Pub Date : 2022-06-01DOI: 10.4208/nmtma.oa-2021-0131
Maryam Bashirizadeh, M. Hajarian
Abstract. Linear complementarity problems have drawn considerable attention in recent years due to their wide applications. In this article, we introduce the two-step two-sweep modulus-based matrix splitting (TSTM) iteration method and two-sweep modulus-based matrix splitting type II (TM II) iteration method which are a combination of the two-step modulus-based method and the two-sweep modulus-based method, as two more effective ways to solve the linear complementarity problems. The convergence behavior of these methods is discussed when the system matrix is either a positive-definite or an H+-matrix. Finally, numerical experiments are given to show the efficiency of our proposed methods. AMS subject classifications: 65F10, 65F15
{"title":"Two-Step Two-Sweep Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems","authors":"Maryam Bashirizadeh, M. Hajarian","doi":"10.4208/nmtma.oa-2021-0131","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2021-0131","url":null,"abstract":"Abstract. Linear complementarity problems have drawn considerable attention in recent years due to their wide applications. In this article, we introduce the two-step two-sweep modulus-based matrix splitting (TSTM) iteration method and two-sweep modulus-based matrix splitting type II (TM II) iteration method which are a combination of the two-step modulus-based method and the two-sweep modulus-based method, as two more effective ways to solve the linear complementarity problems. The convergence behavior of these methods is discussed when the system matrix is either a positive-definite or an H+-matrix. Finally, numerical experiments are given to show the efficiency of our proposed methods. AMS subject classifications: 65F10, 65F15","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41979920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-01DOI: 10.4208/nmtma.oa-2021-0126
Fang Geng, Li-Xiao Duan null, Guo‐Feng Zhang
The Kaczmarz algorithm is a common iterative method for solving linear systems. As an effective variant of Kaczmarz algorithm, the greedy Kaczmarz algorithm utilizes the greedy selection strategy. The two-subspace projection method performs an optimal intermediate projection in each iteration. In this paper, we introduce a new greedy Kaczmarz method, which give full play to the advantages of the two improved Kaczmarz algorithms, so that the generated iterative sequence can exponentially converge to the optimal solution. The theoretical analysis reveals that our algorithm has a smaller convergence factor than the greedy Kaczmarz method. Experimental results confirm that our new algorithm is more effective than the greedy Kaczmarz method for coherent systems and the two-subspace projection method for appropriate scale systems. AMS subject classifications: 15A06, 65F10, 65F20, 65F25, 65F50
{"title":"Greedy Kaczmarz Algorithm Using Optimal Intermediate Projection Technique for Coherent Linear Systems","authors":"Fang Geng, Li-Xiao Duan null, Guo‐Feng Zhang","doi":"10.4208/nmtma.oa-2021-0126","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2021-0126","url":null,"abstract":"The Kaczmarz algorithm is a common iterative method for solving linear systems. As an effective variant of Kaczmarz algorithm, the greedy Kaczmarz algorithm utilizes the greedy selection strategy. The two-subspace projection method performs an optimal intermediate projection in each iteration. In this paper, we introduce a new greedy Kaczmarz method, which give full play to the advantages of the two improved Kaczmarz algorithms, so that the generated iterative sequence can exponentially converge to the optimal solution. The theoretical analysis reveals that our algorithm has a smaller convergence factor than the greedy Kaczmarz method. Experimental results confirm that our new algorithm is more effective than the greedy Kaczmarz method for coherent systems and the two-subspace projection method for appropriate scale systems. AMS subject classifications: 15A06, 65F10, 65F20, 65F25, 65F50","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47291932","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-01DOI: 10.4208/nmtma.oa-2022-0012
Yu-ling Guo, Jianguo Huang
{"title":"A Posteriori Error Analysis of a $P_2$-CDG Space-Time Finite Element Method for the Wave Equation","authors":"Yu-ling Guo, Jianguo Huang","doi":"10.4208/nmtma.oa-2022-0012","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2022-0012","url":null,"abstract":"","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48014385","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-01DOI: 10.4208/nmtma.oa-2022-0007s
Hui-yuan Li, Ruiqing Liu null, Lilian Wang
In this paper, we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains. We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian. As a result, the “stiffness” matrix can be fast computed and assembled via the four-point stable recursive algorithm with O(N2) arithmetic operations. Moreover, the singular factor in a typical kernel function can be fully absorbed by the basis. With the aid of Fourier analysis, we can prove the convergence of the scheme. We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions. We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.
