Research

14 pages, 818 KiB

Open AccessArticle

An Artificial Neural Network Approach for Solving Space Fractional Differential Equations

by Pingfei Dai and Xiangyu Yu

Symmetry 2022, 14(3), 535; https://doi.org/10.3390/sym14030535 - 06 Mar 2022

Cited by 7 | Viewed by 2707

The linear algebraic system generated by the discretization of fractional differential equations has asymmetry, and the numerical solution of this kind of problems is more complex than that of symmetric problems due to the nonlocality of fractional order operators. In this paper, we [...] Read more.

The linear algebraic system generated by the discretization of fractional differential equations has asymmetry, and the numerical solution of this kind of problems is more complex than that of symmetric problems due to the nonlocality of fractional order operators. In this paper, we propose the artificial neural network (ANN) algorithm to approximate the solutions of the fractional differential equations (FDEs). First, we apply truncated series expansion terms to replace unknown function in equations, then we use the neural network to get series coefficients, and the obtained series solution can make the norm value of loss function reach a satisfactory error. In the part of numerical experiments, the results verify that the proposed ANN algorithm can make the numerical results achieve high accuracy and good stability. Full article

(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)

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15 pages, 6261 KiB

Open AccessArticle

Image Encryption Based on Arnod Transform and Fractional Chaotic

by Chao Chen, Hongying Zhang and Bin Wu

Symmetry 2022, 14(1), 174; https://doi.org/10.3390/sym14010174 - 17 Jan 2022

Cited by 4 | Viewed by 1378

Abstract

An image encryption and decryption algorithm based on Arnod transform and fractional chaos is proposed in this work for solving the problem that the encrypted image is easily cracked and the content of the decrypted image is distorted. To begin with, the Arnold [...] Read more.

An image encryption and decryption algorithm based on Arnod transform and fractional chaos is proposed in this work for solving the problem that the encrypted image is easily cracked and the content of the decrypted image is distorted. To begin with, the Arnold transform is used to encrypt, so that the spatial confidence of the original image has been comprehensively disturbed. Secondly, the XOR involving the fractional order chaotic sequence is used to encrypt. The key sequence is dynamically generated to ensure the randomness and difference of key generation. When decryption is required, the first decryption is performed using the key and XOR. Then, the second decryption is carried out by using the inverse Arnold transform, and finally the decrypted image is obtained. Experimental results show that the improved algorithm has achieved a better performance in encryption and decryption. Full article

(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)

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19 pages, 1549 KiB

Open AccessArticle

A Dynamically Adjusted Subspace Gradient Method and Its Application in Image Restoration

by Jun Huo, Yuping Wu, Guoen Xia and Shengwei Yao

Symmetry 2021, 13(12), 2450; https://doi.org/10.3390/sym13122450 - 20 Dec 2021

Viewed by 1971

Abstract

In this paper, a new subspace gradient method is proposed in which the search direction is determined by solving an approximate quadratic model in which a simple symmetric matrix is used to estimate the Hessian matrix in a three-dimensional subspace. The obtained algorithm [...] Read more.

In this paper, a new subspace gradient method is proposed in which the search direction is determined by solving an approximate quadratic model in which a simple symmetric matrix is used to estimate the Hessian matrix in a three-dimensional subspace. The obtained algorithm has the ability to automatically adjust the search direction according to the feedback from experiments. Under some mild assumptions, we use the generalized line search with non-monotonicity to obtain remarkable results, which not only establishes the global convergence of the algorithm for general functions, but also R-linear convergence for uniformly convex functions is further proved. The numerical performance for both the traditional test functions and image restoration problems show that the proposed algorithm is efficient. Full article

(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)

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13 pages, 508 KiB

Open AccessArticle

A Proximal Algorithm with Convergence Guarantee for a Nonconvex Minimization Problem Based on Reproducing Kernel Hilbert Space

by Hong-Xia Dou and Liang-Jian Deng

Symmetry 2021, 13(12), 2393; https://doi.org/10.3390/sym13122393 - 12 Dec 2021

Cited by 1 | Viewed by 1801

Abstract

The underlying function in reproducing kernel Hilbert space (RKHS) may be degraded by outliers or deviations, resulting in a symmetry ill-posed problem. This paper proposes a nonconvex minimization model with

ℓ_{0}

-quasi norm based on RKHS to depict this degraded problem. The [...] Read more.

