Efficient Iterative Technique Research Articles

An iterative technique for a class of highly nonlinear BVP arising in a one-dimensional corneal shape model

PurposeThe aim of this work is to establish a computationally efficient iterative technique for solving the highly nonlinear derivative-dependent boundary value problem (DDBVP). The proposed iterative technique is tested on a one-dimensional mathematical model of the shape of the human cornea, which leads to the highly nonlinear DDBVP.Design/methodology/approachThe approach is a combined venture of quasilinearization along with Picard’s iterative technique. The nonlinear DDBVP is simplified to a sequence of linear problems through a quasilinearization technique. Making use of Picard’s iteration approach, an analogous integral form for the quasilinearized DDBVPs is developed in terms of Green’s function and the convergence controller parameter. The study also covered the convergence analysis of the iterative technique.FindingsNumerical illustrations are presented to evaluate the applicability, efficiency and accuracy of the technique. The proposed technique provides approximate numerical solutions to the corneal shape model with a variety of constant parameters that can arise in different physical situations. In comparison with existing works, the proposed coupled iterative approach has been found to be more accurate and less computationally complex.Originality/valueThis study introduces an efficient coupled iterative technique based on quasilinearization and Picard’s iterative technique to solve a one-dimensional model of a human cornea. The proposed approach is tested on various physical situations that affect the human cornea. Numerical findings are compared with other published results to demonstrate the method’s efficiency and ease of usage.

A “Consistent” Quasidiffusion Method for Solving Particle Transport Problems

The quasidiffusion (QD) method is an established and efficient iterative technique for solving particle transport problems. Each QD iteration consists of a high-order SN sweep, followed by a low-order QD calculation. QD has two defining characteristics: (1) its iterations converge rapidly for any spatial grid and (2) the converged scalar fluxes from the high-order SN sweep and the low-order QD calculation differ, by spatial truncation errors, from each other and from the scalar flux solution of the SN equations. In this paper, we show that by including a transport consistency factor in the low-order equation, the converged high-order and low-order scalar fluxes become equal to each other and to the converged SN scalar flux. However, the inclusion of the transport consistency factor has a negative impact on the convergence rate. We present numerical results that demonstrate the effect of the transport consistency factor on stability.

An efficient iterative pseudo point elimination technique to represent the shape of the digital image boundary

An efficient iterative pseudo point elimination technique to represent the shape of the digital image boundary

An efficient recursive technique with Padé approximation for a kind of Lane–Emden type equations emerging in various physical phenomena

The study numerically examined a class of nonlinear singular differential problems known as the Lane–Emden differential equation, which emerges in numerous real-world situations. The primary goal of this work is to formulate a computationally efficient iterative technique for solving the nonlinear Lane–Emden initial value problems. The proposed approach is a hybrid of the homotopy perturbation method and the Padé approximation. The nonlinear singular Lane–Emden initial value problem (SLEIVP) is transformed into an equivalent recursive integral employing the Picard’s approach. To resolve the singularity and nonlinearity, the recursive integral equation is transformed into a system of integral equations by using the homotopy notion. Furthermore, to enhance the convergence rate of the technique, Padé approximation is taken into account. The convergence analysis for the proposed approach is also conducted. The present technique is tested on SLEIVPs and numerical findings are compared with the existing techniques, to demonstrate the accuracy, effectiveness and ease of use.

A Sufficient Descent Dai-Liao Type Conjugate Gradient Update Parameter

In recent years, conjugate gradient methods have gained popularity as efficient iterative techniques for unconstrained optimization problems without the need for matrix storage. Based on the Dai-Laio conjugacy condition, this article presents a new hybrid conjugate gradient method that combines features of the Dai-Yuan and Dai-Laio methods. The proposed method addresses the numerical instability and slow convergence of the Dai-Yuan method as well as the potential poor performance of the Dai-Laio method in highly non-linear optimization problems. The hybrid method solves optimization problems with faster convergence rates and greater stability by combining the advantages of both methods. The resulting algorithm is shown to be more effective and reliable, and theoretical analysis reveals that it has sufficient descent properties. The proposed method's competitive performance is shown through a number of benchmark tests and comparisons with other approaches, indicating that it has the potential to be an effective approach for complex, unconstrained optimization.

On OAMP: Impact of the Orthogonal Principle

Approximate Message Passing (AMP) is an efficient iterative parameter-estimation technique for certain high-dimensional linear systems with non-Gaussian distributions, such as sparse systems. In AMP, a so-called Onsager term is added to keep estimation errors approximately Gaussian. Orthogonal AMP (OAMP) does not require this Onsager term, relying instead on an orthogonalization procedure to keep the current errors uncorrelated with (i.e., orthogonal to) past errors. In this paper, we show the generality and significance of the orthogonality in ensuring that errors are “asymptotically independently and identically distributed Gaussian” (AIIDG). This AIIDG property, which is essential for the attractive performance of OAMP, holds for separable functions. We present a simple and versatile procedure to establish the orthogonality through Gram-Schmidt (GS) orthogonalization, which is applicable to any prototype. We show that different AMP-type algorithms, such as expectation propagation (EP), turbo, AMP and OAMP, can be unified under the orthogonal principle. The simplicity and generality of OAMP provide efficient solutions for estimation problems beyond the classical linear models. As an example, we study the optimization of OAMP via the GS model and GS orthogonalization. More related applications will be discussed in a companion paper where new algorithms are developed for problems with multiple constraints and multiple measurement variables.

