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    SOURCE TERM IDENTIFICATION WITH DISCONTINUOUS DUAL RECIPROCITY APPROXIMATION AND QUASI-NEWTON METHOD FROM BOUNDARY OBSERVATIONS
    El Madkouri Abdessamad, Ellabib Abdellatif
    Journal of Computational Mathematics    2021, 39 (3): 311-332.   DOI: 10.4208/jcm.1912-m2019-0121
    Abstract33)      PDF      
    This paper deals with discontinuous dual reciprocity boundary element method for solving an inverse source problem. The aim of this work is to determine the source term in elliptic equations for nonhomogenous anisotropic media, where some additional boundary measurements are required. An equivalent formulation to the primary inverse problem is established based on the minimization of a functional cost, where a regularization term is employed to eliminate the oscillations of the noisy data. Moreover, an efficient algorithm is presented and tested for some numerical examples.
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    A NEW ADAPTIVE SUBSPACE MINIMIZATION THREE-TERM CONJUGATE GRADIENT ALGORITHM FOR UNCONSTRAINED OPTIMIZATION
    Keke Zhang, Hongwei Liu, Zexian Liu
    Journal of Computational Mathematics    2021, 39 (2): 159-177.   DOI: 10.4208/jcm.1907-m2018-0173
    Abstract33)      PDF      
    A new adaptive subspace minimization three-term conjugate gradient algorithm with nonmonotone line search is introduced and analyzed in this paper. The search directions are computed by minimizing a quadratic approximation of the objective function on special subspaces, and we also proposed an adaptive rule for choosing different searching directions at each iteration. We obtain a significant conclusion that the each choice of the search directions satisfies the sufficient descent condition. With the used nonmonotone line search, we prove that the new algorithm is globally convergent for general nonlinear functions under some mild assumptions. Numerical experiments show that the proposed algorithm is promising for the given test problem set.
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    QUADRATURE METHODS FOR HIGHLY OSCILLATORY SINGULAR INTEGRALS
    Jing Gao, Marissa Condon, Arieh Iserles, Benjamin Gilvey, Jon Trevelyan
    Journal of Computational Mathematics    2021, 39 (2): 227-260.   DOI: 10.4208/jcm.1911-m2019-0044
    Abstract32)      PDF      
    We address the evaluation of highly oscillatory integrals, with power-law and logarithmic singularities. Such problems arise in numerical methods in engineering. Notably, the evaluation of oscillatory integrals dominates the run-time for wave-enriched boundary integral formulations for wave scattering, and many of these exhibit singularities. We show that the asymptotic behaviour of the integral depends on the integrand and its derivatives at the singular point of the integrand, the stationary points and the endpoints of the integral. A truncated asymptotic expansion achieves an error that decays faster for increasing frequency. Based on the asymptotic analysis, a Filon-type method is constructed to approximate the integral. Unlike an asymptotic expansion, the Filon method achieves high accuracy for both small and large frequency. Complex-valued quadrature involves interpolation at the zeros of polynomials orthogonal to a complex weight function. Numerical results indicate that the complex-valued Gaussian quadrature achieves the highest accuracy when the three methods are compared. However, while it achieves higher accuracy for the same number of function evaluations, it requires significant additional cost of computation of orthogonal polynomials and their zeros.
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    ACCELERATED OPTIMIZATION WITH ORTHOGONALITY CONSTRAINTS
    Jonathan W. Siegel
    Journal of Computational Mathematics    2021, 39 (2): 207-226.   DOI: 10.4208/jcm.1911-m2018-0242
    Abstract25)      PDF      
    We develop a generalization of Nesterov's accelerated gradient descent method which is designed to deal with orthogonality constraints. To demonstrate the effectiveness of our method, we perform numerical experiments which demonstrate that the number of iterations scales with the square root of the condition number, and also compare with existing state-of-the-art quasi-Newton methods on the Stiefel manifold. Our experiments show that our method outperforms existing state-of-the-art quasi-Newton methods on some large, ill-conditioned problems.
