622 results
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2. Some remarks on the paper “Strong convergence of a self-adaptive method for the spilt feasibility problem”
- Author
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Zhou, Haiyun, primary and Wang, Peiyuan, additional
- Published
- 2014
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3. Waveform relaxation of partial differential equations
- Author
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Zhen Miao and Yao-Lin Jiang
- Subjects
Partial differential equation ,Applied Mathematics ,Numerical analysis ,Short paper ,Relaxation (iterative method) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Theory of computation ,Convergence (routing) ,Applied mathematics ,Waveform ,0101 mathematics ,Energy (signal processing) ,Mathematics - Abstract
This short paper concludes a general waveform relaxation (WR) method at the PDE level for semi-linear reaction-diffusion equations. For the case of multiple coupled PDE(s), new Jacobi WR and Gauss-Seidel WR are provided to accelerate the convergence result of classical WR. The convergence conditions are proved based on energy estimate. Numerical experiments are demonstrated with several WR methods in parallel to verify the effectiveness of the general WR method.
- Published
- 2018
4. Complex moment-based methods for differential eigenvalue problems
- Author
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Imakura, Akira, Morikuni, Keiichi, and Takayasu, Akitoshi
- Subjects
Applied Mathematics ,FOS: Mathematics ,Mathematics - Numerical Analysis ,Numerical Analysis (math.NA) - Abstract
This paper considers computing partial eigenpairs of differential eigenvalue problems (DEPs) such that eigenvalues are in a certain region on the complex plane. Recently, based on a "solve-then-discretize" paradigm, an operator analogue of the FEAST method has been proposed for DEPs without discretization of the coefficient operators. Compared to conventional "discretize-then-solve" approaches that discretize the operators and solve the resulting matrix problem, the operator analogue of FEAST exhibits much higher accuracy; however, it involves solving a large number of ordinary differential equations (ODEs). In this paper, to reduce the computational costs, we propose operation analogues of Sakurai-Sugiura-type complex moment-based eigensolvers for DEPs using higher-order complex moments and analyze the error bound of the proposed methods. We show that the number of ODEs to be solved can be reduced by a factor of the degree of complex moments without degrading accuracy, which is verified by numerical results. Numerical results demonstrate that the proposed methods are over five times faster compared with the operator analogue of FEAST for several DEPs while maintaining almost the same high accuracy. This study is expected to promote the "solve-then-discretize" paradigm for solving DEPs and contribute to faster and more accurate solutions in real-world applications., Comment: 26 pages, 9 figures
- Published
- 2022
5. A class of C2 quasi-interpolating splines free of Gibbs phenomenon
- Author
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Sergio Amat, David Levin, Juan Ruiz-Álvarez, Juan C. Trillo, Dionisio F. Yáñez, Universidad Politécnica de Cartagena, and Universidad de Valencia
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Splines ,Computer aided design (modeling of curves) ,12 Matemáticas ,C2 regularity ,Applied Mathematics ,MathematicsofComputing_NUMERICALANALYSIS ,Matemática Aplicada ,Adaption to discontinuities ,Quasi-interpolation ,ComputingMethodologies_COMPUTERGRAPHICS - Abstract
In many applications, it is useful to use piecewise polynomials that satisfy certain regularity conditions at the joint points. Cubic spline functions emerge as good candidates having C2 regularity. On the other hand, if the data points present discontinuities, the classical spline approximations produce Gibbs oscillations. In a recent paper, we have introduced a new nonlinear spline approximation avoiding the presence of these oscillations. Unfortunately, this new reconstruction loses the C2 regularity. This paper introduces a new nonlinear spline that preserves the regularity at all the joint points except at the end points of an interval containing a discontinuity, and that avoids the Gibbs oscillations. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was funded by the Programa de Apoyo a la investigación de la fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia 20928/PI/18, by the national research project MMTM2015-64382-P and PID2019-108336GB-I00 (MINECO/FEDER), by grant MTM2017-83942 funded by Spanish MINECO and by grant PID2020-117211GB-I00 funded by MCIN/AEI/10.13039/501100011033.
- Published
- 2022
6. Optimal convergence of three iterative methods based on nonconforming finite element discretization for 2D/3D MHD equations
- Author
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Haiyan Su, Zhilin Li, and Jiali Xu
- Subjects
Piecewise linear function ,Nonlinear system ,Discretization ,Iterative method ,Applied Mathematics ,Numerical analysis ,Convergence (routing) ,Applied mathematics ,Stability (probability) ,Finite element method ,Mathematics::Numerical Analysis ,Mathematics - Abstract
The main purpose of this paper is to analyze nonconforming iterative finite element methods for 2D/3D stationary incompressible magneto-hydrodynamics equations. First, the Crouzeix-Raviart–type finite element is used to approximate the velocity and the conforming piecewise linear element P1 is used for the pressure. Since the finite element method for the velocity field and the pressure is unstable, a simple locally stabilization term is added to satisfy the weak inf-sup condition. Then, the well-posedness and the optimal error estimates of the continuous and discrete problems are analyzed with the nonlinear terms being iteratively updated. Three effective iterative methods are proposed and their stability and convergence analyses are carried out. Finally, the theoretical analysis presented in this paper is verified by numerical experiments.
- Published
- 2021
7. On the best achievable quality of limit points of augmented Lagrangian schemes
- Author
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Gabriel Haeser, Roberto Andreani, Leonardo D. Secchin, Alberto Ramos, and Leonardo M. Mito
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Constraint (information theory) ,Mathematical optimization ,Augmented Lagrangian method ,Applied Mathematics ,Numerical analysis ,media_common.quotation_subject ,Theory of computation ,Convergence (routing) ,Limit point ,Quality (business) ,Algebra over a field ,Mathematics ,media_common - Abstract
The optimization literature is vast in papers dealing with improvements on the global convergence of augmented Lagrangian schemes. Usually, the results are based on weak constraint qualifications, or, more recently, on sequential optimality conditions obtained via penalization techniques. In this paper, we propose a somewhat different approach, in the sense that the algorithm itself is used in order to formulate a new optimality condition satisfied by its feasible limit points. With this tool at hand, we present several new properties and insights on limit points of augmented Lagrangian schemes, in particular, characterizing the strongest possible global convergence result for the safeguarded augmented Lagrangian method.
- Published
- 2021
8. A Tseng extragradient method for solving variational inequality problems in Banach spaces
- Author
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H. A. Abass, A. A. Mebawondu, Olawale Kazeem Oyewole, and K. O. Aremu
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Sequence ,Applied Mathematics ,Numerical analysis ,Variational inequality ,Convergence (routing) ,Banach space ,Applied mathematics ,Lipschitz continuity ,Constant (mathematics) ,Projection (linear algebra) ,Mathematics - Abstract
This paper presents an inertial Tseng extragradient method for approximating a solution of the variational inequality problem. The proposed method uses a single projection onto a half space which can be easily evaluated. The method considered in this paper does not require the knowledge of the Lipschitz constant as it uses variable stepsizes from step to step which are updated over each iteration by a simple calculation. We prove a strong convergence theorem of the sequence generated by this method to a solution of the variational inequality problem in the framework of a 2-uniformly convex Banach space which is also uniformly smooth. Furthermore, we report some numerical experiments to illustrate the performance of this method. Our result extends and unifies corresponding results in this direction in the literature.
