Research

17 pages, 4164 KiB

Open AccessArticle

Optimized Design for NB-LDPC-Coded High-Order CPM: Power and Iterative Efficiencies

by Rui Xue, Tong Wang, Yanbo Sun and Huaiyu Tang

Symmetry 2020, 12(8), 1353; https://doi.org/10.3390/sym12081353 - 13 Aug 2020

Viewed by 2047

In this paper, a non-binary low-density parity-check (NB-LDPC) coded high-order continuous phase modulation (CPM) system is designed and optimized to improve power and iterative efficiencies. Firstly, the minimum squared normalized Euclidean distance and the 99% double-sided power bandwidth are introduced to design a [...] Read more.

In this paper, a non-binary low-density parity-check (NB-LDPC) coded high-order continuous phase modulation (CPM) system is designed and optimized to improve power and iterative efficiencies. Firstly, the minimum squared normalized Euclidean distance and the 99% double-sided power bandwidth are introduced to design a competitive CPM, improving its power efficiency under a given code rate and spectral efficiency. Secondly, a three-step method based on extrinsic information transfer (EXIT) and entropy theory is used to design NB-LDPC codes, which reduces the convergence threshold approximately 0.42 and 0.58 dB compared with the candidate schemes. Thirdly, an extrinsic information operation is proposed to address the positive feedback issue in iterative detection and decoding and the value of bit error rate (BER) can approximately be reduced by

5 \times 10^{- 3}

. Finally, iteration optimization employing the EXIT chart and mutual information between demodulation and decoding is performed to achieve a suitable tradeoff for the communication reliability and iterative decoding delay. Simulation results show that the resulting scheme provides an approximately 3.95 dB coding gain compared to the uncoded CPM and achieves approximately 0.5 and 0.7 dB advantages compared with the candidate schemes. The resulting NB-LDPC-coded high-order CPM for a given code rate and spectral efficiency converges earlier into a turbo cliff region compared with other competitors and significantly improves power and iterative efficiencies. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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15 pages, 732 KiB

Open AccessFeature PaperArticle

Advanced Algorithms and Common Solutions to Variational Inequalities

by Hasanen A. Hammad, Habib ur Rehman and Manuel De la Sen

Symmetry 2020, 12(7), 1198; https://doi.org/10.3390/sym12071198 - 20 Jul 2020

Cited by 22 | Viewed by 2136

Abstract

The paper aims to present advanced algorithms arising out of adding the inertial technical and shrinking projection terms to ordinary parallel and cyclic hybrid inertial sub-gradient extra-gradient algorithms (for short, PCHISE). Via these algorithms, common solutions of variational inequality problems (CSVIP) and strong [...] Read more.

The paper aims to present advanced algorithms arising out of adding the inertial technical and shrinking projection terms to ordinary parallel and cyclic hybrid inertial sub-gradient extra-gradient algorithms (for short, PCHISE). Via these algorithms, common solutions of variational inequality problems (CSVIP) and strong convergence results are obtained in Hilbert spaces. The structure of this problem is to find a solution to a system of unrelated VI fronting for set-valued mappings. To clarify the acceleration, effectiveness, and performance of our parallel and cyclic algorithms, numerical contributions have been incorporated. In this direction, our results unify and generalize some related papers in the literature. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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17 pages, 321 KiB

Open AccessArticle

Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems

by Kamel Al-Khaled and Ashwaq Hazaimeh

Symmetry 2020, 12(7), 1179; https://doi.org/10.3390/sym12071179 - 16 Jul 2020

Cited by 6 | Viewed by 1938

Abstract

In this paper, we present a comparative study between Sinc–Galerkin method and a modified version of the variational iteration method (VIM) to solve non-linear Sturm–Liouville eigenvalue problem. In the Sinc method, the problem under consideration was converted from a non-linear differential equation to [...] Read more.

