Accelerating the convergence speed of iterative methods for solving nonlinear systems

Xiao, Xiaoyong; Yin, Hongwei

doi:10.1016/j.amc.2018.03.108

Cited by 13 publications

(13 citation statements)

References 21 publications

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“…In the case of the methods that are based on Newton’s, Traub, Jarratt, etc. methods the order of convergence is proved (see for instance [ 8 , 9 , 10 , 11 , 12 , 13 , 14 ]), but in case of the methods that are based on metaheuristic algorithms, like the method presented in this paper, the order of convergence is not derived (see for instance [ 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 ]). This is due the fact that this order greatly varies for different values of the parameters used in the metaheuristic algorithms.…”

Section: The Algorithmmentioning

confidence: 99%

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Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications

Gościniak

Gdawiec

2020

Entropy

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There is a huge group of algorithms described in the literature that iteratively find solutions of a given equation. Most of them require tuning. The article presents root-finding algorithms that are based on the Newton–Raphson method which iteratively finds the solutions, and require tuning. The modification of the algorithm implements the best position of particle similarly to the particle swarm optimisation algorithms. The proposed approach allows visualising the impact of the algorithm’s elements on the complex behaviour of the algorithm. Moreover, instead of the standard Picard iteration, various feedback iteration processes are used in this research. Presented examples and the conducted discussion on the algorithm’s operation allow to understand the influence of the proposed modifications on the algorithm’s behaviour. Understanding the impact of the proposed modification on the algorithm’s operation can be helpful in using it in other algorithms. The obtained images also have potential artistic applications.

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Section: The Algorithmmentioning

confidence: 99%

“…This method uses only one LU factorisation which preserves and reduces computational complexities. An acceleration technique that can be used in many iterative methods was proposed in [ 12 ]. Moreover, the authors introduced a new family of two-step third order methods for solving systems of non-linear equations.…”

Section: Introductionmentioning

confidence: 99%

Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications

Gościniak

Gdawiec

2020

Entropy

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“…The second step is to perform another Newton step that combines the information of a subsequent state(s) and the value of the iterative function ψ x k leading to better convergence. Xiao and Yin [69] extend that method to construct a family of convergent high-order methods.…”

Section: The Accelerated Convergence For the Internal Loopmentioning

confidence: 99%

“…In this research, the ideas of [68,69] and [67] are combined to define a form the function ψ x k leading the classical Newton method M 1 , and the modified Thomas method M T . Details of them are provided concisely.…”

Section: The Accelerated Convergence For the Internal Loopmentioning

confidence: 99%

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An enhanced sequential fully implicit scheme for reservoir geomechanics

et al. 2020

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In this paper, it is proposed an enhanced sequential fully implicit ESFI algorithm with a fixed stress split to approximate robustly poro-elastoplastic solutions related to reservoir geomechanics. The constitutive model considers the total strain effect on porosity/permeability variation and associative plasticity. The sequential fully implicit algorithm SFI is a popular solution to approximate solutions of a coupled system. Generally, the SFI consists of an outer loop to solve the coupled system, in which there are two inner iterative loops for each equation to implicitly solve the equations. The SFI algorithm occasionally suffers from slow convergence or even convergence failure. In order to improve the convergence (robustness) associated with SFI, a new nonlinear acceleration technique is proposed employing Shanks transformations in vector-valued variables to enhance the outer loop convergence, with a Quasi-Newton method considering the modified Thomas method for the internal loops. In this ESFI algorithm, the fluid flow formulation is defined by Darcy's law including nonlinear permeability based on Petunin model. The rock deformation includes a linear part being analyzed based on Biot's theory and a nonlinear part being established using Mohr-Coulomb associative plasticity for geomechanics. Temporal derivatives are approximated by an implicit Euler method and spatial discretizations are adopted using finite element in two different formulations. For the spatial discretization, two weak statements are obtained: the first one uses a continuous Galerkin for poro-elastoplastic and Darcy's flow; the second one uses a continuous Galerkin for poro-elastoplastic and a mixed finite element for Darcy's flow. Several numerical simulations are presented to evaluate the efficiency of ESFI algorithm in reducing the number of iterations. Distinct poromechanical problems in 1D, 2D, and 3D are approximated with linear and nonlinear settings. Where appropriate, the results were verified with analytic solutions and approximated solutions with an explicit Runge-Kutta solver for 2D axisymmetric poroelastoplastic problems.

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A simple yet efficient two-step fifth-order weighted-Newton method for nonlinear models

2022

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Accelerating the convergence speed of iterative methods for solving nonlinear systems

Cited by 13 publications

References 21 publications

Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications

Visual Analysis of Dynamics Behaviour of an Iterative Method Depending on Selected Parameters and Modifications

An enhanced sequential fully implicit scheme for reservoir geomechanics

A simple yet efficient two-step fifth-order weighted-Newton method for nonlinear models

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