- Published
- 2021
9. A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues
- Author
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Sven-Erik Ekström and P. Vassalos
- Subjects
Beräkningsmatematik ,Applied Mathematics ,010102 general mathematics ,Generating function ,Order (ring theory) ,Asymptotic expansion ,Spectral analysis ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Function (mathematics) ,Type (model theory) ,01 natural sciences ,Toeplitz matrix ,Combinatorics ,Computational Mathematics ,Matrix (mathematics) ,Toeplitz matrices ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Structured matrices ,Eigenvalues and eigenvectors ,Mathematics - Abstract
It is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.
- Published
- 2021
10. Stability analysis of the method of fundamental solutions with smooth closed pseudo-boundaries for Laplace’s equation: better pseudo-boundaries
- Author
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Li-Ping Zhang, Zi-Cai Li, Ming-Gong Lee, and Hung-Tsai Huang
- Subjects
Laplace's equation ,Polynomial ,Laplace transform ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Bounded function ,Applied mathematics ,Method of fundamental solutions ,0101 mathematics ,Condition number ,Circulant matrix ,Mathematics - Abstract
Consider Laplace’s equation in a bounded simply-connected domain S, and use the method of fundamental solutions (MFS). The error and stability analysis is made for circular/elliptic pseudo-boundaries in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020), and polynomial convergence rates and exponential growth rates of the condition number (Cond) are obtained. General pseudo-boundaries are suggested for more complicated solution domains in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020, Section 5). Since the ill-conditioning is severe, the success in computation by the MFS mainly depends on stability. This paper is devoted to stability analysis for smooth closed pseudo-boundaries of source nodes. Bounds of the Cond are derived, and exponential growth rates are also obtained. This paper is the first time to explore stability analysis of the MFS for non-circular/non-elliptic pseudo-boundaries. Circulant matrices are often employed for stability analysis of the MFS; but the stability analysis in this paper is explored based on new techniques without using circulant matrices as in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020). To pursue better pseudo-boundaries, the sensitivity index is proposed from growth/convergence rates of stability via accuracy. Better pseudo-boundaries in the MFS can be found by trial computations, to develop the study in Dou et al. (J. Comp. Appl. Math. 377:112861, 2020) for the selection of pseudo-boundaries. For highly smooth and singular solutions, better pseudo-boundaries are different; an analysis of the sensitivity index is explored. Circular/elliptic pseudo-boundaries are optimal for highly smooth solutions, but not for singular solutions. In this paper, amoeba-like domains are chosen in computation. Several useful types of pseudo-boundaries are developed and their algorithms are simple without using nonlinear solutions. For singular solutions, numerical comparisons are made for different pseudo-boundaries via the sensitivity index.
- Published
- 2021
11. Numerical study on Moore-Penrose inverse of tensors via Einstein product
- Author
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Baohua Huang
- Subjects
Applied Mathematics ,Numerical analysis ,Inverse ,Space (mathematics) ,General Relativity and Quantum Cosmology ,symbols.namesake ,Product (mathematics) ,Conjugate gradient method ,symbols ,Tensor ,Einstein ,Moore–Penrose pseudoinverse ,Mathematical physics ,Mathematics - Abstract
The notation of Moore-Penrose inverse of matrices has been extended from matrix space to even-order tensor space with Einstein product. In this paper, we give the numerical study on the Moore-Penrose inverse of tensors via the Einstein product. More precisely, we transform the calculation of Moore-Penrose inverse of tensors via the Einstein product into solving a class of tensor equations via the Einstein product. Then, by means of the conjugate gradient method, we obtain the approximate Moore-Penrose inverse of tensors via the Einstein product. Finally, we report some numerical examples to show the efficiency of the proposed methods and testify the conclusion suggested in this paper.
- Published
- 2021
12. Weak Galerkin finite element methods with or without stabilizers
- Author
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Xiaoshen Wang, Xiu Ye, and Shangyou Zhang
- Subjects
010101 applied mathematics ,Applied Mathematics ,Numerical analysis ,Convergence (routing) ,Theory of computation ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,Algebra over a field ,Galerkin method ,01 natural sciences ,Finite element method ,Mathematics - Abstract
The purpose of this paper is to investigate the connections between the weak Galerkin (WG) methods with and without stabilizers. The choices of stabilizers directly affect the convergence rates of the corresponding WG methods in general. However, we observed that the convergence rates are independent of the choices of stabilizers for these WG elements with stabilizers being optional. In this paper, we will verify such phenomena theoretically as well as numerically.
- Published
- 2021
13. Highly efficient schemes for time-fractional Allen-Cahn equation using extended SAV approach
- Author
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Chuanju Xu, Hongyi Zhu, Dianming Hou, Institut de Mécanique et d'Ingénierie (I2M), Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Arts et Métiers Sciences et Technologies, and HESAM Université (HESAM)-HESAM Université (HESAM)
- Subjects
Discretization ,Applied Mathematics ,Numerical analysis ,Scalar (physics) ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,[SPI.MAT]Engineering Sciences [physics]/Materials ,010101 applied mathematics ,Nonlinear system ,Theory of computation ,FOS: Mathematics ,Order (group theory) ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Allen–Cahn equation ,Mathematics - Abstract
In this paper, we propose and analyze high-order efficient schemes for the time-fractional Allen-Cahn equation. The proposed schemes are based on the L1 discretization for the time-fractional derivative and the extended scalar auxiliary variable (SAV) approach developed very recently to deal with the nonlinear terms in the equation. The main contributions of the paper consist of (1) constructing first- and higher order unconditionally stable schemes for different mesh types, and proving the unconditional stability of the constructed schemes for the uniform mesh; (2) carrying out numerical experiments to verify the efficiency of the schemes and to investigate the coarsening dynamics governed by the time-fractional Allen-Cahn equation. In particular, the influence of the fractional order on the coarsening behavior is carefully examined. Our numerical evidence shows that the proposed schemes are more robust than the existing methods, and their efficiency is less restricted to particular forms of the nonlinear potentials.
- Published
- 2021
14. Centrality measures for node-weighted networks via line graphs and the matrix exponential
- Author
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Mona Matar, Lothar Reichel, and Omar De la Cruz Cabrera
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Discrete mathematics ,Applied Mathematics ,Node (networking) ,Directed graph ,law.invention ,law ,Matrix function ,Theory of computation ,Line graph ,Adjacency matrix ,Matrix exponential ,Centrality ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
This paper is concerned with the identification of important nodes in node-weighted graphs by applying matrix functions, in particular the matrix exponential. Many tools that use an adjacency matrix for a graph have been developed to study the importance of the nodes in unweighted or edge-weighted networks. However, adjacency matrices for node-weighted graphs have not received much attention. The present paper proposes using a line graph associated with a node-weighted graph to construct an edge-weighted graph that can be analyzed with available methods. Both undirected and directed graphs with positive node weights are considered. We show that when the weight of a node increases, the importance of this node in the graph increases as well, provided that the adjacency matrix is suitably scaled. Applications to real-life problems are presented.