In this paper, we present a comparative study between Sinc–Galerkin method and a modified version of the variational iteration method (VIM) to solve non-linear Sturm–Liouville eigenvalue problem. In the Sinc method, the problem under consideration was converted from a non-linear differential equation to a non-linear system of equations, that we were able to solve it via the use of some iterative techniques, like Newton’s method. The other method under consideration is the VIM, where the VIM has been modified through the use of the Laplace transform, and another effective modification has also been made to the VIM by replacing the non-linear term in the integral equation resulting from the use of the well-known VIM with the Adomian’s polynomials. In order to explain the advantages of each method over the other, several issues have been studied, including one that has an application in the field of spectral theory. The results in solutions to these problems, which were included in tables, showed that the improved VIM is better than the Sinc method, while the Sinc method addresses some advantages over the VIM when dealing with singular problems. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

15 pages, 360 KiB

Open AccessArticle

Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations

by Ioannis K. Argyros, Stepan Shakhno and Halyna Yarmola

Symmetry 2020, 12(7), 1093; https://doi.org/10.3390/sym12071093 - 01 Jul 2020

Cited by 1 | Viewed by 1389

Abstract

Solving equations in abstract spaces is important since many problems from diverse disciplines require it. The solutions of these equations cannot be obtained in a form closed. That difficulty forces us to develop ever improving iterative methods. In this paper we improve the [...] Read more.

Solving equations in abstract spaces is important since many problems from diverse disciplines require it. The solutions of these equations cannot be obtained in a form closed. That difficulty forces us to develop ever improving iterative methods. In this paper we improve the applicability of such methods. Our technique is very general and can be used to expand the applicability of other methods. We use two methods of linear interpolation namely the Secant as well as the Kurchatov method. The investigation of Kurchatov’s method is done under rather strict conditions. In this work, using the majorant principle of Kantorovich and our new idea of the restricted convergence domain, we present an improved semilocal convergence of these methods. We determine the quadratical order of convergence of the Kurchatov method and order

\frac{1 + \sqrt{5}}{2}

for the Secant method. We find improved a priori and a posteriori estimations of the method’s error. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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10 pages, 262 KiB

Open AccessFeature PaperArticle

Direct Comparison between Two Third Convergence Order Schemes for Solving Equations

by Samundra Regmi, Ioannis K. Argyros and Santhosh George

Symmetry 2020, 12(7), 1080; https://doi.org/10.3390/sym12071080 - 01 Jul 2020

Cited by 3 | Viewed by 1393

Abstract

We provide a comparison between two schemes for solving equations on Banach space. A comparison between same convergence order schemes has been given using numerical examples which can go in favor of either scheme. However, we do not know in advance and under [...] Read more.

We provide a comparison between two schemes for solving equations on Banach space. A comparison between same convergence order schemes has been given using numerical examples which can go in favor of either scheme. However, we do not know in advance and under the same set of conditions which scheme has the largest ball of convergence, tighter error bounds or best information on the location of the solution. We present a technique that allows us to achieve this objective. Numerical examples are also given to further justify the theoretical results. Our technique can be used to compare other schemes of the same convergence order. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

9 pages, 311 KiB

Open AccessArticle

Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems

by Ioannis K. Argyros, Stepan Shakhno and Halyna Yarmola

Symmetry 2020, 12(6), 978; https://doi.org/10.3390/sym12060978 - 08 Jun 2020

Cited by 1 | Viewed by 1737

Abstract

A vital role in the dynamics of physical systems is played by symmetries. In fact, these studies require the solution for systems of equations on abstract spaces including on the finite-dimensional Euclidean, Hilbert, or Banach spaces. Methods of iterative nature are commonly used [...] Read more.

