Fixed-point Iteration Scheme Research Articles

Numerical Solution of Nonlinear Second Order Singular BVPs Based on Green’s Functions and Fixed-Point Iterative Schemes

This article discusses a numerical iterative scheme for the solution of a class of nonlinear singular boundary value problems. It introduces a recent approach, based on Green’s functions and Picard’s and Mann’s fixed-point iterations procedures, to tackle such problems. The convergence analysis of the proposed method is presented to verify its efficiency. A number of examples are given to demonstrate the applicability of the method. The numerical experiments show that this approach is better than many other existing techniques and that it is reliable, accurate and less time consuming.

Fixed-point iteration Gaussian sum filtering estimator with unknown time-varying non-Gaussian measurement noise

Abstract This paper addresses the state estimation problem in the presence of unknown time-varying non-Gaussian measurement noise (NGMN) modeled by Gaussian mixture distribution(GMD). Recently, the variational Bayesian (VB)-based estimators have been successfully used in the state estimation problem in the presence of unknown measurement noise. However, we reveal that when the state-space model is non-Gaussian, exploiting the traditional VB method, the joint posterior probability density function of the state and parameters of non-Gaussian measurement noise has no analytical result. To fill this gap, a fixed-point iteration Gaussian sum filtering (FPI-GSF) estimator is proposed to jointly estimate the state and the unknown time-varying NGMN online. The proposed estimator utilizes a fixed-point iteration scheme to iteratively calculate the state posterior distribution through the GSF and noise parameters posterior distribution via the VB method, thus it avoids calculating the joint posterior distribution. Simulation results demonstrate the effectiveness of the proposed estimator.

Equivalence results for implicit Jungck–Kirk type iterations

We show that the implicit Jungck–Kirk-multistep, implicit Jungck–Kirk–Noor, implicit Jungck–Kirk–Ishikawa, and implicit Jungck–Kirk–Mann iteration schemes are equivalently used to approximate the common fixed points of a pair of weakly compatible generalized contractive-like operators defined on normed linear spaces. Our results contribute to the existing results on the equivalence of fixed point iteration schemes by extending them to pairs of maps. An example to show the applicability of the main results is included.

An isogeometric analysis formulation for red blood cell electro-deformation modeling

An isogeometric analysis formulation for simulating red blood cell (RBC) electro-deformationis presented. Electrically-induced cell deformation experiments are receiving increasing attention as an attractive strategy for single-cell mechanical phenotyping. As the RBC structure consists in a very thin biological membrane enclosing a nearly-incompressible fluid, (i) a surface shell kinematic model and (ii) the imposition of the shell enclosed-volume conservation constraint are proposed within the isogeometric analysis framework. With regard to the electro-deformation, an accurate evaluation of the electric-field induced forces is achieved by the Maxwell stress tensor approach. A staggered fixed-point iteration scheme is then proposed for performing the electro-mechanical coupling, in order to use reliable mechanical and electrical problem solvers sequentially. Supported by the comparison with experimental results and reference solutions, numerical simulations concerning the large deformation of a RBC by optical tweezers and an in silico electro-deformation experiment prove the accuracy and the effectiveness of the proposed formulation.

Preliminary coupling of the Thermal/Hydraulic solvers in the Monte Carlo code MCS for practical LWR analysis

A Thermal/Hydraulic (T/H) coupling interface has been developed in the Monte Carlo code MCS for coupling with different T/H solvers. The single channel T/H solver TH1D and the Sub-Channel T/H solver COBRA-TF (CTF) both have been coupled to MCS through this interface. A fixed-point iteration scheme is applied in this coupling interface to enable the T/H feedback to be fully updated for Monte Carlo neutron simulation. This paper describes in detail the single rod case to demonstrate the accuracy and efficiency of these two coupled T/H solvers, with the discussion of the results for fuel temperature, coolant temperature and coolant density distributions. On the other hand, the paper also depicts the preliminary application of MCS T/H coupling in BEAVRS whole core benchmark to demonstrate the feasibility of this coupling system for use in practical LWRs multi-physics coupling analysis.

Global optimality condition and fixed point continuation algorithm for non-Lipschitz ℓ p regularized matrix minimization

Regularized minimization problems with nonconvex, nonsmooth, even non-Lipschitz penalty functions have attracted much attention in recent years, owing to their wide applications in statistics, control, system identification and machine learning. In this paper, the non-Lipschitz $\ell_p~(0<p<1)$ regularized matrix minimization problem is studied. A global necessary optimality condition for this non-Lipschitz optimization problem is firstly obtained, specifically, the global optimal solutions for the problem are fixed points of the so-called $p$-thresholding operator which is matrix-valued and set-valued. Then a fixed point iterative scheme for the non-Lipschitz model is proposed, and the convergence analysis is also addressed in detail. Moreover, some acceleration techniques are adopted to improve the performance of this algorithm. The effectiveness of the proposed $p$-thresholding fixed point continuation ($p$-FPC) algorithm is demonstrated by numerical experiments on randomly generated and real matrix completion problems.

