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    A MODIFIED LEVENBERG-MARQUARDT ALGORITHM FOR SINGULAR SYSTEM OF NONLINEAR EQUATIONS$*1$
    Jin Yan FAN
    Journal of Computational Mathematics   
    Abstract809)      PDF      
    Based on the work of paprer [1], we propose a modified Levenberg-Marquardt algoithm for solving singular system of nonlinear equations $F(x)=0$, where $F(x):R^n\rightarrow R^n$ is continuously differentiable and $F'(x)$ is Lipschitz continuous. The algorithm is equivalent to a trust region algorithm in some sense , and the global convergence result is given. The sequence generated by the algorithm converges to the solution quadratically, if $\|F(x)\|_2$provides a local error bound for the system of nonlinear equations. Numerical results show that the algorithm performs well.
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    A TRUST-REGION-BASED ALTERNATING LEAST-SQUARES ALGORITHM FOR TENSOR DECOMPOSITIONS
    Fan Jiang, Deren Han, Xiaofei Zhang
    Journal of Computational Mathematics    2018, 36 (3): 351-373.   DOI: 10.4208/jcm.1605-m2016-0828
    Abstract199)      PDF      
    Tensor canonical decomposition (shorted as CANDECOMP/PARAFAC or CP) decomposes a tensor as a sum of rank-one tensors, which finds numerous applications in signal processing, hypergraph analysis, data analysis, etc. Alternating least-squares (ALS) is one of the most popular numerical algorithms for solving it. While there have been lots of efforts for enhancing its efficiency, in general its convergence can not been guaranteed.
    In this paper, we cooperate the ALS and the trust-region technique from optimization field to generate a trust-region-based alternating least-squares (TRALS) method for CP. Under mild assumptions, we prove that the whole iterative sequence generated by TRALS converges to a stationary point of CP. This thus provides a reasonable way to alleviate the swamps, the notorious phenomena of ALS that slow down the speed of the algorithm. Moreover, the trust region itself, in contrast to the regularization alternating least-squares (RALS) method, provides a self-adaptive way in choosing the parameter, which is essential for the efficiency of the algorithm. Our theoretical result is thus stronger than that of RALS in[26], which only proved the cluster point of the iterative sequence generated by RALS is a stationary point. In order to accelerate the new algorithm, we adopt an extrapolation scheme. We apply our algorithm to the amino acid fluorescence data decomposition from chemometrics, BCM decomposition and rank-( L r, L r, 1) decomposition arising from signal processing, and compare it with ALS and RALS. The numerical results show that TRALS is superior to ALS and RALS, both from the number of iterations and CPU time perspectives.
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    COMPACT FOURTH-ORDER FINITEDIFFERENCE SCHEMES FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS
    Fu Yiping
    Journal of Computational Mathematics   
    Abstract637)      PDF      
    In this paper, two fourth-order accurate compact difference schemes
    are presented for solving the Helmholtz equation in two space
    dimensions when the corresponding wave numbers are large. The main
    idea is to derive and to study a fourth-order accurate compact
    difference scheme whose leading truncation term, namely, the
    $\mathcal O(h^4)$ term, is independent of the wave number and the
    solution of the Helmholtz equation. The convergence property of the
    compact schemes are analyzed and the implementation of solving the
    resulting linear algebraic system based on a FFT approach is
    considered. Numerical results are presented, which support our
    theoretical predictions.
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    IMPLICIT-EXPLICIT SCHEME FOR THE ALLEN-CAHN EQUATION PRESERVES THE MAXIMUM PRINCIPLE
    Tao Tang, Jiang Yang
    Journal of Computational Mathematics    2016, 34 (5): 451-461.   DOI: 10.4208/jcm.1603-m2014-0017
    Abstract317)      PDF      
    It is known that the Allen-Chan equations satisfy the maximum principle. Is this true for numerical schemes? To the best of our knowledge, the state-of-art stability framework is the nonlinear energy stability which has been studied extensively for the phase field type equations. In this work, we will show that a stronger stability under the infinity norm can be established for the implicit-explicit discretization in time and central finite difference in space. In other words, this commonly used numerical method for the Allen-Cahn equation preserves the maximum principle.
