Special Issue "Iterative Algorithms for Nonlinear Problems: Convergence and Stability"

A special issue of Algorithms (ISSN 1999-4893). This special issue belongs to the section "Analysis of Algorithms and Complexity Theory".

Deadline for manuscript submissions: closed (15 November 2020).

Special Issue Editors

Special Issue Information

Dear Colleagues,

Many areas of Science and Technology involve the non-trivial task of solving nonlinear problems. Usually, it is not affordable in a direct way and iterative algorithms play a fundamental role in their approach. This area of research has enjoyed a period of an exponential growth in the last number of years.

This Special Issue is mainly dedicated, but not exclusively, to the design, analysis of convergence and stability of new iterative algorithms for solving nonlinear problems. Moreover, their application to practical problems of Engineering and Basic Sciences are of singular interest. The set of algorithms includes, but is not limited to, methods with and without memory, with derivatives of derivative-free, the real or complex dynamics associated to them and an analysis of their convergence that can be local, semilocal or global.

Prof. Dr. Alicia Cordero
Prof. Dr. Juan R. Torregrosa
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Algorithms is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • nonlinear systems
  • nonlinear matrix equations
  • transcendent equations
  • iterative algorithms
  • convergence
  • efficiency
  • chaotic behavior
  • complex or real dynamics
  • fractional nonlinear analysis

Published Papers (7 papers)