A vital role in the dynamics of physical systems is played by symmetries. In fact, these studies require the solution for systems of equations on abstract spaces including on the finite-dimensional Euclidean, Hilbert, or Banach spaces. Methods of iterative nature are commonly used to determinate the solution. In this article, such methods of higher convergence order are studied. In particular, we develop a two-step iterative method to solve large scale systems that does not require finding an inverse operator. Instead of the operator’s inverting, it uses a two-step Schultz approximation. The convergence is investigated using Lipschitz condition on the first-order derivatives. The cubic order of convergence is established and the results of the numerical experiment are given to determine the real benefits of the proposed method. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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14 pages, 642 KiB

Open AccessArticle

Fast Convergence Methods for Hyperbolic Systems of Balance Laws with Riemann Conditions

by Kamel Al-Khaled and Nid’a M. Rababah

Symmetry 2020, 12(5), 757; https://doi.org/10.3390/sym12050757 - 06 May 2020

Cited by 1 | Viewed by 1547

Abstract

In this paper, we develop an accurate technique via the use of the Adomian decomposition method (ADM) to solve analytically a

2 \times 2

systems of partial differential equation that represent balance laws of hyperbolic-elliptic type. We prove that the sequence of iteration [...] Read more.

In this paper, we develop an accurate technique via the use of the Adomian decomposition method (ADM) to solve analytically a

2 \times 2

systems of partial differential equation that represent balance laws of hyperbolic-elliptic type. We prove that the sequence of iteration obtained by ADM converges strongly to the exact solution by establishing a construction of fixed points. For comparison purposes, we also use the Sinc function methodology to establish a new procedure to solve numerically the same system. It is shown that approximation by Sinc function converges to the exact solution exponentially, also handles changes in type. A numerical example is presented to demonstrate the theoretical results. It is noted that the two methods show the symmetry in the approximate solution. The results obtained by both methods reveal that they are reliable and convenient for solving balance laws where the initial conditions are of the Riemann type. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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27 pages, 1239 KiB

Open AccessArticle

A Self-Adaptive Extra-Gradient Methods for a Family of Pseudomonotone Equilibrium Programming with Application in Different Classes of Variational Inequality Problems

by Habib ur Rehman, Poom Kumam, Ioannis K. Argyros, Nasser Aedh Alreshidi, Wiyada Kumam and Wachirapong Jirakitpuwapat

Symmetry 2020, 12(4), 523; https://doi.org/10.3390/sym12040523 - 02 Apr 2020

Cited by 23 | Viewed by 2207

Abstract

The main objective of this article is to propose a new method that would extend Popov’s extragradient method by changing two natural projections with two convex optimization problems. We also show the weak convergence of our designed method by taking mild assumptions on [...] Read more.

The main objective of this article is to propose a new method that would extend Popov’s extragradient method by changing two natural projections with two convex optimization problems. We also show the weak convergence of our designed method by taking mild assumptions on a cost bifunction. The method is evaluating only one value of the bifunction per iteration and it is uses an explicit formula for identifying the appropriate stepsize parameter for each iteration. The variable stepsize is going to be effective for enhancing iterative algorithm performance. The variable stepsize is updating for each iteration based on the previous iterations. After numerical examples, we conclude that the effect of the inertial term and variable stepsize has a significant improvement over the processing time and number of iterations. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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24 pages, 1336 KiB

Open AccessArticle

Inertial Extra-Gradient Method for Solving a Family of Strongly Pseudomonotone Equilibrium Problems in Real Hilbert Spaces with Application in Variational Inequality Problem

by Habib ur Rehman, Poom Kumam, Ioannis K. Argyros, Wejdan Deebani and Wiyada Kumam

Symmetry 2020, 12(4), 503; https://doi.org/10.3390/sym12040503 - 01 Apr 2020

Cited by 40 | Viewed by 2721

Abstract

In this paper, we propose a new method, which is set up by incorporating an inertial step with the extragradient method for solving a strongly pseudomonotone equilibrium problems. This method had to comply with a strongly pseudomonotone property and a certain Lipschitz-type condition [...] Read more.

In this paper, we propose a new method, which is set up by incorporating an inertial step with the extragradient method for solving a strongly pseudomonotone equilibrium problems. This method had to comply with a strongly pseudomonotone property and a certain Lipschitz-type condition of a bifunction. A strong convergence result is provided under some mild conditions, and an iterative sequence is accomplished without previous knowledge of the Lipschitz-type constants of a cost bifunction. A sufficient explanation is that the method operates with a slow-moving stepsize sequence that converges to zero and non-summable. For numerical explanations, we analyze a well-known equilibrium model to support our well-established convergence result, and we can see that the proposed method seems to have a significant consistent improvement over the performance of the existing methods. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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14 pages, 2145 KiB

Open AccessArticle

Convergence and Dynamics of a Higher-Order Method

by Alejandro Moysi, Ioannis K. Argyros, Samundra Regmi, Daniel González, Á. Alberto Magreñán and Juan Antonio Sicilia

Symmetry 2020, 12(3), 420; https://doi.org/10.3390/sym12030420 - 05 Mar 2020

Cited by 3 | Viewed by 1904

Abstract

Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That [...] Read more.