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    LOCAL GAUSSIAN-COLLOCATION SCHEME TO APPROXIMATE THE SOLUTION OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS USING VOLTERRA INTEGRAL EQUATIONS
    Pouria Assari, Fatemeh Asadi-Mehregan, Mehdi Dehghan
    Journal of Computational Mathematics    2021, 39 (2): 261-282.   DOI: 10.4208/jcm.1912-m2019-0072
    Abstract22)      PDF      
    This work describes an accurate and effective method for numerically solving a class of nonlinear fractional differential equations. To start the method, we equivalently convert these types of differential equations to nonlinear fractional Volterra integral equations of the second kind by integrating from both sides of them. Afterward, the solution of the mentioned Volterra integral equations can be estimated using the collocation method based on locally supported Gaussian functions. The local Gaussian-collocation scheme estimates the unknown function utilizing a small set of data instead of all points in the solution domain, so the proposed method uses much less computer memory and volume computing in comparison with global cases. We apply the composite non-uniform Gauss-Legendre quadrature formula to estimate singular-fractional integrals in the method. Because of the fact that the proposed scheme requires no cell structures on the domain, it is a meshless method. Furthermore, we obtain the error analysis of the proposed method and demonstrate that the convergence rate of the approach is arbitrarily high. Illustrative examples clearly show the reliability and efficiency of the new technique and confirm the theoretical error estimates.
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    TWO NOVEL GRADIENT METHODS WITH OPTIMAL STEP SIZES
    Harry Oviedo, Oscar Dalmau, Rafael Herrera
    Journal of Computational Mathematics    2021, 39 (3): 375-391.   DOI: 10.4208/jcm.2001-m2018-0205
    Abstract20)      PDF      
    In this work we introduce two new Barzilai and Borwein-like steps sizes for the classical gradient method for strictly convex quadratic optimization problems. The proposed step sizes employ second-order information in order to obtain faster gradient-type methods. Both step sizes are derived from two unconstrained optimization models that involve approximate information of the Hessian of the objective function. A convergence analysis of the proposed algorithm is provided. Some numerical experiments are performed in order to compare the efficiency and effectiveness of the proposed methods with similar methods in the literature. Experimentally, it is observed that our proposals accelerate the gradient method at nearly no extra computational cost, which makes our proposal a good alternative to solve large-scale problems.
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    ERROR ESTIMATES FOR TWO-SCALE COMPOSITE FINITE ELEMENT APPROXIMATIONS OF NONLINEAR PARABOLIC EQUATIONS
    Tamal Pramanick
    Journal of Computational Mathematics    2021, 39 (4): 493-517.   DOI: 10.4208/jcm.2001-m2019-0117
    Abstract20)      PDF (1089KB)(27)      
    We study spatially semidiscrete and fully discrete two-scale composite finite element method for approximations of the nonlinear parabolic equations with homogeneous Dirichlet boundary conditions in a convex polygonal domain in the plane. This new class of finite elements, which is called composite finite elements, was first introduced by Hackbusch and Sauter [Numer. Math., 75 (1997), pp. 447-472] for the approximation of partial differential equations on domains with complicated geometry. The aim of this paper is to introduce an efficient numerical method which gives a lower dimensional approach for solving partial differential equations by domain discretization method. The composite finite element method introduces two-scale grid for discretization of the domain, the coarse-scale and the fine-scale grid with the degrees of freedom lies on the coarse-scale grid only. While the fine-scale grid is used to resolve the Dirichlet boundary condition, the dimension of the finite element space depends only on the coarse-scale grid. As a consequence, the resulting linear system will have a fewer number of unknowns. A continuous, piecewise linear composite finite element space is employed for the space discretization whereas the time discretization is based on both the backward Euler and the Crank-Nicolson methods. We have derived the error estimates in the L ( L 2)-norm for both semidiscrete and fully discrete schemes. Moreover, numerical simulations show that the proposed method is an efficient method to provide a good approximate solution.