{"title":"Efficient Hermite Spectral-Galerkin Methods for Nonlocal Diffusion Equations in Unbounded Domains","authors":"Hui-yuan Li, Ruiqing Liu null, Lilian Wang","doi":"10.4208/nmtma.oa-2022-0007s","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2022-0007s","url":null,"abstract":"In this paper, we develop an efficient Hermite spectral-Galerkin method for nonlocal diffusion equations in unbounded domains. We show that the use of the Hermite basis can de-convolute the troublesome convolutional operations involved in the nonlocal Laplacian. As a result, the “stiffness” matrix can be fast computed and assembled via the four-point stable recursive algorithm with O(N2) arithmetic operations. Moreover, the singular factor in a typical kernel function can be fully absorbed by the basis. With the aid of Fourier analysis, we can prove the convergence of the scheme. We demonstrate that the recursive computation of the entries of the stiffness matrix can be extended to the two-dimensional nonlocal Laplacian using the isotropic Hermite functions as basis functions. We provide ample numerical results to illustrate the accuracy and efficiency of the proposed algorithms.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41644829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-01DOI: 10.4208/nmtma.oa-2021-0129
M. Esmaeilzadeh, R. Barron
{"title":"Numerical Solution of Partial Differential Equations in Arbitrary Shaped Domains Using Cartesian Cut-Stencil Finite Difference Method. Part II: Higher-Order Schemes","authors":"M. Esmaeilzadeh, R. Barron","doi":"10.4208/nmtma.oa-2021-0129","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2021-0129","url":null,"abstract":"","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":"50 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41310514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-01DOI: 10.4208/nmtma.oa-2021-0134
Yunsong Gan, Jie Zhang null, Huibin Chang
. In this paper, we propose new algorithms for multiplicative noise removal based on the Aubert-Aujol (AA) model. By introducing a constraint from the forward model with an auxiliary variable for the noise, the NEMA (short for Noise Estimate based Multiplicative noise removal by alternating direction method of multipliers (ADMM)) is firstly given. To further reduce the computational cost, an additional proximal term is considered for the subproblem with regard to the original variable, the NEMA f (short for a variant of NEMA with fully splitting form) is further proposed. We conduct numerous experiments to show the convergence and perfor-mance of the proposed algorithms. Namely, the restoration results by the proposed algorithms are better in terms of SNRs for image deblurring than other compared methods including two popular algorithms for AA model and three algorithms of its convex variants.
． 本文提出了基于Aubert-Aujol (AA)模型的乘性噪声去除新算法。通过引入正演模型的约束和噪声的辅助变量，首先给出了基于噪声估计的乘法器交替方向乘法去噪方法(NEMA)。为了进一步降低计算成本，考虑了子问题相对于原始变量的另一个近端项，进一步提出了NEMA f (NEMA的一种变体，具有完全分裂形式)。我们进行了大量的实验来证明所提出算法的收敛性和性能。即，本文算法的图像去模糊恢复结果在信噪比方面优于其他比较方法，包括两种常用的AA模型算法及其凸变体的三种算法。
{"title":"New Splitting Algorithms for Multiplicative Noise Removal Based on Aubert-Aujol Model","authors":"Yunsong Gan, Jie Zhang null, Huibin Chang","doi":"10.4208/nmtma.oa-2021-0134","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2021-0134","url":null,"abstract":". In this paper, we propose new algorithms for multiplicative noise removal based on the Aubert-Aujol (AA) model. By introducing a constraint from the forward model with an auxiliary variable for the noise, the NEMA (short for Noise Estimate based Multiplicative noise removal by alternating direction method of multipliers (ADMM)) is firstly given. To further reduce the computational cost, an additional proximal term is considered for the subproblem with regard to the original variable, the NEMA f (short for a variant of NEMA with fully splitting form) is further proposed. We conduct numerous experiments to show the convergence and perfor-mance of the proposed algorithms. Namely, the restoration results by the proposed algorithms are better in terms of SNRs for image deblurring than other compared methods including two popular algorithms for AA model and three algorithms of its convex variants.","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48324182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-01DOI: 10.4208/nmtma.oa-2021-0067
Wang Kong null, Zhongyi Huang
In this paper, we study the numerical solution of the time-fractional telegraph equation on the unbounded domain. We first introduce the artificial boundaries Γ± to get a finite computational domain. On the artificial boundaries Γ±, we use the Laplace transform to construct the exact artificial boundary conditions (ABCs) to reduce the original problem to an initial-boundary value problem on a bounded domain. In addition, we propose a finite difference scheme based on the L1−2 formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem. In order to reduce the effect of unsmoothness of the solution at the initial moment, we use a fine mesh and low-order interpolation to discretize the solution near t = 0. Finally, some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results. AMS subject classifications: 65M10, 78A48
{"title":"Artificial Boundary Conditions for Time-Fractional Telegraph Equation","authors":"Wang Kong null, Zhongyi Huang","doi":"10.4208/nmtma.oa-2021-0067","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2021-0067","url":null,"abstract":"In this paper, we study the numerical solution of the time-fractional telegraph equation on the unbounded domain. We first introduce the artificial boundaries Γ± to get a finite computational domain. On the artificial boundaries Γ±, we use the Laplace transform to construct the exact artificial boundary conditions (ABCs) to reduce the original problem to an initial-boundary value problem on a bounded domain. In addition, we propose a finite difference scheme based on the L1−2 formule for the Caputo fractional derivative in time direction and the central difference scheme for the spatial directional derivative to solve the reduced problem. In order to reduce the effect of unsmoothness of the solution at the initial moment, we use a fine mesh and low-order interpolation to discretize the solution near t = 0. Finally, some numerical results show the efficiency and reliability of the ABCs and validate our theoretical results. AMS subject classifications: 65M10, 78A48","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46912681","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-01DOI: 10.4208/nmtma.oa-2021-0099