Solving problems in various disciplines such as biology, chemistry, economics, medicine, physics, and engineering, to mention a few, reduces to solving an equation. Its solution is one of the greatest challenges. It involves some iterative method generating a sequence approximating the solution. That is why, in this work, we analyze the convergence in a local form for an iterative method with a high order to find the solution of a nonlinear equation. We extend the applicability of previous results using only the first derivative that actually appears in the method. This is in contrast to either works using a derivative higher than one, or ones not in this method. Moreover, we consider the dynamics of some members of the family in order to see the existing differences between them. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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11 pages, 244 KiB

Open AccessArticle

Local Convergence of Solvers with Eighth Order Having Weak Conditions

by Ramandeep Behl and Ioannis K. Argyros

Symmetry 2020, 12(1), 70; https://doi.org/10.3390/sym12010070 - 02 Jan 2020

Viewed by 1576

Abstract

In particular, the problem of approximating a solution of an equation is of extreme importance in many disciplines, since numerous problems from diverse disciplines reduce to solving such equations. The solutions are found using iterative schemes since in general to find closed form [...] Read more.

In particular, the problem of approximating a solution of an equation is of extreme importance in many disciplines, since numerous problems from diverse disciplines reduce to solving such equations. The solutions are found using iterative schemes since in general to find closed form solution is not possible. That is why it is important to study convergence order of solvers. We extended the applicability of an eighth-order convergent solver for solving Banach space valued equations. Earlier considerations adopting suppositions up to the ninth Fŕechet-derivative, although higher than one derivatives are not appearing on these solvers. But, we only practiced supposition on Lipschitz constants and the first-order Fŕechet-derivative. Hence, we extended the applicability of these solvers and provided the computable convergence radii of them not given in the earlier works. We only showed improvements for a certain class of solvers. But, our technique can be used to extend the applicability of other solvers in the literature in a similar fashion. We used a variety of numerical problems to show that our results are applicable to solve nonlinear problems but not earlier ones. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

12 pages, 944 KiB

Open AccessArticle

Nonparametric Tensor Completion Based on Gradient Descent and Nonconvex Penalty

by Kai Xu and Zhi Xiong

Symmetry 2019, 11(12), 1512; https://doi.org/10.3390/sym11121512 - 12 Dec 2019

Viewed by 1974

Abstract

Existing tensor completion methods all require some hyperparameters. However, these hyperparameters determine the performance of each method, and it is difficult to tune them. In this paper, we propose a novel nonparametric tensor completion method, which formulates tensor completion as an unconstrained optimization [...] Read more.

Existing tensor completion methods all require some hyperparameters. However, these hyperparameters determine the performance of each method, and it is difficult to tune them. In this paper, we propose a novel nonparametric tensor completion method, which formulates tensor completion as an unconstrained optimization problem and designs an efficient iterative method to solve it. In each iteration, we not only calculate the missing entries by the aid of data correlation, but consider the low-rank of tensor and the convergence speed of iteration. Our iteration is based on the gradient descent method, and approximates the gradient descent direction with tensor matricization and singular value decomposition. Considering the symmetry of every dimension of a tensor, the optimal unfolding direction in each iteration may be different. So we select the optimal unfolding direction by scaled latent nuclear norm in each iteration. Moreover, we design formula for the iteration step-size based on the nonconvex penalty. During the iterative process, we store the tensor in sparsity and adopt the power method to compute the maximum singular value quickly. The experiments of image inpainting and link prediction show that our method is competitive with six state-of-the-art methods. Full article

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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