A Novel Class of Energy-Preserving Runge-Kutta Methods for the Korteweg-de Vries Equation

In this paper, we present a quadratic auxiliary variable approach to develop a new class of energy-preserving Runge-Kutta methods for the Korteweg-de Vries equation. The quadratic auxiliary variable approach is first proposed to reformulate the original model into an equivalent system, which transforms the energy conservation law of the Korteweg-de Vries equation into two quadratic invariants of the reformulated system. Then the symplectic Runge-Kutta methods are directly employed for the reformulated model to arrive at a new kind of time semi-discrete schemes for the original problem. Under the consistent initial condition, the proposed methods are rigorously proved to maintain the original energy conservation law of the Korteweg-de Vries equation. In addition, the Fourier pseudo-spectral method is used for spatial discretization, resulting in fully discrete energy-preserving schemes. To implement the proposed methods effectively, we present a very efficient iterative technique, which not only greatly saves the calculation cost, but also achieves the purpose of practically preserving structure. Ample numerical results are addressed to confirm the expected order of accuracy, conservative property and efficiency of the proposed algorithms.

Fast and Accurate Proper Orthogonal Decomposition using Efficient Sampling and Iterative Techniques for Singular Value Decomposition

In this article, we propose a computationally efficient iterative algorithm for proper orthogonal decomposition (POD) using random sampling based techniques. In this algorithm, additional rows and columns are sampled and a merging technique is used to update the dominant POD modes in each iteration. We derive bounds for the spectral norm of the error introduced by a series of merging operations. We use an existing theorem to get an approximate measure of the quality of subspaces obtained on convergence of the iteration. Results on various datasets indicate that the POD modes and/or the subspaces are approximated with excellent accuracy with a significant runtime improvement over computing the truncated SVD. We also propose a method to compute the POD modes of large matrices that do not fit in the RAM using this iterative sampling and merging algorithms.

A nonlinear modeling of fractional order based variational model in optical flow estimation

In this paper, a nonlinear fractional order variational (NFOV) model is introduced to estimate the optical flow. In particular, the presented model can be scaled to the existing integer order variational models. The proposed variational functional is designed by combining a non-quadratic Charbonnier norm based data term and Marchaud fractional derivative based regularization term. This non-quadratic penalty is robust against outliers, whereas the non-local character of Marchaud fractional derivative leads the model to deal with the discontinuous information about texture and edges, and yields a dense flow field. The numerical discretization of the Marchaud fractional derivative is carried out with the help of Grünwald–Letnikov fractional derivative. The discretized nonlinear system is further reduced to a linear system of equations by employing an outer fixed point iteration scheme, and finally solved with an efficient iteration technique. The experiments are performed over a variety of datasets. The performance of the model is tested using different error measures (AAE, AEE, AENG, WE) depending upon the availability of the ground truth (GT) flow. The robustness of the NFOV model is shown under different amounts of noise along with the variation of smoothing parameter. A detailed comparison with the recently published works is presented to demonstrate the efficiency and accuracy of the proposed model.

An Efficient Two-Step Iterative Technique for Solving Non-Linear Equations

We use the homotopy perturbation method (HPM) to construct a new iterative system for solving non-linear equations in this article. The criteria for convergence in the scheme developed are also imposed. To show the validity and reliability of our process, we compare our regime with other current procedures by looking at various test problems.

An efficient analytical iterative technique for solving nonlinear differential equations

An efficient analytical iterative technique for solving nonlinear differential equations

MULTIPLE USE OF BACKTRACKING LINE SEARCH IN UNCONSTRAINED OPTIMIZATION

The gradient method is a very efficient iterative technique for solving unconstrained optimization problems. Motivated by recent modifications of some variants of the SM method, this study proposed two methods that are globally convergent as well as computationally efficient. Each of the methods is globally convergent under the influence of a backtracking line search. Results obtained from the numerical implementation of these methods and performance profiling show that the methods are very competitive with well-known traditional methods.