The underlying function in reproducing kernel Hilbert space (RKHS) may be degraded by outliers or deviations, resulting in a symmetry ill-posed problem. This paper proposes a nonconvex minimization model with

ℓ_{0}

-quasi norm based on RKHS to depict this degraded problem. The underlying function in RKHS can be represented by the linear combination of reproducing kernels and their coefficients. Thus, we turn to estimate the related coefficients in the nonconvex minimization problem. An efficient algorithm is designed to solve the given nonconvex problem by the mathematical program with equilibrium constraints (MPEC) and proximal-based strategy. We theoretically prove that the sequences generated by the designed algorithm converge to the nonconvex problem’s local optimal solutions. Numerical experiment also demonstrates the effectiveness of the proposed method. Full article

(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)

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17 pages, 1260 KiB

Open AccessArticle

Symplectic All-at-Once Method for Hamiltonian Systems

by Bei-Bei Zhu and Yong-Liang Zhao

Symmetry 2021, 13(10), 1930; https://doi.org/10.3390/sym13101930 - 14 Oct 2021

Cited by 1 | Viewed by 1374

Abstract

The all-at-once technique has attracted many researchers’ interest in recent years. In this paper, we combine this technique with a classical symplectic and symmetric method for solving Hamiltonian systems. The solutions at all time steps are obtained at one-shot. In order to reduce [...] Read more.

The all-at-once technique has attracted many researchers’ interest in recent years. In this paper, we combine this technique with a classical symplectic and symmetric method for solving Hamiltonian systems. The solutions at all time steps are obtained at one-shot. In order to reduce the computational cost of solving the all-at-once system, a fast algorithm is designed. Numerical experiments of Hamiltonian systems with degrees of freedom

n \leq 3

are provided to show that our method is more efficient than the classical symplectic method. Full article

(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)

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15 pages, 290 KiB

Open AccessArticle

Generalized Higher Order Preinvex Functions and Equilibrium-like Problems

by Jiangming Ma, Muhammad Aslam Noor and Khalida Inayat Noor

Symmetry 2021, 13(10), 1875; https://doi.org/10.3390/sym13101875 - 05 Oct 2021

Viewed by 1049

Abstract

Equilibrium problems and variational inequalities are connected to the symmetry concepts, which play important roles in many fields of sciences. Some new preinvex functions, which are called generalized preinvex functions, with the bifunction

ζ (., .)

and an arbitrary function [...] Read more.

Equilibrium problems and variational inequalities are connected to the symmetry concepts, which play important roles in many fields of sciences. Some new preinvex functions, which are called generalized preinvex functions, with the bifunction

ζ (., .)

and an arbitrary function k, are introduced and studied. Under the normed spaces, new parallelograms laws are taken as an application of the generalized preinvex functions. The equilibrium-like problems are represented as the minimum values of generalized preinvex functions under the

k ζ

-invex sets. Some new inertial methods are proposed and researched to solve the higher order directional equilibrium-like problem, Convergence criteria of the our methods is discussed, along with some unresolved issues. Full article

(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)

15 pages, 367 KiB

Open AccessArticle

A Preconditioned Variant of the Refined Arnoldi Method for Computing PageRank Eigenvectors

by Zhao-Li Shen, Hao Yang, Bruno Carpentieri, Xian-Ming Gu and Chun Wen

Symmetry 2021, 13(8), 1327; https://doi.org/10.3390/sym13081327 - 23 Jul 2021

Cited by 3 | Viewed by 1406

Abstract

The PageRank model computes the stationary distribution of a Markov random walk on the linking structure of a network, and it uses the values within to represent the importance or centrality of each node. This model is first proposed by Google for ranking [...] Read more.