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    CAN A CUBIC SPLINE CURVE BE G 3
    Wujie Liu, Xin Li
    Journal of Computational Mathematics    2021, 39 (2): 178-191.   DOI: 10.4208/jcm.1910-m2019-0119
    Abstract19)      PDF      
    This paper proposes a method to construct an G 3 cubic spline curve from any given open control polygon. For any two inner Bézier points on each edge of a control polygon, we can define each Bézier junction point such that the spline curve is G 2-continuous. Then by suitably choosing the inner Bézier points, we can construct a global G 3 spline curve. The curvature combs and curvature plots show the advantage of the G 3 cubic spline curve in contrast with the traditional C 2 cubic spline curve.
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    WELL-CONDITIONED FRAMES FOR HIGH ORDER FINITE ELEMENT METHODS
    Kaibo Hu, Ragnar Winther
    Journal of Computational Mathematics    2021, 39 (3): 333-357.   DOI: 10.4208/jcm.2001-m2018-0078
    Abstract18)      PDF      
    The purpose of this paper is to discuss representations of high order C 0 finite element spaces on simplicial meshes in any dimension. When computing with high order piecewise polynomials the conditioning of the basis is likely to be important. The main result of this paper is a construction of representations by frames such that the associated L 2 condition number is bounded independently of the polynomial degree. To our knowledge, such a representation has not been presented earlier. The main tools we will use for the construction is the bubble transform, introduced previously in[1], and properties of Jacobi polynomials on simplexes in higher dimensions. We also include a brief discussion of preconditioned iterative methods for the finite element systems in the setting of representations by frames.
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    NUMERICAL ANALYSIS OF CRANK-NICOLSON SCHEME FOR THE ALLEN-CAHN EQUATION
    Qianqian Chu, Guanghui Jin, Jihong Shen, Yuanfeng Jin
    Journal of Computational Mathematics    2021, 39 (5): 655-665.   DOI: 10.4208/jcm.2002-m2019-0213
    Abstract17)      PDF (534KB)(14)      
    We consider numerical methods to solve the Allen-Cahn equation using the secondorder Crank-Nicolson scheme in time and the second-order central difference approach in space. The existence of the finite difference solution is proved with the help of Browder fixed point theorem. The difference scheme is showed to be unconditionally convergent in L norm by constructing an auxiliary Lipschitz continuous function. Based on this result, it is demonstrated that the difference scheme preserves the maximum principle without any restrictions on spatial step size and temporal step size. The numerical experiments also verify the reliability of the method.
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    SCHWARZ METHOD FOR FINANCIAL ENGINEERING
    Guangbao Guo, Weidong Zhao
    Journal of Computational Mathematics    2021, 39 (4): 538-555.   DOI: 10.4208/jcm.2003-m2018-0115
    Abstract16)      PDF (398KB)(16)      
    Schwarz method is put forward to solve second order backward stochastic differential equations (2BSDEs) in this work. We will analyze uniqueness, convergence, stability and optimality of the proposed method. Moreover, several simulation results are presented to demonstrate the effectiveness; several applications of the 2BSDEs are investigated. It is concluded from these results that the proposed the method is powerful to calculate the 2BSDEs listing from the financial engineering.
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    CHARACTERISATION OF RATIONAL AND NURBS DEVELOPABLE SURFACES IN COMPUTER AIDED DESIGN
    Leonardo Fernández-Jambrina
    Journal of Computational Mathematics    2021, 39 (4): 556-573.   DOI: 10.4208/jcm.2003-m2019-0226
    Abstract15)      PDF (472KB)(8)      
    In this paper we provide a characterisation of rational developable surfaces in terms of the blossoms of the bounding curves and three rational functions Λ, M, ν. Properties of developable surfaces are revised in this framework. In particular, a closed algebraic formula for the edge of regression of the surface is obtained in terms of the functions Λ, M, ν, which are closely related to the ones that appear in the standard decomposition of the derivative of the parametrisation of one of the bounding curves in terms of the director vector of the rulings and its derivative. It is also shown that all rational developable surfaces can be described as the set of developable surfaces which can be constructed with a constant Λ, M, ν . The results are readily extended to rational spline developable surfaces.