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    An Anisotropic Nonconforming Finite Element with Some Superconvergence Results
    Dong-yang Shi, Shi-peng Mao, Shao-chun Chen
    Journal of Computational Mathematics   
    Abstract837)      PDF      
    The main aim of this paper is to study the error estimates of a nonconforming finite element with some superconvergence results under anisotropic meshes. The anisotropic interpolation error and consistency error estimates are obtained by using some novel approaches and techniques, respectively. Furthermore, the superclose and a superconvergence estimate on the central points of elements are also obtained without the regularity assumption and quasi-uniform assumption requirement on the meshes. Finally, a numerical test is carried out, which coincides with our theoretical analysis.
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    EXTENDED LEVENBERG-MARQUARDT METHOD FOR COMPOSITE FUNCTION MINIMIZATION
    Jianchao Huang, Zaiwen Wen, Xiantao Xiao
    Journal of Computational Mathematics    2017, 35 (4): 529-546.   DOI: 10.4208/jcm.1702-m2016-0699
    Abstract181)      PDF      
    In this paper, we propose an extended Levenberg-Marquardt (ELM) framework that generalizes the classic Levenberg-Marquardt (LM) method to solve the unconstrained minimization problem min ρ ( r( x)), where r:R n → Rm and ρ:R m → R. We also develop a few inexact variants which generalize ELM to the cases where the inner subproblem is not solved exactly and the Jacobian is simplified, or perturbed. Global convergence and local superlinear convergence are established under certain suitable conditions. Numerical results show that our methods are promising.
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    A SHIFT-SPLITTING PRECONDITIONER FOR NON-HERMITIAN POSITIVE DEFINITE MATRICES
    Zhong-zhi Bai,Jun-feng Yin,Yang-feng Su
    Journal of Computational Mathematics   
    Abstract733)      PDF      
    A shift splitting concept is introduced and, correspondingly,
    a shift-splitting iteration scheme and a shift-splitting
    preconditioner are presented,
    for solving the large sparse system of linear equations of which the
    coefficient matrix is an ill-conditioned non-Hermitian
    positive definite matrix.
    The convergence property of the shift-splitting iteration method
    and the eigenvalue distribution of the shift-splitting
    preconditioned matrix are discussed in depth,
    and the best possible choice of the shift is investigated
    in detail. Numerical computations show that
    the shift-splitting preconditioner can induce accurate, robust
    and effective preconditioned Krylov subspace iteration methods
    for solving the large sparse non-Hermitian positive definite
    systems of linear equations.
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    APPROXIMATION, STABILITY AND FAST EVALUATION OF EXACT ARTIFICIAL BOUNDARY CONDITION FOR THE ONE-DIMENSIONAL HEAT EQUATION
    Chunxiong Zheng
    Journal of Computational Mathematics   
    Abstract639)      PDF      
    In this paper we consider the numerical solution of the
    one-dimensional heat equation on unbounded domains. First an exact
    semi-discrete artificial boundary condition is derived by
    discretizing the time variable with the Crank-Nicolson method. The
    semi-discretized heat equation equipped with this boundary condition
    is then proved to be unconditionally stable, and its solution is
    shown to have second-order accuracy. In order to reduce the
    computational cost, we develop a new fast evaluation method for the
    convolution operation involved in the exact semi-discrete artificial
    boundary condition. A great advantage of this method is that the
    unconditional stability held by the semi-discretized heat equation
    is preserved. An error estimate is also given to show the dependence
    of numerical errors on the time step and the approximation accuracy
    of the convolution kernel. Finally, a simple numerical example is
    presented to validate the theoretical results.
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    Symmetric Quadrature Rules on Triangles and Tetrahedra
    Linbo Zhang, Tao Cui, Hui Liu
    Journal of Computational Mathematics   
    Abstract856)      PDF      
    We present a program for computing symmetric quadrature rules on
    triangles and tetrahedra. A set of rules are obtained by using this
    program. Quadrature rules up to order 21 on triangles and up to
    order 14 on tetrahedra have been obtained which are useful for use
    in finite element computations. All rules presented here have
    positive weights with points lying within the integration domain.