- Published
- 2021
15. Tensor extrapolation methods with applications
- Author
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Rachid Sadaka, Khalide Jbilou, Fatemeh Panjeh Ali Beik, and A. El Ichi
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Sequence ,Applied Mathematics ,Numerical analysis ,Theory of computation ,Singular value decomposition ,Extrapolation ,Applied mathematics ,Tensor ,Algebra over a field ,Mathematics ,Matrix polynomial - Abstract
In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for tensors corresponding to T-product. Furthermore, we discuss on the solution of least-squares problem associated with a tensor equation using Tensor Singular Value Decomposition (TSVD). Motivated by the effectiveness of some proposed vector extrapolation methods in earlier papers, we describe how an extrapolation technique can be also implemented on the sequence of tensors produced by truncated TSVD (TTSVD) for solving possibly ill-posed tensor equations.
- Published
- 2020
16. A nonnegativity preserving algorithm for multilinear systems with nonsingular ${\mathcal M}$-tensors
- Author
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Hongjin He, Guanglu Zhou, Chen Ling, and Xueli Bai
- Subjects
Sequence ,Multilinear map ,Applied Mathematics ,Numerical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,System of linear equations ,01 natural sciences ,law.invention ,010101 applied mathematics ,Invertible matrix ,law ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Theory of computation ,Tensor ,0101 mathematics ,Algorithm ,Mathematics ,Numerical partial differential equations - Abstract
This paper addresses multilinear systems of equations which arise in various applications such as data mining and numerical partial differential equations. When the multilinear system under consideration involves a nonsingular ${\mathscr{M}}$ -tensor and a nonnegative right-hand side vector, it may have multiple nonnegative solutions. In this paper, we propose an algorithm which can always preserve the nonnegativity of solutions. Theoretically, we show that the sequence generated by the proposed algorithm is a nonnegative componentwise nonincreasing sequence and converges to a nonnegative solution of the system. Numerical results further support the novelty of the proposed method.
- Published
- 2020
17. On the split common fixed point problem for strict quasi-ϕ-pseudocontractive mappings in Banach spaces
- Author
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Xindong Liu, Zili Chen, and Jinxing Liu
- Subjects
Sequence ,Pure mathematics ,Iterative method ,Applied Mathematics ,Numerical analysis ,Theory of computation ,Banach space ,Common fixed point ,Null point ,Algebra over a field ,Mathematics - Abstract
The purpose of this paper is to propose an algorithm for solving the split common fixed point problem for strict quasi-ϕ-pseudocontractive mappings in Banach spaces. It is proved that the sequence generated by the proposed iterative algorithm converges strongly to a solution of the split common fixed point problem. Then, the main result is used to study the split common null point problem and the split quasi-inclusion problem. Finally, a numerical example is provided to illustrate our main result. The results presented in this paper extend and improve some recent corresponding results.
- Published
- 2020
18. An implicit-explicit preconditioned direct method for pricing options under regime-switching tempered fractional partial differential models
- Author
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Xu Chen, Wenfei Wang, Siu-Long Lei, and Deng Ding
- Subjects
Partial differential equation ,Direct sum ,Applied Mathematics ,Direct method ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Toeplitz matrix ,010101 applied mathematics ,Valuation of options ,Applied mathematics ,Partial derivative ,0101 mathematics ,Coefficient matrix ,Mathematics - Abstract
Recently, fractional partial differential equations have been widely applied in option pricing problems, which better explains many important empirical facts of financial markets, but rare paper considers the multi-state options pricing problem based on fractional diffusion models. Thus, multi-state European option pricing problem under regime-switching tempered fractional partial differential equation is considered in this paper. Due to the expensive computational cost caused by the implicit finite difference scheme, a novel implicit-explicit finite difference scheme has been developed with consistency, stability, and convergence guarantee. Since the resulting coefficient matrix equals to the direct sum of several Toeplitz matrices, a preconditioned direct method has been proposed with ${\mathcal O}(\bar {S}N\log N+\bar {S}^{2} N)$ operation cost on each time level with adaptability analysis, where $\bar {S}$ is the number of states and N is the number of grid points. Related numerical experiments including an empirical example have been presented to demonstrate the effectiveness and accuracy of the proposed numerical method.
- Published
- 2020
19. Analysis of optimal superconvergence of the local discontinuous Galerkin method for nonlinear fourth-order boundary value problems
- Author
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Mahboub Baccouch
- Subjects
Degree (graph theory) ,Discontinuous Galerkin method ,Applied Mathematics ,Order (ring theory) ,Applied mathematics ,Boundary value problem ,Superconvergence ,Type (model theory) ,Finite element method ,Prime (order theory) ,Mathematics::Numerical Analysis ,Mathematics - Abstract
This paper is concerned with the convergence and superconvergence of the local discontinuous Galerkin (LDG) finite element method for nonlinear fourth-order boundary value problems of the type $u^{(4)}=f(x,u,u^{\prime },u^{\prime \prime },u^{\prime \prime \prime })$ , x ∈ [a,b] with classical boundary conditions at the endpoints. Convergence properties for the solution and for all three auxiliary variables are established. More specifically, we use the duality argument to prove that the errors between the LDG solutions and the exact solutions in the L2 norm achieve optimal (p + 1)th-order convergence, when polynomials of degree p are used. We also prove that the derivatives of the errors between the LDG solutions and Gauss-Radau projections of the exact solutions in the L2 norm are superconvergent with order p + 1. Furthermore, a (2p + 1)th-order superconvergent for the errors of the numerical fluxes at mesh nodes as well as for the cell averages is also obtained under quasi-uniform meshes. Finally, we prove that the LDG solutions are superconvergent with an order of p + 2 toward particular projections of the exact solutions. The error analysis presented in this paper is valid for p ≥ 1. Numerical experiments indicate that our theoretical findings are optimal.
- Published
- 2020
20. Continuous and discrete zeroing dynamics models using JMP function array and design formula for solving time-varying Sylvester-transpose matrix inequality
- Author
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Huanchang Huang, Xiao Liu, Min Yang, Yunong Zhang, and Yihong Ling
- Subjects
Discretization ,Truncation error (numerical integration) ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,010101 applied mathematics ,Matrix (mathematics) ,Transpose ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Mathematics ,Variable (mathematics) - Abstract
Zeroing dynamics (ZD) has shown great performance to solve various time-varying problems. In this paper, the problem of time-varying Sylvester-transpose matrix inequality is first investigated. Since it is difficult to solve a matrix inequality with a matrix variable and its transpose by traditional methods, this paper proposes a continuous ZD (CZD) model by employing ZD design formula and JMP function array to solve this challenging problem. Furthermore, for better implementation on digital computers, three discrete ZD (DZD) models are proposed by using three different discretization formulas with different precision, i.e., the Euler-forward formula, the 6-instant Zhang et al discretization (ZeaD) formula and the 7-instant ZeaD formula. What is more, theoretical truncation error analyses and numerical experiments substantiate the convergence, efficacy and superiority of the DZD models for solving time-varying Sylvester-transpose matrix inequality.