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    MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS
    Xiaocui Li, Xu You
    Journal of Computational Mathematics    2021, 39 (1): 130-146.   DOI: 10.4208/jcm.1911-m2018-0153
    Abstract13)      PDF      
    This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations. The proposed method is based on the mixed finite element method in space and a finite difference scheme in time. The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail. Furthermore, We give the convergence analysis for both semidiscrete and fully discrete schemes and then prove that the numerical solution converges the exact one with order O( h 2 + k), where h and k respectively denote the space step size and the time step size. Finally, numerical examples are presented to demonstrate the effectiveness of our numerical methods.
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    CONVERGENCE OF NUMERICAL SCHEMES FOR A CONSERVATION EQUATION WITH CONVECTION AND DEGENERATE DIFFUSION
    R. Eymard, C. Guichard, Xavier Lhébrard
    Journal of Computational Mathematics    2021, 39 (3): 428-452.   DOI: 10.4208/jcm.2002-m2018-0287
    Abstract13)      PDF      
    The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a θ-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions which can be chosen are restricted by the weak regularity hypotheses on the convection field, owing to the application of a discrete Gronwall lemma and a general result for the time translate in the gradient discretisation setting. Some numerical examples, using both the Control Volume Finite Element method and the Vertex Approximate Gradient scheme, show the role of θ for stabilising the scheme.
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    CONSTRAINT-PRESERVING ENERGY-STABLE SCHEME FOR THE 2D SIMPLIFIED ERICKSEN-LESLIE SYSTEM
    Xuelian Bao, Rui Chen, Hui Zhang
    Journal of Computational Mathematics    2021, 39 (1): 1-21.   DOI: 10.4208/jcm.1906-m2018-0144
    Abstract12)      PDF      
    Here we consider the numerical approximations of the 2D simplified Ericksen-Leslie system. We first rewrite the system and get a new system. For the new system, we propose an easy-to-implement time discretization scheme which preserves the sphere constraint at each node, enjoys a discrete energy law, and leads to linear and decoupled elliptic equations to be solved at each time step. A discrete maximum principle of the scheme in the finite element form is also proved. Some numerical simulations are performed to validate the scheme and simulate the dynamic motion of liquid crystals.
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    ERROR ESTIMATES FOR SPARSE OPTIMAL CONTROL PROBLEMS BY PIECEWISE LINEAR FINITE ELEMENT APPROXIMATION
    Xiaoliang Song, Bo Chen, Bo Yu
    Journal of Computational Mathematics    2021, 39 (3): 471-492.   DOI: 10.4208/jcm.2003-m2017-0213
    Abstract12)      PDF      
    Optimization problems with L 1-control cost functional subject to an elliptic partial differential equation (PDE) are considered. However, different from the finite dimensional l 1-regularization optimization, the resulting discretized L 1-norm does not have a decoupled form when the standard piecewise linear finite element is employed to discretize the continuous problem. A common approach to overcome this difficulty is employing a nodal quadrature formula to approximately discretize the L 1-norm. In this paper, a new discretized scheme for the L 1-norm is presented. Compared to the new discretized scheme for L 1-norm with the nodal quadrature formula, the advantages of our new discretized scheme can be demonstrated in terms of the order of approximation. Moreover, finite element error estimates results for the primal problem with the new discretized scheme for the L 1-norm are provided, which confirms that this approximation scheme will not change the order of error estimates. To solve the new discretized problem, a symmetric Gauss-Seidel based majorized accelerated block coordinate descent(sGS-mABCD) method is introduced to solve it via its dual. The proposed sGS-mABCD algorithm is illustrated at two numerical examples. Numerical results not only confirm the finite element error estimates, but also show that our proposed algorithm is efficient.