The PageRank model computes the stationary distribution of a Markov random walk on the linking structure of a network, and it uses the values within to represent the importance or centrality of each node. This model is first proposed by Google for ranking web pages, then it is widely applied as a centrality measure for networks arising in various fields such as in chemistry, bioinformatics, neuroscience and social networks. For example, it can measure the node centralities of the gene-gene annotation network to evaluate the relevance of each gene with a certain disease. The networks in some fields including bioinformatics are undirected, thus the corresponding adjacency matrices are symmetry. Mathematically, the PageRank model can be stated as finding the unit positive eigenvector corresponding to the largest eigenvalue of a transition matrix built upon the linking structure. With rapid development of science and technology, the networks in real applications become larger and larger, thus the PageRank model always desires numerical algorithms with reduced algorithmic or memory complexity. In this paper, we propose a novel preconditioning approach for solving the PageRank model. This approach transforms the original PageRank eigen-problem into a new one that is more amenable to solve. We then present a preconditioned version of the refined Arnoldi method for solving this model. We demonstrate theoretically that the preconditioned Arnoldi method has higher execution efficiency and parallelism than the refined Arnoldi method. In plenty of numerical experiments, this preconditioned method exhibits noticeably faster convergence speed over its standard counterpart, especially for difficult cases with large damping factors. Besides, this superiority maintains when this technique is applied to other variants of the refined Arnoldi method. Overall, the proposed technique can give the PageRank model a faster solving process, and this will possibly improve the efficiency of researches, engineering projects and services where this model is applied. Full article

(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)

21 pages, 1809 KiB

Open AccessArticle

A Special Multigrid Strategy on Non-Uniform Grids for Solving 3D Convection–Diffusion Problems with Boundary/Interior Layers

by Tianlong Ma, Lin Zhang, Fujun Cao and Yongbin Ge

Symmetry 2021, 13(7), 1123; https://doi.org/10.3390/sym13071123 - 24 Jun 2021

Cited by 1 | Viewed by 1824

Abstract

Boundary or interior layer problems of high-dimensional convection–diffusion equations have distinct asymmetry. Consequently, computational grid distributions and linear algebraic systems arising from finite difference schemes for them are also asymmetric. Numerical solutions for these kinds of problems are more complicated than those symmetric [...] Read more.

Boundary or interior layer problems of high-dimensional convection–diffusion equations have distinct asymmetry. Consequently, computational grid distributions and linear algebraic systems arising from finite difference schemes for them are also asymmetric. Numerical solutions for these kinds of problems are more complicated than those symmetric problems. In this paper, we extended our previous work on the partial semi-coarsening multigrid method combined with the high-order compact (HOC) difference scheme for solving the two-dimensional (2D) convection–diffusion problems on non-uniform grids to the three-dimensional (3D) cases. The main merit of the present method is that the multigrid method on non-uniform grids can be performed with a different number of grids in different coordinate axes, which is more efficient than the multigrid method on non-uniform grids with the same number of grids in different coordinate axes. Numerical experiments are carried out to validate the accuracy and efficiency of the present method. It is shown that, without losing the high precision, the present method is very effective to reduce computing cost by cutting down the number of grids in the direction(s) which does/do not contain boundary or interior layer(s). Full article

(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)

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13 pages, 7633 KiB

Open AccessArticle

Anderson Acceleration of the Arnoldi-Inout Method for Computing PageRank

by Xia Tang, Chun Wen, Xian-Ming Gu and Zhao-Li Shen

Symmetry 2021, 13(4), 636; https://doi.org/10.3390/sym13040636 - 10 Apr 2021

Cited by 1 | Viewed by 1552

Abstract

Anderson(

m_{0}

) extrapolation, an accelerator to a fixed-point iteration, stores

m_{0} + 1

prior evaluations of the fixed-point iteration and computes a linear combination of those evaluations as a new iteration. The computational cost of the Anderson(

m_{0}

) [...] Read more.

Anderson(

m_{0}

) extrapolation, an accelerator to a fixed-point iteration, stores

m_{0} + 1

prior evaluations of the fixed-point iteration and computes a linear combination of those evaluations as a new iteration. The computational cost of the Anderson(

m_{0}

) acceleration becomes expensive with the parameter

m_{0}

increasing, thus

m_{0}

is a common choice in most practice. In this paper, with the aim of improving the computations of PageRank problems, a new method was developed by applying Anderson(1) extrapolation at periodic intervals within the Arnoldi-Inout method. The new method is called the AIOA method. Convergence analysis of the AIOA method is discussed in detail. Numerical results on several PageRank problems are presented to illustrate the effectiveness of our proposed method. Full article

(This article belongs to the Special Issue PDE, Optimization Modeling and Symmetry in Multi-Dimensional Data and Low-Level Vision Tasks)

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