- Published
- 2020
21. Newton’s method with fractional derivatives and various iteration processes via visual analysis
- Author
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Krzysztof Gdawiec, Agnieszka Lisowska, and Wiesław Kotarski
- Subjects
Polynomial ,Applied Mathematics ,Numerical analysis ,Stability (learning theory) ,Fractional derivative ,01 natural sciences ,Fractional calculus ,010101 applied mathematics ,symbols.namesake ,Newton method ,Fixed-point iteration ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,Convergence (routing) ,symbols ,Polynomiography ,Applied mathematics ,Iterations ,0101 mathematics ,010301 acoustics ,Complex plane ,Newton's method ,Mathematics - Abstract
The aim of this paper is to visually investigate the dynamics and stability of the process in which the classic derivative is replaced by the fractional Riemann–Liouville or Caputo derivatives in the standard Newton root-finding method. Additionally, instead of the standard Picard iteration, the Mann, Khan, Ishikawa and S iterations are used. This process when applied to polynomials on complex plane produces images showing basins of attractions for polynomial zeros or images representing the number of iterations required to achieve any polynomial root. The images are called polynomiographs. In this paper, we use the colouring according to the number of iterations which reveals the speed of convergence and dynamic properties of processes visualised by polynomiographs. Moreover, to investigate the stability of the methods, we use basins of attraction. To compare numerically the modified root-finding methods among them, we demonstrate their action for polynomialz3− 1 on a complex plane.
- Published
- 2020
22. On equivalence of three-parameter iterative methods for singular symmetric saddle-point problem
- Author
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M. Tzoumas and Apostolos Hadjidimos
- Subjects
Iterative method ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Symmetric case ,01 natural sciences ,law.invention ,010101 applied mathematics ,Invertible matrix ,law ,Saddle point ,Theory of computation ,Applied mathematics ,0101 mathematics ,Equivalence (measure theory) ,Saddle ,Mathematics - Abstract
There have been a couple of papers for the solution of the nonsingular symmetric saddle-point problem using three-parameter iterative methods. In most of them, regions of convergence for the parameters are found, while in three of them, optimal parameters are determined, and in one of the latter, many more cases, than in all the others, are distinguished, analyzed, and studied. It turns out that two of the optimal parameters coincide making the optimal three-parameter methods be equivalent to the optimal two-parameter known ones. Our aim in this work is manifold: (i) to show that the iterative methods we present are equivalent, (ii) to slightly change some statements in one of the main papers, (iii) to complete the analysis in another one, (iv) to explain how the transition from any of the methods to the others is made, (v) to extend the iterative method to cover the singular symmetric case, and (vi) to present a number of numerical examples in support of our theory. It would be an omission not to mention that the main material which all researchers in the area have inspired from and used is based on the one of the most cited papers by Bai et al. (Numer. Math. 102:1–38, 2005).
- Published
- 2020
23. Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces
- Author
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Oluwatosin Temitope Mewomo, Timilehin Opeyemi Alakoya, and Adeolu Taiwo
- Subjects
Iterative and incremental development ,Applied Mathematics ,Regular polygon ,Banach space ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Monotone polygon ,Corollary ,Theory of computation ,Convergence (routing) ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.
- Published
- 2020
24. Superconvergence in H1-norm of a difference finite element method for the heat equation in a 3D spatial domain with almost-uniform mesh
- Author
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Ruijian He, Zhangxin Chen, and Xinlong Feng
- Subjects
Backward differentiation formula ,Computational complexity theory ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,010103 numerical & computational mathematics ,Superconvergence ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Norm (mathematics) ,Bounded function ,Heat equation ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose a novel difference finite element (DFE) method based on the P1-element for the 3D heat equation on a 3D bounded domain. One of the novel ideas of this paper is to use the second-order backward difference formula (BDF) combining DFE method to overcome the computational complexity of conventional finite element (FE) method for the high-dimensional parabolic problem. First, we design a fully discrete difference FE solution ${u^{n}_{h}}$ by the second-order backward difference formula in the temporal t-direction, the center difference scheme in the spatial z-direction, and the P1-element on a almost-uniform mesh Jh in the spatial (x, y)-direction. Next, the H1-stability of ${u_{h}^{n}}$ and the second-order H1-convergence of the interpolation post-processing function on ${u_{h}^{n}}$ with respect to u(tn) are provided. Finally, numerical tests are presented to show the second-order H1-convergence results of the proposed DFE method for the heat equation in a 3D spatial domain.
- Published
- 2020
25. Analysis and application of the interpolating element-free Galerkin method for extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling
- Author
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Mohammad Ilati
- Subjects
Partial differential equation ,Applied Mathematics ,Numerical analysis ,Finite difference ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Nonlinear system ,Convergence (routing) ,Biharmonic equation ,Fisher–Kolmogorov equation ,Applied mathematics ,0101 mathematics ,Galerkin method ,Mathematics - Abstract
In this paper, the interpolating element-free Galerkin method is applied for solving the nonlinear biharmonic extended Fisher–Kolmogorov equation which arises in brain tumor dynamics modeling. At first, a finite difference formula is utilized for obtaining a time-discrete scheme. The unconditional stability and convergence of the time-discrete method are proved by the energy method. Then, we use the interpolating element-free Galerkin method to approximate the spatial derivatives. An error analysis of the interpolating element-free Galerkin method is proposed for this nonlinear equation. Moreover, this method is compared with some other meshless local weak-form techniques. The main aim of this paper is to show that the interpolating element-free Galerkin is a suitable technique for solving the nonlinear fourth-order partial differential equations especially extended Fisher–Kolmogorov equation. The numerical experiments confirm the analytical results and show the good efficiency of the interpolating element-free Galerkin method for solving this nonlinear biharmonic equation.
- Published
- 2019
26. Multiscale radial kernels with high-order generalized Strang-Fix conditions
- Author
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Wenwu Gao and Xuan Zhou
- Subjects
Polynomial ,Applied Mathematics ,Numerical analysis ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Convolution ,010101 applied mathematics ,Moment (mathematics) ,Linear form ,Theory of computation ,Applied mathematics ,0101 mathematics ,ComputingMethodologies_COMPUTERGRAPHICS ,Variable (mathematics) ,Mathematics - Abstract
The paper provides a general and simple approach for explicitly constructing multiscale radial kernels with high-order generalized Strang-Fix conditions from a given univariate generator. The resulting kernels are constructed by taking a linear functional to the scaled f -form of the generator with respect to the scale variable. Equivalent divided difference forms of the kernels are also derived; based on which, a pyramid-like algorithm for fast and stable computation of multiscale radial kernels is proposed. In addition, characterizations of the kernels in both the spatial and frequency domains are given, which show that the generalized Strang-Fix condition, the moment condition, and the condition of polynomial reproduction in the convolution sense are equivalent to each other. Hence, as a byproduct, the paper provides a unified view of these three classical concepts. These kernels can be used to construct quasi-interpolation with high approximation accuracy and construct convolution operators with high approximation orders, to name a few. As an example, we construct a quasi-interpolation scheme for irregularly spaced data and derived its error estimates and choices of scale parameters of multiscale radial kernels. Numerical results of approximating a bivariate Franke function using our quasi-interpolation are presented at the end of the paper. Both theoretical and numerical results show that quasi-interpolation with multiscale radial kernels satisfying high-order generalized Strang-Fix conditions usually provides high approximation orders.