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    STRONG CONVERGENCE OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR A CLASS OF SEMILINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH MULTIPLICATIVE NOISE
    Xiaobing Feng, Yukun Li, Yi Zhang
    Journal of Computational Mathematics    2021, 39 (4): 574-598.   DOI: 10.4208/jcm.2003-m2019-0250
    Abstract11)      PDF (2163KB)(10)      
    This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-sided Lipschitz condition. These assumptions are the same ones as the cases where numerical methods for general nonlinear stochastic ordinary differential equations (SODEs) under “minimum assumptions” were studied. As a result, the semilinear SPDEs considered in this paper are a direct generalization of these nonlinear SODEs. There are several difficulties which need to be overcome for this generalization. First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty. It turns out a spatial discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix. In this paper we use a finite element interpolation technique to discretize the nonlinear drift term. Second, in order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the H 1-seminorm of the numerical solution must be established, which are difficult and delicate. A judicious combination of the properties of the drift and diffusion terms and some nontrivial techniques is used in this paper to achieve the goal. Finally, stability estimates for the second and higher order moments of the L 2-norm of the numerical solution are also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant. This is done by utilizing the interpolation theory and the higher moment estimates for the H 1-seminorm of the numerical solution. After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence. Numerical experiment results are also presented to validate the theoretical results and to demonstrate the efficiency of the proposed numerical method.
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    SUB-OPTIMAL CONVERGENCE OF DISCONTINUOUS GALERKIN METHODS WITH CENTRAL FLUXES FOR LINEAR HYPERBOLIC EQUATIONS WITH EVEN DEGREE POLYNOMIAL APPROXIMATIONS
    Yong Liu, Chi-Wang Shu, Mengping Zhang
    Journal of Computational Mathematics    2021, 39 (4): 518-537.   DOI: 10.4208/jcm.2003-m2018-0305
    Abstract11)      PDF (219KB)(11)      
    In this paper, we theoretically and numerically verify that the discontinuous Galerkin (DG) methods with central fluxes for linear hyperbolic equations on non-uniform meshes have sub-optimal convergence properties when measured in the L 2-norm for even degree polynomial approximations. On uniform meshes, the optimal error estimates are provided for arbitrary number of cells in one and multi-dimensions, improving previous results. The theoretical findings are found to be sharp and consistent with numerical results.
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    IMPLICIT-EXPLICIT RUNGE-KUTTA-ROSENBROCK METHODS WITH ERROR ANALYSIS FOR NONLINEAR STIFF DIFFERENTIAL EQUATIONS
    Bin Huang, Aiguo Xiao, Gengen Zhang
    Journal of Computational Mathematics    2021, 39 (4): 599-620.   DOI: 10.4208/jcm.2005-m2019-0238
    Abstract10)      PDF (234KB)(6)      
    Implicit-explicit Runge-Kutta-Rosenbrock methods are proposed to solve nonlinear stiff ordinary differential equations by combining linearly implicit Rosenbrock methods with explicit Runge-Kutta methods. First, the general order conditions up to order 3 are obtained. Then, for the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, the corresponding errors of the implicit-explicit methods are analysed. At last, some numerical examples are given to verify the validity of the obtained theoretical results and the effectiveness of the methods.
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    CONVERGENCE OF LAPLACIAN SPECTRA FROM RANDOM SAMPLES
    Wenqi Tao, Zuoqiang Shi
    Journal of Computational Mathematics    2020, 38 (6): 952-984.   DOI: 10.4208/jcm.2008-m2018-0232
    Abstract10)      PDF (442KB)(13)      
    Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear dimensionality reduction. Graph Laplacian is one widely used discrete laplacian on point cloud. It was previously proved by Belkin and Niyogithat the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Point Integral method (PIM) to solve elliptic equations and corresponding eigenvalue problem on point clouds. In this paper, we prove that the eigenvectors and eigenvalues obtained by PIM converge in the limit of infinitely many random samples. Moreover, estimation of the convergence rate is also given.
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