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Research

Article
Improved Sliding Mode Finite-Time Synchronization of Chaotic Systems with Unknown Parameters
Algorithms 2020, 13(12), 346; https://doi.org/10.3390/a13120346 - 20 Dec 2020
Cited by 1 | Viewed by 650
Abstract
This work uses the sliding mode control method to conduct the finite-time synchronization of chaotic systems. The utilized parameter selection principle differs from conventional methods. The designed controller selects the unknown parameters independently from the system model. These parameters enable tracking and prediction [...] Read more.
This work uses the sliding mode control method to conduct the finite-time synchronization of chaotic systems. The utilized parameter selection principle differs from conventional methods. The designed controller selects the unknown parameters independently from the system model. These parameters enable tracking and prediction of the additional variables that affect the chaotic motion but are difficult to measure. Consequently, the proposed approach avoids the limitations of selecting the unknown parameters that are challenging to measure or modeling the parameters solely within the relevant system. This paper proposes a novel nonsingular terminal sliding surface and demonstrates its finite-time convergence. Then, the adaptive law of unknown parameters is presented. Next, the adaptive sliding mode controller based on the finite-time control idea is proposed, and its finite-time convergence and stability are discussed. Finally, the paper presents numerical simulations of chaotic systems with either the same or different structures, thus verifying the proposed method’s applicability and effectiveness. Full article
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Article
Development of a Family of Jarratt-Like Sixth-Order Iterative Methods for Solving Nonlinear Systems with Their Basins of Attraction
Algorithms 2020, 13(11), 303; https://doi.org/10.3390/a13110303 - 20 Nov 2020
Cited by 1 | Viewed by 550
Abstract
We develop a family of three-step sixth order methods with generic weight functions employed in the second and third sub-steps for solving nonlinear systems. Theoretical and computational studies are of major concern for the convergence behavior with applications to special cases of rational [...] Read more.
We develop a family of three-step sixth order methods with generic weight functions employed in the second and third sub-steps for solving nonlinear systems. Theoretical and computational studies are of major concern for the convergence behavior with applications to special cases of rational weight functions. A number of numerical examples are illustrated to confirm the convergence behavior of local as well as global character of the proposed and existing methods viewed through the basins of attraction. Full article
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Article
On the Solutions of Second-Order Differential Equations with Polynomial Coefficients: Theory, Algorithm, Application
Algorithms 2020, 13(11), 286; https://doi.org/10.3390/a13110286 - 09 Nov 2020
Cited by 1 | Viewed by 766
Abstract
The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n [...] Read more.
The analysis of many physical phenomena is reduced to the study of linear differential equations with polynomial coefficients. The present work establishes the necessary and sufficient conditions for the existence of polynomial solutions to linear differential equations with polynomial coefficients of degree n, n1, and n2 respectively. We show that for n3 the necessary condition is not enough to ensure the existence of the polynomial solutions. Applying Scheffé’s criteria to this differential equation we have extracted n generic equations that are analytically solvable by two-term recurrence formulas. We give the closed-form solutions of these generic equations in terms of the generalized hypergeometric functions. For arbitrary n, three elementary theorems and one algorithm were developed to construct the polynomial solutions explicitly along with the necessary and sufficient conditions. We demonstrate the validity of the algorithm by constructing the polynomial solutions for the case of n=4. We also demonstrate the simplicity and applicability of our constructive approach through applications to several important equations in theoretical physics such as Heun and Dirac equations. Full article
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Article
Simple Iterative Method for Generating Targeted Universal Adversarial Perturbations
Algorithms 2020, 13(11), 268; https://doi.org/10.3390/a13110268 - 22 Oct 2020
Cited by 2 | Viewed by 943
Abstract
Deep neural networks (DNNs) are vulnerable to adversarial attacks. In particular, a single perturbation known as the universal adversarial perturbation (UAP) can foil most classification tasks conducted by DNNs. Thus, different methods for generating UAPs are required to fully evaluate the vulnerability of [...] Read more.
Deep neural networks (DNNs) are vulnerable to adversarial attacks. In particular, a single perturbation known as the universal adversarial perturbation (UAP) can foil most classification tasks conducted by DNNs. Thus, different methods for generating UAPs are required to fully evaluate the vulnerability of DNNs. A realistic evaluation would be with cases that consider targeted attacks; wherein the generated UAP causes the DNN to classify an input into a specific class. However, the development of UAPs for targeted attacks has largely fallen behind that of UAPs for non-targeted attacks. Therefore, we propose a simple iterative method to generate UAPs for targeted attacks. Our method combines the simple iterative method for generating non-targeted UAPs and the fast gradient sign method for generating a targeted adversarial perturbation for an input. We applied the proposed method to state-of-the-art DNN models for image classification and proved the existence of almost imperceptible UAPs for targeted attacks; further, we demonstrated that such UAPs can be easily generated. Full article
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Article
Perturbative-Iterative Computation of Inertial Manifolds of Systems of Delay-Differential Equations with Small Delays
Algorithms 2020, 13(9), 209; https://doi.org/10.3390/a13090209 - 27 Aug 2020
Viewed by 1042
Abstract
Delay-differential equations belong to the class of infinite-dimensional dynamical systems. However, it is often observed that the solutions are rapidly attracted to smooth manifolds embedded in the finite-dimensional state space, called inertial manifolds. The computation of an inertial manifold yields an ordinary differential [...] Read more.
Delay-differential equations belong to the class of infinite-dimensional dynamical systems. However, it is often observed that the solutions are rapidly attracted to smooth manifolds embedded in the finite-dimensional state space, called inertial manifolds. The computation of an inertial manifold yields an ordinary differential equation (ODE) model representing the long-term dynamics of the system. Note in particular that any attractors must be embedded in the inertial manifold when one exists, therefore reducing the study of these attractors to the ODE context, for which methods of analysis are well developed. This contribution presents a study of a previously developed method for constructing inertial manifolds based on an expansion of the delayed term in small powers of the delay, and subsequent solution of the invariance equation by the Fraser functional iteration method. The combined perturbative-iterative method is applied to several variations of a model for the expression of an inducible enzyme, where the delay represents the time required to transcribe messenger RNA and to translate that RNA into the protein. It is shown that inertial manifolds of different dimensions can be computed. Qualitatively correct inertial manifolds are obtained. Among other things, the dynamics confined to computed inertial manifolds display Andronov–Hopf bifurcations at similar parameter values as the original DDE model. Full article
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Article
Local Convergence of an Efficient Multipoint Iterative Method in Banach Space
Algorithms 2020, 13(1), 25; https://doi.org/10.3390/a13010025 - 15 Jan 2020
Viewed by 1655
Abstract
We discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fréchet-derivative only. The approaches of using Taylor expansions, [...] Read more.
We discuss the local convergence of a derivative-free eighth order method in a Banach space setting. The present study provides the radius of convergence and bounds on errors under the hypothesis based on the first Fréchet-derivative only. The approaches of using Taylor expansions, containing higher order derivatives, do not provide such estimates since the derivatives may be nonexistent or costly to compute. By using only first derivative, the method can be applied to a wider class of functions and hence its applications are expanded. Numerical experiments show that the present results are applicable to the cases wherein previous results cannot be applied. Full article
Article
Image Restoration Using a Fixed-Point Method for a TVL2 Regularization Problem
Algorithms 2020, 13(1), 1; https://doi.org/10.3390/a13010001 - 18 Dec 2019
Cited by 3 | Viewed by 1861
Abstract
In this paper, we first propose a new TVL2 regularization model for image restoration, and then we propose two iterative methods, which are fixed-point and fixed-point-like methods, using CGLS (Conjugate Gradient Least Squares method) for solving the new proposed TVL2 problem. We also [...] Read more.
In this paper, we first propose a new TVL2 regularization model for image restoration, and then we propose two iterative methods, which are fixed-point and fixed-point-like methods, using CGLS (Conjugate Gradient Least Squares method) for solving the new proposed TVL2 problem. We also provide convergence analysis for the fixed-point method. Lastly, numerical experiments for several test problems are provided to evaluate the effectiveness of the proposed two iterative methods. Numerical results show that the new proposed TVL2 model is preferred over an existing TVL2 model and the proposed fixed-point-like method is well suited for the new TVL2 model. Full article
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