- Published
- 2019
27. An adaptive local discontinuous Galerkin method for nonlinear two-point boundary-value problems
- Author
-
Mahboub Baccouch
- Subjects
Discretization ,Adaptive mesh refinement ,Applied Mathematics ,Estimator ,010103 numerical & computational mathematics ,Superconvergence ,01 natural sciences ,Prime (order theory) ,010101 applied mathematics ,Rate of convergence ,Discontinuous Galerkin method ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
In this paper, we propose an adaptive mesh refinement (AMR) strategy based on a posteriori error estimates for the local discontinuous Galerkin (LDG) method for nonlinear two-point boundary-value problems (BVPs) of the form $u^{\prime \prime }=f(x,u),\ x\in [a,b]$ subject to some suitable boundary conditions at the endpoint of the interval [a, b]. We first use the superconvergence results proved in the first part of this paper as reported by Baccouch (Numer. Algorithm. 79(3), 697–718 2018) to show that the significant parts of the local discretization errors are proportional to (p + 1)-degree Radau polynomials, when polynomials of total degree not exceeding p are used. These new results allow us to construct a residual-based a posteriori error estimators which are obtained by solving a local residual problem with no boundary conditions on each element. The proposed error estimates are efficient, reliable, and asymptotically exact. We prove that, for smooth solutions, the proposed a posteriori error estimates converge to the exact errors in the L2-norm with order of convergence p + 3/2. Finally, we present a local AMR procedure that makes use of our local and global a posteriori error estimates. Our proofs are valid for arbitrary regular meshes and for Pp polynomials with p ≥ 1. Several numerical results are presented to validate the theoretical results and to show the efficiency of the grid refinement strategy.
- Published
- 2019
28. Family weak conjugate gradient algorithms and their convergence analysis for nonconvex functions
- Author
-
Zhou Sheng, Gonglin Yuan, and Xiaoliang Wang
- Subjects
Trust region ,Line search ,Applied Mathematics ,Numerical analysis ,Structure (category theory) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Simple (abstract algebra) ,Conjugate gradient method ,Convergence (routing) ,Theory of computation ,0101 mathematics ,Algorithm ,Mathematics - Abstract
It is well-known that conjugate gradient algorithms are widely applied in many practical fields, for instance, engineering problems and finance models, as they are straightforward and characterized by a simple structure and low storage. However, challenging problems remain, such as the convergence of the PRP algorithms for nonconvexity under an inexact line search, obtaining a sufficient descent for all conjugate gradient methods, and other theory properties regarding global convergence and the trust region feature for nonconvex functions. This paper studies family conjugate gradient formulas based on the six classic formulas, PRP, HS, CD, FR, LS, and DY, where the family conjugate gradient algorithms have better theory properties than those of the formulas by themselves. Furthermore, this technique of the presented conjugate gradient formulas can be extended to any two-term conjugate gradient formula. This paper designs family conjugate gradient algorithms for nonconvex functions, which have the following features without other conditions: (i) the sufficient descent property holds, (ii) the trust region feature is true, and (iii) the global convergence holds under normal assumptions. Numerical results show that the given conjugate gradient algorithms are competitive with those of normal methods.
- Published
- 2019
29. Reconstruction algorithms of an inverse source problem for the Helmholtz equation
- Author
-
Ji-Chuan Liu and Xiao-Chen Li
- Subjects
Nonlinear system ,Helmholtz equation ,Salient ,Applied Mathematics ,Numerical analysis ,Theory of computation ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Boundary (topology) ,Cauchy distribution ,Algorithm ,Regularization (mathematics) ,Mathematics - Abstract
In this paper, we study an inverse source problem for the Helmholtz equation from measurements. The purpose of this paper is to reconstruct the salient features of the hidden sources within a body. We propose three stable reconstruction algorithms to detect the number, the location, the size, and the shape of the hidden sources along with compact support from a single measurement of near-field Cauchy data on the external boundary. This problem is nonlinear and ill-posed; thus, we should consider regularization techniques in reconstruction algorithms. We give several numerical experiments to demonstrate the viability of our proposed reconstruction algorithms.
- Published
- 2019
30. Interval methods of Adams-Bashforth type with variable step sizes
- Author
-
Andrzej Marciniak and Malgorzata A. Jankowska
- Subjects
Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Exact solutions in general relativity ,Theory of computation ,Initial value problem ,Interval (graph theory) ,Applied mathematics ,0101 mathematics ,Constant (mathematics) ,Mathematics ,Variable (mathematics) ,Linear multistep method - Abstract
In a number of our previous papers, we have proposed interval versions of multistep methods (explicit and implicit), including interval predictor-corrector methods, in which the step size was constant. In this paper, we present interval versions of Adams-Bashforth methods with a possibility to change step sizes. This possibility can be used to obtain interval enclosures of the exact solution with a width given beforehand.
- Published
- 2019
31. Split-step cubic B-spline collocation methods for nonlinear Schrödinger equations in one, two, and three dimensions with Neumann boundary conditions
- Author
-
Luming Zhang and Shanshan Wang
- Subjects
Nonlinear system ,symbols.namesake ,Collocation ,Applied Mathematics ,B-spline ,Numerical analysis ,Neumann boundary condition ,symbols ,Finite difference ,Applied mathematics ,Nonlinear Schrödinger equation ,Schrödinger equation ,Mathematics - Abstract
In this paper, split-step cubic B-spline collocation (SS3BC) schemes are constructed by combining the split-step approach with the cubic B-spline collocation (3BC) method for the nonlinear Schrodinger (NLS) equation in one, two, and three dimensions with Neumann boundary conditions. Unfortunately, neither of the advantages of the two methods can be maintained for the multi-dimensional problems, if one combines them in the usual manner. For overcoming the difficulty, new medium quantities are introduced in this paper to successfully reduce the multi-dimensional problems into one-dimensional ones, which are essential for the SS3BC methods. Numerical tests are carried out, and the schemes are verified to be convergent with second-order both in time and space. The proposed method is also compared with the split-step finite difference (SSFD) scheme. Finally, the present method is applied to two problems of the Bose-Einstein condensate. The proposed SS3BC method is numerically verified to be effective and feasible.
- Published
- 2019
32. A priori error estimates of a Jacobi spectral method for nonlinear systems of fractional boundary value problems and related Volterra-Fredholm integral equations with smooth solutions
- Author
-
Mahmoud A. Zaky and Ibrahem G. Ameen
- Subjects
Nonlinear system ,Collocation ,Rate of convergence ,Applied Mathematics ,Numerical analysis ,Applied mathematics ,Interval (mathematics) ,Boundary value problem ,Spectral method ,Integral equation ,Mathematics - Abstract
Our aim in this paper is to develop a Legendre-Jacobi collocation approach for a nonlinear system of two-point boundary value problems with derivative orders at most two on the interval (0,T). The scheme is constructed based on the reduction of the system considered to its equivalent system of Volterra-Fredholm integral equations. The spectral rate of convergence for the proposed method is established in both L2- and $ L^{\infty } $- norms. The resulting spectral method is capable of achieving spectral accuracy for problems with smooth solutions and a reasonable order of convergence for non-smooth solutions. Moreover, the scheme is easy to implement numerically. The applicability of the method is demonstrated on a variety of problems of varying complexity. To the best of our knowledge, the spectral solution of such a nonlinear system of fractional differential equations and its associated nonlinear system of Volterra-Fredholm integral equations has not yet been studied in literature in detail. This gap in the literature is filled by the present paper.