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    Cited: Baidu(89)
    EXPONENTIAL FOURIER COLLOCATION METHODS FOR SOLVING FIRST-ORDER DIFFERENTIAL EQUATIONS
    Bin Wang, Xinyuan Wu, Fanwei Meng, Yonglei Fang
    Journal of Computational Mathematics    2017, 35 (6): 711-736.   DOI: 10.4208/jcm.1611-m2016-0596
    Abstract175)      PDF      
    In this paper, a novel class of exponential Fourier collocation methods (EFCMs) is presented for solving systems of first-order ordinary differential equations. These so-called exponential Fourier collocation methods are based on the variation-of-constants formula, incorporating a local Fourier expansion of the underlying problem with collocation methods. We discuss in detail the connections of EFCMs with trigonometric Fourier collocation methods (TFCMs), the well-known Hamiltonian Boundary Value Methods (HBVMs), Gauss methods and Radau ⅡA methods. It turns out that the novel EFCMs are an essential extension of these existing methods. We also analyse the accuracy in preserving the quadratic invariants and the Hamiltonian energy when the underlying system is a Hamiltonian system. Other properties of EFCMs including the order of approximations and the convergence of fixed-point iterations are investigated as well. The analysis given in this paper proves further that EFCMs can achieve arbitrarily high order in a routine manner which allows us to construct higher-order methods for solving systems of firstorder ordinary differential equations conveniently. We also derive a practical fourth-order EFCM denoted by EFCM(2,2) as an illustrative example. The numerical experiments using EFCM(2,2) are implemented in comparison with an existing fourth-order HBVM, an energy-preserving collocation method and a fourth-order exponential integrator in the literature. The numerical results demonstrate the remarkable efficiency and robustness of the novel EFCM(2,2).
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    PRIMAL PERTURBATION SIMPLEX ALGORITHMS FOR LINEAR PROGRAMMING
    Ping Qi PAN
    Journal of Computational Mathematics   
    Abstract726)      PDF      
    In this paper,we propose two new perturbation simplex variants.Solving linear programming problems without introducing artificial variables,each of the two uses the dual pivot rule to achieve primal feasibility,and then the primal pivot rule two achieve optimality.The second algorithm,a modification of the first,is designed to handle highly degenerate problems more efficiently.Some interesting results concerning merit of the perturbation are established.Numerical results from preliminary tests are also reported.
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    ON KORN'S INEQUALITY
    Lie Heng WANG
    Journal of Computational Mathematics   
    Abstract623)      PDF      
    This paper is devoted to give a new proof of Korn's inequality in LT - norm (1 < γ < ∞).
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    MULTISYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHRSCHRODINGER EQUATIONS WITH WAVE OPERATOR
    Jian Wang
    Journal of Computational Mathematics   
    Abstract643)      PDF      
    In this paper, the multisymplectic Fourier pseudospectral scheme
    for initial-boundary value problems of nonlinear Schr\"{o}dinger
    equations with wave operator is considered. We investigate the
    local and global conservation properties of the multisymplectic
    discretization based on Fourier pseudospectral approximations. The
    local and global spatial conservation of energy is proved. The
    error estimates of local energy conservation law are also derived.
    Numerical experiments are presented to verify the theoretical
    predications.
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    PARALLEL IMPLEMENTATIONS OF THE FAST SWEEPING METHOD
    Hongkai Zhao
    Journal of Computational Mathematics   
    Abstract640)      PDF      
    The fast sweeping method is an efficient iterative method for
    hyperbolic
    problems.
    It combines Gauss-Seidel iterations with alternating sweeping orderings.
    In this paper several parallel implementations of the fast sweeping method
    are presented. These parallel algorithms are simple and efficient due
    to the causality of the underlying partial
    different equations. Numerical examples are used to verify our algorithms.
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    FAST ALGORITHMS FOR HIGHER-ORDER SINGULAR VALUE DECOMPOSITION FROM INCOMPLETE DATA
    Yangyang Xu
    Journal of Computational Mathematics    2017, 35 (4): 397-422.   DOI: 10.4208/jcm.1608-m2016-0641
    Abstract214)      PDF      
    Higher-order singular value decomposition (HOSVD) is an efficient way for data reduction and also eliciting intrinsic structure of multi-dimensional array data. It has been used in many applications, and some of them involve incomplete data. To obtain HOSVD of the data with missing values, one can first impute the missing entries through a certain tensor completion method and then perform HOSVD to the reconstructed data. However, the two-step procedure can be inefficient and does not make reliable decomposition. In this paper, we formulate an incomplete HOSVD problem and combine the two steps into solving a single optimization problem, which simultaneously achieves imputation of missing values and also tensor decomposition.