Xiakai Wang, Zhongyi Huang null, Wei Zhu
In this paper, we propose a novel PDE-based model for the multi-phase segmentation problem by using a complex version of Cahn-Hilliard equations. Specifically, we modify the original complex system of Cahn-Hilliard equations by adding the mean curvature term and the fitting term to the evolution of its real part, which helps to render a piecewisely constant function at the steady state. By applying the K-means method to this function, one could achieve the desired multiphase segmentation. To solve the proposed system of equations, a semi-implicit finite difference scheme is employed. Numerical experiments are presented to demonstrate the feasibility of the proposed model and compare our model with other related ones. AMS subject classifications: 68U10, 65K10, 65N06
{"title":"Multi-Phase Segmentation Using Modified Complex Cahn-Hilliard Equations","authors":"Xiakai Wang, Zhongyi Huang null, Wei Zhu","doi":"10.4208/nmtma.oa-2021-0099","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2021-0099","url":null,"abstract":"In this paper, we propose a novel PDE-based model for the multi-phase segmentation problem by using a complex version of Cahn-Hilliard equations. Specifically, we modify the original complex system of Cahn-Hilliard equations by adding the mean curvature term and the fitting term to the evolution of its real part, which helps to render a piecewisely constant function at the steady state. By applying the K-means method to this function, one could achieve the desired multiphase segmentation. To solve the proposed system of equations, a semi-implicit finite difference scheme is employed. Numerical experiments are presented to demonstrate the feasibility of the proposed model and compare our model with other related ones. AMS subject classifications: 68U10, 65K10, 65N06","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43415111","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-01DOI: 10.4208/nmtma.oa-2021-0097
G. Sun, S. Gan, H. null, Zaijiu Shang
Symmetric and symplectic methods are classical notions in the theory of numerical methods for solving ordinary differential equations. They can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase space. This article is mainly concerned with the symmetric-adjoint and symplectic-adjoint Runge-Kutta methods as well as their applications. It is a continuation and an extension of the study in [14], where the authors introduced the notion of symplectic-adjoint method of a Runge-Kutta method and provided a simple way to construct symplectic partitioned Runge-Kutta methods via the symplectic-adjoint method. In this paper, we provide a more comprehensive and systematic study on the properties of the symmetric-adjoint and symplecticadjoint Runge-Kutta methods. These properties reveal some intrinsic connections among some classical Runge-Kutta methods. Moreover, those properties can be used to significantly simplify the order conditions and hence can be applied to the construction of high-order Runge-Kutta methods. As a specific and illustrating application, we construct a novel class of explicit Runge-Kutta methods of stage 6 and order 5. Finally, with the help of symplectic-adjoint method, we thereby obtain a new simple proof of the nonexistence of explicit Runge-Kutta method with stage 5 and order 5. AMS subject classifications: 65L06, 37M15, 65P10
{"title":"Symmetric-Adjoint and Symplectic-Adjoint Runge-Kutta Methods and Their Applications","authors":"G. Sun, S. Gan, H. null, Zaijiu Shang","doi":"10.4208/nmtma.oa-2021-0097","DOIUrl":"https://doi.org/10.4208/nmtma.oa-2021-0097","url":null,"abstract":"Symmetric and symplectic methods are classical notions in the theory of numerical methods for solving ordinary differential equations. They can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase space. This article is mainly concerned with the symmetric-adjoint and symplectic-adjoint Runge-Kutta methods as well as their applications. It is a continuation and an extension of the study in [14], where the authors introduced the notion of symplectic-adjoint method of a Runge-Kutta method and provided a simple way to construct symplectic partitioned Runge-Kutta methods via the symplectic-adjoint method. In this paper, we provide a more comprehensive and systematic study on the properties of the symmetric-adjoint and symplecticadjoint Runge-Kutta methods. These properties reveal some intrinsic connections among some classical Runge-Kutta methods. Moreover, those properties can be used to significantly simplify the order conditions and hence can be applied to the construction of high-order Runge-Kutta methods. As a specific and illustrating application, we construct a novel class of explicit Runge-Kutta methods of stage 6 and order 5. Finally, with the help of symplectic-adjoint method, we thereby obtain a new simple proof of the nonexistence of explicit Runge-Kutta method with stage 5 and order 5. AMS subject classifications: 65L06, 37M15, 65P10","PeriodicalId":51146,"journal":{"name":"Numerical Mathematics-Theory Methods and Applications","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42447380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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