- Published
- 2019
33. A Matlab software for approximate solution of 2D elliptic problems by means of the meshless Monte Carlo random walk method
- Author
-
Sławomir Milewski
- Subjects
Discretization ,business.industry ,Applied Mathematics ,Numerical analysis ,Monte Carlo method ,010103 numerical & computational mathematics ,System of linear equations ,Random walk ,01 natural sciences ,010101 applied mathematics ,Software ,Applied mathematics ,Meshfree methods ,Boundary value problem ,0101 mathematics ,business ,Mathematics - Abstract
This paper is devoted to the development of an innovative Matlab software, dedicated to the numerical analysis of two-dimensional elliptic problems, by means of the probabilistic approach. This approach combines features of the Monte Carlo random walk method with discretization and approximation techniques, typical for meshless methods. It allows for determination of an approximate solution of elliptic equations at the specified point (or group of points), without a necessity to generate large system of equations for the entire problem domain. While the procedure is simple and fast, the final solution may suffer from both stochastic and discretization errors. The attached Matlab software is based on several original author’s concepts. It permits the use of arbitrarily irregular clouds of nodes, non-homogeneous right-hand side functions, mixed type of boundary conditions as well as variable material coefficients (of anisotropic materials). The paper is illustrated with results of analysis of selected elliptic problems, obtained by means of this software.
- Published
- 2019
34. Unconditional optimal error estimate of the projection/Lagrange-Galerkin finite element method for the Boussinesq equations
- Author
-
Zhiyong Si, Zhang Tong, and Yanfang Lei
- Subjects
Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,01 natural sciences ,Term (time) ,010101 applied mathematics ,Error function ,Exact solutions in general relativity ,Convergence (routing) ,Projection method ,Applied mathematics ,0101 mathematics ,Constant (mathematics) ,Projection (set theory) ,Mathematics - Abstract
This paper provides an unconditional optimal convergence of a fractional-step method for solving the Boussinesq equations. In this method, the convection is treated by the Lagrange-Galerkin technique, whereas the diffusion and the incompressibility are treated by the projection method. There are lots of authors who worked on this method, and some authors gave the error estimate of this method. But, to our best knowledge, the error estimate for this method is under certain time-step restrictions. In this paper, we prove that the methods are stable almost unconditionally, i.e., when τ and h are smaller than a given constant. The basic idea of our analysis is splitting the error function into three terms, one term between the finite element solution and the projection, the other term between the projection and the time-discrete solution, the third term between the time-discrete solution and the exact solution, and giving the error estimates for each term respectively. Then, we obtain the optimal error estimates in L2 and H1-norm for the velocity and L2-norm for the pressure. In order to show the efficiency of our method, some numerical results are presented.
- Published
- 2019
35. Variable order and stepsize in general linear methods
- Author
-
Winston L. Sweatman, Saghir Ahmad, and John C. Butcher
- Subjects
Set (abstract data type) ,General linear methods ,Applied Mathematics ,Numerical analysis ,Theory of computation ,Ode ,Applied mathematics ,Estimator ,Heuristics ,Mathematics ,Variable (mathematics) - Abstract
This paper describes the implementation of a class of IRKS methods (Wright 2002). These GLM algorithms are practical with reliable error estimators (Butcher and Podhaisky, Appl. Numer. Math. 56, 345–357 2006). The current robust ODE solvers in variable stepsize as well as in variable-order mode are based upon heuristics. In this paper, we examine an optimisation approach, based on Euler-Lagrange theory (Butcher, IMA J. Numer. Anal. 6, 433–438 1986), (Butcher, Computing 44, 209–220 1990), to control the stepsize as well as the order and implement the GLMs in an efficient manner. A set of nonstiff to mildly stiff problems have been used to investigate this approach in fixed-order and variable-order modes.
- Published
- 2019
36. Energy-preserving trigonometrically fitted continuous stage Runge-Kutta-Nyström methods for oscillatory Hamiltonian systems
- Author
-
Jiyong Li and Yachao Gao
- Subjects
Runge–Kutta methods ,Applied Mathematics ,Numerical analysis ,Order (ring theory) ,Applied mathematics ,Initial value problem ,Prime (order theory) ,Symmetry (physics) ,Hamiltonian system ,Mathematics ,Separable space - Abstract
Recently, continuous-stage Runge-Kutta-Nystrom (CSRKN) methods for solving numerically second-order initial value problem $q^{\prime \prime }= f(q)$ have been proposed and developed by Tang and Zhang (Appl. Math. Comput. 323, 204–219, 2018). This problem is equivalent to a separable Hamiltonian system when f(q) = −∇U(q) with smooth function U(q). Symplecticity-preserving discretizations of this system were studied in that paper. However, as an important representation of the Hamiltonian system, energy preservation has not been studied. In addition, many Hamiltonian systems in practical applications often have oscillatory characteristics so we should design special algorithms adapted to this feature. In this paper, we propose and study energy-preserving trigonometrically fitted CSRKN methods for oscillatory Hamiltonian systems. We extend the theory of trigonometrical fitting to CSRKN methods and derive sufficient conditions for energy preservation. We also study the symmetry and stability of the methods. Two symmetric and energy-preserving trigonometrically fitted schemes of order two and four, respectively, are constructed. Some numerical experiments are provided to confirm the theoretical expectations.
- Published
- 2019
37. A numerical algorithm based on a new kind of tension B-spline function for solving Burgers-Huxley equation
- Author
-
Nastaran Alinia and Mohammad Zarebnia
- Subjects
Polynomial ,Tension (physics) ,Applied Mathematics ,Numerical analysis ,B-spline ,Convergence (routing) ,Hyperbolic function ,Function (mathematics) ,Trigonometry ,Algorithm ,Mathematics - Abstract
In this paper, a numerical algorithm based on a new kind of tension B-spline, named hyperbolic-trigonometric tension B-spline method, is applied for solving Burgers-Huxley equation. This method is generated over the space span {sin(tt),cos(tt),sinh(tt),cosh(tt),1,t,...,tn-?5},n =?5, where t is the tension parameter. Properties of it are the same in most of the properties of the usual polynomial B-splines and benefit from some other advantages, as well. Therefore, in this paper, we apply three methods consisting of trigonometric method, hyperbolic tension B-spline method, and our new hyperbolic-trigonometric tension B-spline method, to solve Burgers-Huxley equation. The convergence analysis is discussed. Then, we use some numerical examples to illustrate the accuracy and implementation of the proposed algorithm.
- Published
- 2019
38. Optimal parameter selections for a general Halpern iteration
- Author
-
Tao Wu, Songnian He, Themistocles M. Rassias, and Yeol Je Cho
- Subjects
Sequence ,Applied Mathematics ,Numerical analysis ,Hilbert space ,Field (mathematics) ,010103 numerical & computational mathematics ,Fixed point ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Rate of convergence ,Variational inequality ,symbols ,0101 mathematics ,Real number ,Mathematics - Abstract
Let C be a closed affine subset of a real Hilbert space H and $T:C \rightarrow C$ be a nonexpansive mapping. In this paper, for any fixed u ∈ C, a general Halpern iteration process: $$\left\{\begin{array}{ll} x_{0} \in C,\\ x_{n + 1}=t_{n}u+(1-t_{n})Tx_{n},n\geq 0, \end{array}\right. $$ is considered for finding a fixed point of T nearest to u, where the parameter sequence {tn} is selected in the real number field, $\mathbb {R}$. The core problem to be addressed in this paper is to find the optimal parameter sequence so that this iteration process has the optimal convergence rate and to give some numerical results showing advantages of our algorithms. Also, we study the problem of selecting the optimal parameters for a general viscosity approximation method and apply the results obtained from this study to solve a class of variational inequalities.
