point of a nonexpansive mapping which solving the variational inequality generated by two strongly positive bounded linear operators. Strong convergence theorems of the proposed iterative methods are obtained in a reflexive Banach space which admits a weakly continuous duality mapping. The results presented in this paper improve and extend the corresponding results announced by Marino and Xu (2006), Wangkeeree et al. (in press), and Ceng et al. (2009).

New Iterative Approximation Methods for a Countable Family of Nonexpansive Mappings in Banach Spaces

We introduce new general iterative approximation methods for finding a common fixed point of a countable family of nonexpansive mappings. Strong convergence theorems are established in the framework of reflexive Banach spaces which admit a weakly continuous duality mapping. Finally, we apply our results to solve the the equilibrium problems and the problem of finding a zero of an accretive operator. The results presented in this paper mainly improve on the corresponding results reported by many others.

Iterative approximation method for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems

In this paper, a new iterative scheme by hybrid method is constructed. Strong convergence of the scheme to a common element of the set of common fixed points of a finite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth is proved. Our results extend important recent results.

The general iterative methods for nonexpansive mappings in Banach spaces

In this paper, we introduce a general iterative approximation method for finding a common fixed point of a countable family of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a reflexive Banach space which admits a weakly continuous duality mapping. As applications, at the end of the paper, we apply our results to the problem of finding a zero of an accretive operator. The main result extends various results existing in the current literature.

Pretension Process Analysis of Arch-Supported Prestressed Grid Structures Based on Member Initial Deformation

Arch-supported prestressed grid structures, composed of steel arches, latticed shells, cables and hangers, are capable of covering large spans with good mechanical performance. Because of the many prestressed cables and hangers, structural analysis of the whole pretensioning process is rather difficult and important. Based on the structural characteristics, this paper develops the stiffness equation, and brings forth mixed influence matrix and iterative approximation methods to determine the initial deformations of cables and hangers. A sequential analysis method is then developed to analyze both batched tension and repeated tension schemes, so as to determine the active and passive forces in the cables and hangers during the pretensioning process. A case study, the Yangtze River Flood-control Model-testing Hall, proves that the proposed methods can efficiently model the whole pretension process for arch-supported prestressed grid structures.

Iterative methods for the force-based quasicontinuum approximation: Analysis of a 1D model problem

Force-based atomistic-continuum hybrid methods are the only known pointwise consistent methods for coupling a general atomistic model to a finite-element continuum model. For this reason, and due to their algorithmic simplicity, force-based coupling methods have become a popular class of atomistic-continuum hybrid models as well as other types of multiphysics models. However, the recently discovered unusual stability properties of the linearized force-based quasicontinuum (QCF) approximation, especially its indefiniteness, present a challenge to the development of efficient and reliable iterative methods. We present analytic and computational results for the generalized minimal residual (GMRES) solution of the linearized QCF equilibrium equations. We show that the GMRES method accurately reproduces the stability of the force-based approximation and conclude that an appropriately preconditioned GMRES method results in a reliable and efficient solution method.

Fast and accurate three‐dimensional controlled source electromagnetic modelling†

ABSTRACTWe present a fast approximate method for three‐dimensional low frequency controlled source electro‐magnetic modeling. We apply the method to a synthetic model in a typical marine controlled source electromagnetic scenario, where conductivity and permittivity are different from the known background medium. For 3D configurations, fast computational methods are relevant for both forward and inverse modelling studies. Since this problem involves a large number of unknowns, it has to be solved efficiently to obtain results in a timely manner, without compromising accuracy. For this reason, the Born approximation, extended Born approximation and iterative extended Born approximation are implemented and compared with the full solution of the conjugate gradient fast Fourier transformation method. These methods are based on an electric field domain integral equation formulation. It is shown here how well the iterative extended Born approximation method performs in terms of both accuracy and speed with different configurations and different source positions. The improved accuracy comes at virtually no additional computational cost. With the help of this method, it is now possible to perform sensitivity analysis using 3D modelling in a timely manner, which is vital for controlled source electromagnetic applications. For forward modeling the solution at the sea‐bottom is of interest, because that is where the receivers are usually located. For inverse modeling, the accuracy of the solution in the target zone is important to obtain reasonably accurate conductivity values from the inversion using this approximate solution method. Our modelling studies show that the iterative extended Born approximation method is fast and accurate for both forward and inverse modelling. Sensitivity analysis as a function of the source position and different reservoir sizes validate the accuracy of the iterative extended Born approximation.

Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations

In this study it is shown that the numerical solutions of linear Fredholm integro-differential equations obtained by using Legendre polynomials can also be found by using the variational iteration method. Furthermore the numerical solutions of the given problems which are solved by the variational iteration method obviously converge rapidly to exact solutions better than the Legendre polynomial technique. Additionally, although the powerful effect of the applied processes in Legendre polynomial approach arises in the situations where the initial approximation value is unknown, it is shown by the examples that the variational iteration method produces more certain solutions where the first initial function approximation value is estimated. In this paper, the Legendre polynomial approximation (LPA) and the variational iteration method (VIM) are implemented to obtain the solutions of the linear Fredholm integro-differential equations and the numerical solutions with respect to these methods are compared.

10.1007/s11470-008-3005-z

A duality theory is developed to describe iterative methods for polyhedral approximation of convex bodies. The various types of approximation problems requiring the application of the duality theory are considered. Based on the theory, approximation methods can be designed for bodies with a dual description (in terms of the support/distance function) and methods can be developed that are optimal in terms of dual complexity characteristics of approximating polytopes (vertices/facets). New optimal methods based on the theory are formulated.

An iterative approximation method for solving a general system of variational inequality problems and mixed equilibrium problems

In this paper, we introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a general system of variational inequalities for a cocoercive mapping in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets. Our results extend and improve the corresponding results of Ceng, Wang, and Yao [L.C. Ceng, C.Y. Wang, J.C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Methods Oper. Res. 67 (2008) 375–390], Ceng and Yao [L.C. Ceng, J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. doi:10.1016/j.cam.2007.02.022], Takahashi and Takahashi [S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007) 506–515] and many others.

Calculation of water equivalent thickness of materials of arbitrary density, elemental composition and thickness in proton beam irradiation

In proton therapy, the radiological thickness of a material is commonly expressed in terms of water equivalent thickness (WET) or water equivalent ratio (WER). However, the WET calculations required either iterative numerical methods or approximate methods of unknown accuracy. The objective of this study was to develop a simple deterministic formula to calculate WET values with an accuracy of 1 mm for materials commonly used in proton radiation therapy. Several alternative formulas were derived in which the energy loss was calculated based on the Bragg–Kleeman rule (BK), the Bethe–Bloch equation (BB) or an empirical version of the Bethe–Bloch equation (EBB). Alternative approaches were developed for targets that were ‘radiologically thin’ or ‘thick’. The accuracy of these methods was assessed by comparison to values from an iterative numerical method that utilized evaluated stopping power tables. In addition, we also tested the approximate formula given in the International Atomic Energy Agency's dosimetry code of practice (Technical Report Series No 398, 2000, IAEA, Vienna) and stopping power ratio approximation. The results of these comparisons revealed that most methods were accurate for cases involving thin or low-Z targets. However, only the thick-target formulas provided accurate WET values for targets that were radiologically thick and contained high-Z material.

A capacity scaling heuristic for the multicommodity capacitated network design problem

In this paper, we propose a capacity scaling heuristic using a column generation and row generation technique to address the multicommodity capacitated network design problem. The capacity scaling heuristic is an approximate iterative solution method for capacitated network problems based on changing arc capacities, which depend on flow volumes on the arcs. By combining a column and row generation technique and a strong formulation including forcing constraints, this heuristic derives high quality results, and computational effort can be reduced considerably. The capacity scaling heuristic offers one of the best current results among approximate solution algorithms designed to address the multicommodity capacitated network design problem.

Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of stochastic calcium excitability (i.e., calcium puffs and sparks). Calcium release site models are stochastic automata networks that involve many functional transitions, that is, the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. We present a Kronecker-structured representation for calcium release site models and perform benchmark stationary distribution calculations using both exact and approximate iterative numerical solution techniques that leverage this structure. When it is possible to obtain an exact solution, response measures such as the number of channels in a particular state converge more quickly using the iterative numerical methods than occupation measures calculated via Monte Carlo simulation. In particular, multi-level methods provide excellent convergence with modest additional memory requirements for the Kronecker representation of calcium release site models. When an exact solution is not feasible, iterative approximate methods based on the power method may be used, with performance similar to Monte Carlo estimates. This suggests approximate methods with multi-level iterative engines as a promising avenue of future research for large-scale calcium release site models.