    We also present one algorithm for solving the problem based on block coordinate update (BCU). Global convergence of the algorithm is shown under mild assumptions and implies that of the popular higher-order orthogonality iteration (HOOI) method, and thus we, for the first time, give global convergence of HOOI.
    In addition, we compare the proposed method to state-of-the-art ones for solving incomplete HOSVD and also low-rank tensor completion problems and demonstrate the superior performance of our method over other compared ones. Furthermore, we apply it to face recognition and MRI image reconstruction to show its practical performance.
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    OPTIMAL APPROXIMATE SOLUTION OF THE MATRIX EQUATIONAXB=C OVER SYMMETRIC MATRICES
    Anping Liao, Yuan Lei
    Journal of Computational Mathematics   
    Abstract608)      PDF      
    Let $S_E$ denote the least-squares symmetric solution set of the
    matrix equation $AXB=C$, where A, B and C are given matrices of
    suitable size. To find the optimal approximate solution in the set
    $S_E$ to a given matrix, we give a new feasible method based on the
    projection theorem, the generalized SVD and the canonical correction
    decomposition.
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    Cited: Baidu(19)
    A FINITE ELEMENT METHOD WITH PERFECTLY MATCHED ABSORBINGLAYERS FOR THE WAVE SCATTERING BY A PERIODIC CHIRAL STRUCTURE
    Deyue Zhang , Fuming Ma
    Journal of Computational Mathematics   
    Abstract652)      PDF      
    Consider the diffraction of a time-harmonic wave incident upon a
    periodic chiral structure. The diffraction problem may be simplified
    to a two-dimensional one. In this paper, the diffraction problem is
    solved by a finite element method with perfectly matched absorbing
    layers (PMLs). We use the PML technique to truncate the unbounded
    domain to a bounded one which attenuates the outgoing waves in the
    PML region. Our computational experiments indicate that the proposed
    method is efficient, which is capable of dealing with complicated
    chiral grating structures.
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    VON NEUMANN STABILITY ANALYSIS OF SYMPLECTIC INTEGRATORS APPLIED TO HAMILTONIAN PDEs
    Helen M. Regan
    Journal of Computational Mathematics   
    Abstract831)      PDF      
    Symplectic integration of separable Hamiltonian ordinary and partial differential equations is discussed. A von Neumann analysis is performed to achieve general linear stability criteria for symplectic methods applied to a restricted class of Hamiltonian PDE to form a system of Hamiltonian ODEs to which a symplectic integrator can be applied.In this way stability criteria are achieved by considering the spectra of linearised Hamiltonian PDEs rather than spatisl step size.
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    A CHEBYSHEV-GAUSS SPECTRAL COLLOCATION METHOD FOR ORDINARY DIFFERENTIAL EQUATIONS
    Xi Yang, Zhongqing Wang
    Journal of Computational Mathematics    2015, 33 (1): 59-85.   DOI: 10.4208/jcm.1405-m4368
    Abstract307)      PDF      
    In this paper, we introduce an efficient Chebyshev-Gauss spectral collocation method for initial value problems of ordinary differential equations. We first propose a single interval method and analyze its convergence. We then develop a multi-interval method. The suggested algorithms enjoy spectral accuracy and can be implemented in stable and efficient manners. Some numerical comparisons with some popular methods are given to demonstrate the effectiveness of this approach.
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    MODIFIED BERNOULLI ITERATION METHODS FOR QUADRATIC MATRIXEQUATION
    Zhongzhi Bai, Yonghua Gao
    Journal of Computational Mathematics   
    Abstract602)      PDF      
    We construct a modified Bernoulli iteration method for solving the
    quadratic matrix equation $AX^{2} + BX + C = 0$, where $A$, $B$ and
    $C$ are square matrices. This method is motivated from the
    Gauss-Seidel iteration for solving linear systems and the
    Sherman-Morrison-Woodbury formula for updating matrices. Under
    suitable conditions, we prove the local linear convergence of the
    new method. An algorithm is presented to find the solution of the
    quadratic matrix equation and some numerical results are given to
    show the feasibility and the effectiveness of the algorithm. In
    addition, we also describe and analyze the block version of the
    modified Bernoulli iteration method.
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