- Published
- 2019
39. Trees and B-series
- Author
-
John C. Butcher
- Subjects
Algebra ,General linear methods ,Differential equation ,Applied Mathematics ,Numerical analysis ,Theory of computation ,Order (group theory) ,B series ,Term (logic) ,Mathematics ,Connection (mathematics) - Abstract
The connection between trees and differential equations was pointed out in the classic paper by Cayley (Phil. Mag. 13, 172–176 1857). Trees were also used in the work of Merson (1957), on the order of Runge–Kutta methods. The paper by Hairer and Wanner (Computing 13, 1–15 1974), where the term B-series was introduced, followed papers by the present author (Butcher J. Austral. Math. Soc. 3, 185–201 1963, Math. Comput. 26, 79–106 1972). The present paper will survey the use of trees in the formulation of B-series and illustrate the results by constructing and analysing some examples of general linear methods.
- Published
- 2018
40. A bias-compensated fractional order normalized least mean square algorithm with noisy inputs
- Author
-
Jianmei Shuai, Yiheng Wei, Weidi Yin, Songsong Cheng, and Yong Wang
- Subjects
Applied Mathematics ,Numerical analysis ,Stability (learning theory) ,010103 numerical & computational mathematics ,Variance (accounting) ,01 natural sciences ,Noise (electronics) ,010101 applied mathematics ,Least mean squares filter ,Convergence (routing) ,Theory of computation ,0101 mathematics ,Root-mean-square deviation ,Algorithm ,Mathematics - Abstract
This paper comes up with a stable bias-compensated fractional order normalized least mean square (BC-FONLMS) algorithm with noisy inputs. This kind of bias-compensated algorithm needs the estimation of input noise variance to avoid the bias caused by noisy inputs. Yet, existing algorithms either cause instability because of the method used to estimate input noise variance, or surmount the instability problems at the price of performance diminishment. This paper introduces fractional order calculus into LMS algorithm to be a new BC-FONLMS algorithm. Then, analyze the stability of the BC-FONLMS algorithm through probing the recursive equations of mean deviation (MD) and mean square deviation (MSD). On the basis of the stability analysis, methods to estimate input noise variance and to adjust step size are suggested to stabilize the algorithm and likewise to enhance the performance such as convergence speed and steady-state error. Numerical simulations are given at last, whose results show that the proposed BC-FONLMS algorithm performs well.
- Published
- 2018
41. Weighted and deflated global GMRES algorithms for solving large Sylvester matrix equations
- Author
-
Gang Wu, Najmeh Azizi Zadeh, and Azita Tajaddini
- Subjects
Sylvester matrix ,Applied Mathematics ,Numerical analysis ,Linear system ,MathematicsofComputing_NUMERICALANALYSIS ,010103 numerical & computational mathematics ,Krylov subspace ,Computer Science::Numerical Analysis ,01 natural sciences ,Generalized minimal residual method ,Mathematics::Numerical Analysis ,Weighting ,010101 applied mathematics ,Matrix (mathematics) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0101 mathematics ,Algorithm ,Eigenvalues and eigenvectors ,Mathematics - Abstract
The solution of a large-scale Sylvester matrix equation plays an important role in control and large scientific computations. In this paper, we are interested in the large Sylvester matrix equation with large dimensionA and small dimension B, and a popular approach is to use the global Krylov subspace method. In this paper, we propose three new algorithms for this problem. We first consider the global GMRES algorithm with weighting strategy, which can be viewed as a precondition method. We present three new schemes to update the weighting matrix during iterations. Due to the growth of memory requirements and computational cost, it is necessary to restart the algorithm effectively. The deflation strategy is efficient for the solution of large linear systems and large eigenvalue problems; to the best of our knowledge, little work is done on applying deflation to the (weighted) global GMRES algorithm for large Sylvester matrix equations. We then consider how to combine the weighting strategy with deflated restarting, and propose a weighted global GMRES algorithm with deflation for solving large Sylvester matrix equations. In particular, we are interested in the global GMRES algorithm with deflation, which can be viewed as a special case when the weighted matrix is chosen as the identity. Theoretical analysis is given to show rationality of the new algorithms. Numerical experiments illustrate the numerical behavior of the proposed algorithms.
- Published
- 2018
42. A 5-instant finite difference formula to find discrete time-varying generalized matrix inverses, matrix inverses, and scalar reciprocals
- Author
-
Mingzhi Mao, Jian Li, Yunong Zhang, and Frank Uhlig
- Subjects
Discretization ,Applied Mathematics ,Numerical analysis ,Scalar (mathematics) ,Finite difference ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Discrete time and continuous time ,Theory of computation ,Euler's formula ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
Finite difference schemes have been widely studied because of their fundamental role in numerical analysis. However, most finite difference formulas in the literature are not suitable for discrete time-varying problems because of intrinsic limitations and their relatively low precision. In this paper, a high-precision 1-step-ahead finite difference formula is developed. This 5-instant finite difference (5-IFD) formula is used to approximate and discretize first-order derivatives, and it helps us to compute discrete time-varying generalized matrix inverses. Furthermore, as special cases of generalized matrix inverses, time-varying matrix inversion, and scalar reciprocals are generally deemed as independent problems and studied separately, which are solved unitedly in this paper. The precision of the 5-IFD formula and the convergence behavior of the corresponding discrete-time models are derived theoretically and shown in numerical experiments. Conventional useful formulas, such as the Euler forward finite difference (EFFD) formula and the 4-instant finite difference (4-IFD) formula are also used for comparisons and to show the superiority of the 5-IFD formula.
- Published
- 2018
43. Long-term adaptive symplectic numerical integration of linear stochastic oscillators driven by additive white noise
- Author
-
A. Foroush Bastani, Mohammad Reza Yaghouti, and M. Malzoumati-Khiaban
- Subjects
Stochastic oscillator ,Applied Mathematics ,Numerical analysis ,Second moment of area ,Applied mathematics ,White noise ,Regularization (mathematics) ,Time reversibility ,Brownian motion ,Numerical integration ,Mathematics - Abstract
In this paper, we present an adaptive variable step size numerical scheme for the integration of linear stochastic oscillator equations driven by additive Brownian white noise. We first show that traditional adaptive schemes based on local error estimation destroy the long-time behavior of the underlying method. As a remedy, we extend the idea presented in Hairer and Soderlind (SIAM J. Sci. Comput. 26(6), 1838–1851 2005) to the stochastic setting and show that using step density control mechanisms based on time regularization and local error tracking, we are able to obtain numerical schemes which preserve the important qualitative features of the solution process such as symmetry, time reversibility, symplecticity, linear growth rate of the second moment, and infinite oscillation. Numerical experiments confirm the theoretical findings of the paper.