Antenna Design and Analysis Using the Sequential Loading Method

This article presents a new method for accurately designing and analyzing an antenna or array. The proposed technique operates solely on the transfer admittance matrix of a perfect electrically conducting antenna fabric that is obtained using a full-wave simulator only once. The desired antenna, which must be a subset of the fabric, is then synthesized by progressively loading the fabric and applying a computationally efficient iterative postprocessing technique that transforms the stored admittance matrix into one that corresponds to the desired antenna or array. Given that the admittance matrix of the fabric is used as the basis for all subsequent antennas that can be represented by the fabric, it contains the solutions to an entire class of antennas. The proposed method is demonstrated by applying it to the class of colinear antennas up to five wavelengths long, arranged along a single axis. The method is validated by comparing its results to those obtained using a conventional full-wave simulator. The proposed method is applied first to the case of multiple loads located at arbitrary port locations along the fabric. Then, open-circuited loads are used to decimate the tips of a linear antenna to demonstrate that the method accurately predicts the response of an electrically shorter antenna. Finally, decimation is used to transform the fabric into an array of smaller antennas, where it is shown that mutual coupling effects are accurately predicted.

A New Family of Hybrid Conjugate Gradient Methods for Unconstrained Optimization

The conjugate gradient method is a very efficient iterative technique for solving large-scale unconstrainedoptimization problems. Motivated by recent modifications of some variants of the method and construction of hybrid methods, this study proposed four hybrid methods that are globally convergent as well as computationally efficient. The approach adopted for constructing the hybrid methods entails projecting ten recently modified conjugate gradient methods. Each of the hybrid methods is shown to satisfy the descent property independent of any line search technique and globally convergent under the influence of strong Wolfe line search. Results obtained from numerical implementation of these methods and performance profiling show that the methods are very competitive with well-known traditional methods.

Haar wavelet iteration method for solving time fractional Fisher's equation

In this work, we investigate fractional version of the Fisher equation and solve it by using an efficient iteration technique based on the Haar wavelet operational matrices. In fact, we convert the nonlinear equation into a Sylvester equation by the Haar wavelet collocation iteration method (HWCIM) to obtain the solution. We provide four numerical examples to illustrate the simplicity and efficiency of the technique.

A theoretical calculation for carrier concentration and resistance prediction

Due to lack of analytical solutions for equation of electrical neutrality under thermal equilibrium conditions, a novel and efficient numerical iteration technique was developed. The computational carrier concentration obtained by this new numerical method is in good correlation to commonly cited values and experimental data. Based on this new numerical method effects of some factors on the resistance of silicon piezoresistive pressure sensor were analyzed. Results show that no matter doping concentration is high or low the minority carrier exclusion effect disappears under considerably high current conditions, and increases in bias current induce greater maximum operating temperature; increases in doping concentration can increase the maximum operating temperature and decrease temperature coefficient of resistance.

A New High-Order and Efficient Family of Iterative Techniques for Nonlinear Models

In this paper, we want to construct a new high-order and efficient iterative technique for solving a system of nonlinear equations. For this purpose, we extend the earlier scalar scheme [16] to a system of nonlinear equations preserving the same convergence order. Moreover, by adding one more additional step, we obtain minimum 5th-order convergence for every value of a free parameter, θ∈ℝ, and for θ=−1, the method reaches maximum 6-order convergence. We present an extensive convergence analysis of our scheme. The analytical discussion of the work is upheld by performing numerical experiments on some applied science problems and a large system of nonlinear equations. Furthermore, numerical results demonstrate the validity and reliability of the suggested methods.

An efficient fifth-order iterative scheme for solving a system of nonlinear equations and PDE

This article, introduces an efficient fifth-order iterative technique for solving systems of nonlinear equations. The order of convergence of the proposed method has been verified by the computational order of convergence. Some numerical examples are employed to show the superiority of the proposed iterative method. The computational efficiency index has also been illustrated and analysed. The application of proposed scheme for solving nonlinear PDE has also been discussed here.

An efficient fifth-order iterative scheme for solving a system of nonlinear equations and PDE

This article, introduces an efficient fifth-order iterative technique for solving systems of nonlinear equations. The order of convergence of the proposed method has been verified by the computational order of convergence. Some numerical examples are employed to show the superiority of the proposed iterative method. The computational efficiency index has also been illustrated and analysed. The application of proposed scheme for solving nonlinear PDE has also been discussed here.

High-Order Lax-Friedrichs WENO Fast Sweeping Methods for the SN Neutron Transport Equation

Fast sweeping methods are efficient iterative techniques originally developed to solve the steady-state Hamilton-Jacobi equations and later used for the hyperbolic conservation laws. For these boundary value problems, their solution information propagates along characteristics starting from the boundary. These fast sweeping methods take advantage of this property and achieve very fast convergence based on a Gauss-Seidel–type iteration approach and alternating-direction sweeping strategy. In this paper, we solve the SN neutron transport equation using the high-order Lax-Friedrichs Weighted Essentially Non-Oscillatory (LF-WENO) fast sweeping methods. Our numerical tests in one and two dimensions demonstrate that the proposed new sweeping methods can achieve better accuracy and positivity preserving than the diamond difference method for the SN solution.