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A MODIFIED LEVENBERG-MARQUARDT ALGORITHM FOR SINGULAR SYSTEM OF NONLINEAR EQUATIONS$*1$ Jin Yan FAN Journal of Computational Mathematics    Abstract （876）      PDF       Knowledge map    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. Related Articles | Metrics

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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 Abstract （755）      PDF       Knowledge map    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. Reference | Related Articles | Metrics

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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 Abstract （304）      PDF       Knowledge map    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. Reference | Related Articles | Metrics

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COMPACT FOURTH-ORDER FINITEDIFFERENCE SCHEMES FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS Fu Yiping Journal of Computational Mathematics    Abstract （682）      PDF       Knowledge map    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. Related Articles | Metrics

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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    Abstract （870）      PDF       Knowledge map    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. Related Articles | Metrics Cited: CSCD(110)

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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    Abstract （792）      PDF       Knowledge map    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. Related Articles | Metrics

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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 Abstract （242）      PDF       Knowledge map    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. Reference | Related Articles | Metrics

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PARALLEL IMPLEMENTATIONS OF THE FAST SWEEPING METHOD Hongkai Zhao Journal of Computational Mathematics    Abstract （671）      PDF       Knowledge map    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. Related Articles | Metrics Cited: Baidu(125)

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Symmetric Quadrature Rules on Triangles and Tetrahedra Linbo Zhang, Tao Cui, Hui Liu Journal of Computational Mathematics    Abstract （910）      PDF       Knowledge map    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. Related Articles | Metrics Cited: Baidu(89)

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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    Abstract （673）      PDF       Knowledge map    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. Related Articles | Metrics Cited: Baidu(32)

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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 Abstract （288）      PDF       Knowledge map    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. Reference | Related Articles | Metrics

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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 Abstract （211）      PDF       Knowledge map    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). Reference | Related Articles | Metrics

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ON KORN'S INEQUALITY Lie Heng WANG Journal of Computational Mathematics    Abstract （661）      PDF       Knowledge map    This paper is devoted to give a new proof of Korn's inequality in LT - norm (1 < γ < ∞). Related Articles | Metrics

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PRIMAL PERTURBATION SIMPLEX ALGORITHMS FOR LINEAR PROGRAMMING Ping Qi PAN Journal of Computational Mathematics    Abstract （793）      PDF       Knowledge map    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. Related Articles | Metrics

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MULTISYMPLECTIC FOURIER PSEUDOSPECTRAL METHOD FOR THE NONLINEAR SCHRSCHRODINGER EQUATIONS WITH WAVE OPERATOR Jian Wang Journal of Computational Mathematics    Abstract （679）      PDF       Knowledge map    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. Related Articles | Metrics

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OPTIMAL APPROXIMATE SOLUTION OF THE MATRIX EQUATIONAXB=C OVER SYMMETRIC MATRICES Anping Liao, Yuan Lei Journal of Computational Mathematics    Abstract （658）      PDF       Knowledge map    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. Related Articles | Metrics Cited: Baidu(19)

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A SECOND-ORDER CONVEX SPLITTING SCHEME FOR A CAHN-HILLIARD EQUATION WITH VARIABLE INTERFACIAL PARAMETERS Xiao Li, Zhonghua Qiao, Hui Zhang Journal of Computational Mathematics    2017, 35 (6): 693-710.   DOI: 10.4208/jcm.1611-m2016-0517 Abstract （234）      PDF       Knowledge map    In this paper, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation with a variable interfacial parameter, is solved numerically by using a convex splitting scheme which is second-order in time for the non-stochastic part in combination with the CrankNicolson and the Adams-Bashforth methods. For the non-stochastic case, the unconditional energy stability is obtained in the sense that a modified energy is non-increasing. The scheme in the stochastic version is then obtained by adding the discretized stochastic term. Numerical experiments are carried out to verify the second-order convergence rate for the non-stochastic case, and to show the long-time stochastic evolutions using larger time steps. Reference | Related Articles | Metrics

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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    Abstract （684）      PDF       Knowledge map    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. Related Articles | Metrics

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VON NEUMANN STABILITY ANALYSIS OF SYMPLECTIC INTEGRATORS APPLIED TO HAMILTONIAN PDEs Helen M. Regan Journal of Computational Mathematics    Abstract （881）      PDF       Knowledge map    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. Related Articles | Metrics

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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 Abstract （391）      PDF       Knowledge map    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. Reference | Related Articles | Metrics Cited: Baidu(3)