- Published
- 2018
44. On the split feasibility problem and fixed point problem of quasi-ϕ-nonexpansive mapping in Banach spaces
- Author
-
Zhaoli Ma, Lin Wang, and Shih-sen Chang
- Subjects
Sequence ,Iterative method ,Applied Mathematics ,Numerical analysis ,010102 general mathematics ,Banach space ,Fixed point ,Inverse problem ,01 natural sciences ,010101 applied mathematics ,Fixed point problem ,Theory of computation ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to propose an algorithm to solve the split feasibility and fixed point problem of quasi-ϕ-nonexpansive mappings in Banach spaces. Without the assumption of semi-compactness on the mappings, it is proved that the sequence generated by the proposed iterative algorithm converges strongly to a common solution of the split feasibility and fixed point problems. As applications, the main results presented in this paper are used to study the convexly constrained linear inverse problem and split null point problem. Finally, a numerical example is given to support our results. The results presented in the paper are new and improve and extend some recent corresponding results.
- Published
- 2018
45. Sparse matrix computation for air quality forecast data assimilation
- Author
-
Michael K. Ng and Zhaochen Zhu
- Subjects
0209 industrial biotechnology ,Numerical linear algebra ,Applied Mathematics ,Computation ,Numerical analysis ,02 engineering and technology ,computer.software_genre ,020901 industrial engineering & automation ,Data assimilation ,Theory of computation ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Ensemble Kalman filter ,Data mining ,computer ,Air quality index ,Sparse matrix ,Mathematics - Abstract
In this paper, we study the ensemble Kalman filter (EnKF) method for chemical species simulation in air quality forecast data assimilation. The main contribution of this paper is that we study the sparse observation data and make use of the matrix structure of the EnKF update equations to design an algorithm for the purpose of computing the analysis of chemical species in an air quality forecast system efficiently. The proposed method can also handle the combined observations from multiple chemical species together. We applied the proposed method and tested its performance in real air quality data assimilation. Numerical examples are presented to demonstrate the efficiency of the proposed computation method for EnKF updating and the effectiveness of the proposed method for NO2, NO, CO, SO2, O3, PM2.5, and PM10 prediction in air quality forecast data assimilation.
- Published
- 2018
46. Selective projection methods for solving a class of variational inequalities
- Author
-
Hanlin Tian and Songnian He
- Subjects
021103 operations research ,Current (mathematics) ,Euclidean space ,Applied Mathematics ,0211 other engineering and technologies ,Hilbert space ,Mathematics::General Topology ,02 engineering and technology ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Mathematics::Logic ,symbols.namesake ,Projection (relational algebra) ,Integer ,Rate of convergence ,Variational inequality ,symbols ,0101 mathematics ,Convex function ,Mathematics - Abstract
Very recently, Gibali et al. (Optimization 66, 417–437 2017) proposed a method, called selective projection method (SPM) in this paper, for solving the variational inequality problem (VIP) defined on $C:=\bigcap _{i = 1}^{m} C^{i}\neq \emptyset $ , where m ≥ 1 is an integer and $\{C^{i}\}_{i = 1}^{m}$ is a finite level set family of convex functions on a real Hilbert space H. For the current iterate xn, SPM updates xn+ 1 by projecting onto a half-space $C^{i_{n}}_{n} (\supset C^{i_{n}})$ constructed by using the input data, where in ∈{1,2,⋯ ,m} is selected by a special rule. The prominent advantage of SPM is that it is concise and easy to implement. Gibali et al. proved its convergence in the Euclidean space $H=\mathbb {R}^{d}$ . In this paper, we firstly prove the strong convergence of SPM in a general Hilbert space. The proof given in this paper is very different from that given by Gibali et al. We also extend SPM to solve VIP defined on the common fixed point set of finite nonexpansive self-mappings of H. Then, we estimate the convergence rate of SPM and its extension in the nonasymptotic sense. Finally, we give some preliminary numerical experiments which illustrate the advantage of SPM.
- Published
- 2018
47. A two-grid parallel partition of unity finite element scheme
- Author
-
Guangzhi Du and Liyun Zuo
- Subjects
Two grid ,Applied Mathematics ,Numerical analysis ,010103 numerical & computational mathematics ,Grid ,01 natural sciences ,Finite element method ,010101 applied mathematics ,Partition of unity ,Scheme (mathematics) ,Theory of computation ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
A two-grid parallel partition of unity finite element scheme is proposed and analyzed in this paper for linear elliptic boundary value problems. The interesting features of this scheme contain the following: (1) a partition of unity is constructed to derive the globally continuous finite element solution; (2) errors decay exponentially with patches of diameter kH increase; and (3) a global coarse grid correction is done to improve the L2 − accuracy of the approximation. Numerical experiments are presented at the end of the paper to support our analysis.
- Published
- 2018
48. Convergence of discrete time waveform relaxation methods
- Author
-
Zhencheng Fan
- Subjects
Discretization ,Applied Mathematics ,Numerical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Relaxation (iterative method) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Rate of convergence ,Discrete time and continuous time ,Ordinary differential equation ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Applied mathematics ,Waveform ,0101 mathematics ,Interpolation ,Mathematics - Abstract
This paper concerns the discrete time waveform relaxation (DWR) methods for ordinary differential equations (ODEs). We present a general algorithm of constructing the DWR methods with any order of convergence, which applies any numerical methods of ODEs to the perturbed equations of iterative schemes of continuous time waveform relaxation methods. It is demonstrated that the DWR method presented in this paper has the same convergent order as the numerical method used to discretize perturbed equations. Two classes of interpolation polynomials are given to generate perturbed equations. Finally, numerical experiments are presented in order to check against results obtained.
- Published
- 2018
49. Hybrid iterative method for split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings
- Author
-
Rehan Ali, K. R. Kazmi, and Mohd Furkan
- Subjects
Iterative method ,Applied Mathematics ,Numerical analysis ,010102 general mathematics ,Hilbert space ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Monotone polygon ,Fixed point problem ,Theory of computation ,symbols ,Applied mathematics ,0101 mathematics ,Algebra over a field ,Mathematics - Abstract
In this paper, we propose a hybrid iterative method to approximate a common solution of split monotone variational inclusion problem and hierarchical fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces. We prove that sequences generated by the proposed hybrid iterative method converge strongly to a common solution of these problems. Further, we discuss some applications of the main result. We also discuss a numerical example to demonstrate the applicability of the iterative method. The method and results presented in this paper extend and unify the corresponding known results in this area.
- Published
- 2017
50. On the eigenvalues of the saddle point matrices discretized from Navier–Stokes equations
- Author
-
Na Huang and Changfeng Ma
- Subjects
Discretization ,Applied Mathematics ,Numerical analysis ,Mathematical analysis ,010103 numerical & computational mathematics ,Positive-definite matrix ,Computer Science::Numerical Analysis ,01 natural sciences ,Generalized minimal residual method ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Linearization ,Saddle point ,0101 mathematics ,Navier–Stokes equations ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper, we study the spectral distributions of the saddle point matrices arising from the discretization and linearization of the Navier–Stokes equations, where the (1,1) block is nonsymmetric positive definite. In this paper, we derive the lower and upper bounds of the real and imaginary parts of all the eigenvalues of the saddle point matrices. We then propose a new class of block triangle preconditioners for solving the saddle point problems, and analyze the spectral properties of the preconditioned systems. Some numerical experiments with the preconditioned restarted generalized minimal residual method are reported to demonstrate the effectiveness and feasibility of these block triangle preconditioners.
- Published
- 2017
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