A novel Ishikawa–Green’s fixed point scheme for the solution of BVPs

In this article, a new numerical approach is introduced for the numerical solution of a wide class of boundary value problems (BVPs). The underlying strategy of the algorithm is based on embedding an integral operator, defined in terms of Green’s function, into Ishikawa fixed point iteration scheme. The validity of the method is demonstrated by a number of examples that confirm the applicability and high efficiency of the method. The absolute error or residual error computations show that the current technique provides highly accurate approximations.

A Distributed Multi-Control-Center Dynamic Power Flow Algorithm Based on Asynchronous Iteration Scheme

From the study of dispatcher power flow in a distributed energy management system (DEMS) among multiple control centers, a fixed-point iteration scheme for boundary bus states is constructed to modify the exterior network equivalent through data exchange. Implementation details such as determination of combination parameters, independent choice of reference bus, and distribution of unbalanced power are analyzed. Using this method, each subsystem involved in distributed computing can maintain its method of modeling the exterior network. Adverse effects of communication interference on computing can also be eliminated. Results consistent with integrated power flow are obtained with little loss of accuracy. Thus, the proposed algorithm is especially suitable for application in DEMS, in which control centers can only communicate by WAN. Tests are conducted on the IEEE 118-bus system partitioned into five subsystems, the IEEE 300-bus system, and an actual power system, which verifies the favorable convergence and effectiveness of the method.

A study in the fixed point theory for a new iterative scheme and a class of generalized mappings

In this study, we introduce a new iteration scheme and prove the strong convergence result for this iteration method. We also compare the rate of convergence with the iterative scheme and the fixed point iteration scheme known as Picard-S due to Gursoy. Then we prove that this new iteration method is equivalent to convergence of the iteration schemes given in the introduction section of the manuscript. Moreover, we show the result of its data dependency.

Data dependence for four-step fixed point iterative scheme allocating via contractive-like operators

Generally, by the data dependence for an iterative scheme means that there exist an estimation by which one can find the fixed point of an unknown operator from the fixed point of given operator. In this paper, we have introduced and analyzed the four-step fixed point iterative scheme and established its convergence and data dependence results by using contractive-like operators. A numerical example is also given, in which instead of computing the fixed point of an operator, an approximation has been made about the fixed point of that operator through contractive-like one. This work is an extension and improvement of the corresponding works of Asaduzzaman et al. [10] and other several authors in literature.

A Fixed-Point Iteration Scheme for Model Predictive Torque Control of PMSMs

This paper proposes a tailored fixed-point iteration scheme for the model predictive torque control of permanent magnet synchronous machines. Both spherical voltage and current constraints are taken into account using a projection function and an augmented Lagrangian approach. The fixed-point scheme is a continuous control set approach and is based on the nonlinear dq-model. The inductivities and permanent magnet flux are modelled current-dependently and therefore account for saturation and parasitic effects. Experimental results on a standard dSpace hardware show the performance as well as the very small computational load of the proposed torque control scheme, which is therefore suitable for an embedded implementation.

Study of the non-linear eddy-current response in a ferromagnetic plate: Theoretical analysis for the 2D case

The non-linear induction problem in an infinite ferromagnetic pate is studied theoretically by means of the truncated region eigenfunction expansion (TREE) for the 2D case. The non-linear formulation is linearised using a fixed-point iterative scheme, and the solution of the resulting linear problem is constructed in the Fourier domain following the TREE formalism. The calculation is carried out for the steady-state response under harmonic excitation and the harmonic distortion is derived from the obtained spectrum. This article is meant to be the theoretical part of a study, which will be complemented by the corresponding experimental work in a future communication.

Numerical solution of functional differential equations: a Green's function-based iterative approach

ABSTRACTIn this article, a recently introduced iterative method based on Green's functions and fixed-point iteration schemes is presented for the approximate solutions of delay and functional differential equations. The approach is especially suited for handling boundary value problems (BVPs). The algorithm is illustrated through a number of examples that confirm the high accuracy and efficiency of the strategy. The results of the test examples show excellent agreement with exact solutions and outperforms other existing numerical iterative schemes.

Green’s Function Iterative Approach for Solving Strongly Nonlinear Oscillators

In this paper, a Green’s function based iterative algorithm is proposed to solve strong nonlinear oscillators. The method’s essential part is based on finding an appropriate Green’s function that will be incorporated into a linear integral operator. An application of fixed point iteration schemes such as Picard’s or Mann’s will generate an iterative formula that gives reliable approximations to the true periodic solutions that characterize these kinds of equations. The applicability and stability of the method will be tested through numerical examples. Since exact solutions to these equations usually do not exist, the proposed method will be tested against other popular numerical methods such as the modified homotopy perturbation, the modified differential transformation, and the fourth-order Runge–Kutta methods.

On new faster fixed point iterative schemes for contraction operators and comparison of their rate of convergence in convex metric spaces

In this paper we present new iterative algorithms in convex metric spaces. We show that these iterative schemes are convergent to the fixed point of a single-valued contraction operator. Then we make the comparison of their rate of convergence. Additionally, numerical examples for these iteration processes are given.

A new accelerated proximal technique for regression with high-dimensional datasets

We consider the problem of minimization of the sum of two convex functions, one of which is a smooth function, while another one may be a nonsmooth function. Many high-dimensional learning problems (classification/regression) can be designed using such frameworks, which can be efficiently solved with the help of first-order proximal-based methods. Due to slow convergence of traditional proximal methods, a recent trend is to introduce acceleration to such methods, which increases the speed of convergence. Such proximal gradient methods belong to a wider class of the forward–backward algorithms, which mathematically can be interpreted as fixed-point iterative schemes. In this paper, we design few new proximal gradient methods corresponding to few state-of-the-art fixed-point iterative schemes and compare their performances on the regression problem. In addition, we propose a new accelerated proximal gradient algorithm, which outperforms earlier traditional methods in terms of convergence speed and regression error. To demonstrate the applicability of our method, we conducted experiments for the problem of regression with several publicly available high-dimensional real datasets taken from different application domains. Empirical results exhibit that the proposed method outperforms the previous methods in terms of convergence, accuracy, and objective function values.

A semi-analytical solution of amperometric enzymatic reactions based on Green's functions and fixed point iterative schemes

In this paper, a constructed Green's function coupled with a fixed point iteration scheme will be employed to solve nonlinear dynamical problems that arise in electroanalytical chemistry. More precisely, the method will be used to mathematically model and solve the kinetics of the amperometric enzyme. A main property that makes the proposed method superior to other iterative methods is the way it handles boundary value problems, where both endpoints are taken into consideration while other iterative methods only make account of the initial point and as a result, the approximate solution may deteriorate for values that are far away from the initial point and closer to the other endpoint. Through tests on some known amperometric enzyme kinetics, the proposed method gave more accurate results than many numerical schemes that were employed for this purpose. The method is found to be easily implemented, fast, and computationally economical and attractive.

Generalized Single-Index Models: The EFM Approach

Generalized single-index models are natural extensions of linear models and circumvent the so-called curse of dimensionality. They are becoming increasingly popular in many scientific fields including biostatistics, medicine, economics and finan- cial econometrics. Estimating and testing the model index coefficients beta is one of the most important objectives in the statistical analysis. However, the commonly used assumption on the index coefficients, beta = 1, represents a non-regular problem: the true index is on the boundary of the unit ball. In this paper we introduce the EFM ap- proach, a method of estimating functions, to study the generalized single-index model. The procedure is to first relax the equality constraint to one with (d - 1) components of beta lying in an open unit ball, and then to construct the associated (d - 1) estimating functions by projecting the score function to the linear space spanned by the residuals with the unknown link being estimated by kernel estimating functions. The root-n consistency and asymptotic normality for the estimator obtained from solving the re- sulting estimating equations is achieved, and a Wilk's type theorem for testing the index is demonstrated. A noticeable result we obtain is that our estimator for beta has smaller or equal limiting variance than the estimator of Carroll et al. (1997). A fixed point iterative scheme for computing this estimator is proposed. This algorithm only involves one-dimensional nonparametric smoothers, thereby avoiding the data sparsity problem caused by high model dimensionality. Numerical studies based on simulation and on applications suggest that this new estimating system is quite powerful and easy to implement.

Bratu-like equation arising in electrospinning process: a Green's function fixed-point iteration approach

The aim of this article is to solve a Bratu-like nonlinear differential equation arising in electrospinning and vibration-electrospinning process that was recently introduced by Liu and Wang (2014). Electro-spinning process has been associated to Bratu equation through thermo-electro-hydrodynamics balance equations. The proposed strategy is based on embedding Green's function into Picard's fixed-point iteration scheme. Validation of the novel algorithm is demonstrated through numerical examples which comprise certain selected values of the parameters that appear in the modelled equation.

Bratu-like equation arising in electrospinning process: a Green's function fixed-point iteration approach

The aim of this article is to solve a Bratu-like nonlinear differential equation arising in electrospinning and vibration-electrospinning process that was recently introduced by Liu and Wang (2014). Electro-spinning process has been associated to Bratu equation through thermo-electro-hydrodynamics balance equations. The proposed strategy is based on embedding Green's function into Picard's fixed-point iteration scheme. Validation of the novel algorithm is demonstrated through numerical examples which comprise certain selected values of the parameters that appear in the modelled equation.