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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
    Abstract30)      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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    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
    Abstract27)      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
    Abstract21)      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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    ACCELERATED OPTIMIZATION WITH ORTHOGONALITY CONSTRAINTS
    Jonathan W. Siegel
    Journal of Computational Mathematics    2021, 39 (2): 207-226.   DOI: 10.4208/jcm.1911-m2018-0242
    Abstract17)      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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    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
    Abstract17)      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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    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
    Abstract15)      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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    ACCURATE AND EFFICIENT IMAGE RECONSTRUCTION FROM MULTIPLE MEASUREMENTS OF FOURIER SAMPLES
    T. Scarnati, Anne Gelb
    Journal of Computational Mathematics    2020, 38 (5): 797-826.   DOI: 10.4208/jcm.2002-m2019-0192
    Abstract14)      PDF (2220KB)(39)      
    Several problems in imaging acquire multiple measurement vectors (MMVs) of Fourier samples for the same underlying scene. Image recovery techniques from MMVs aim to exploit the joint sparsity across the measurements in the sparse domain. This is typically accomplished by extending the use of l 1 regularization of the sparse domain in the single measurement vector (SMV) case to using l 2,1 regularization so that the "jointness" can be accounted for. Although effective, the approach is inherently coupled and therefore computationally inefficient. The method also does not consider current approaches in the SMV case that use spatially varying weighted l 1 regularization term. The recently introduced variance based joint sparsity (VBJS) recovery method uses the variance across the measurements in the sparse domain to produce a weighted MMV method that is more accurate and more efficient than the standard l 2,1 approach. The efficiency is due to the decoupling of the measurement vectors, with the increased accuracy resulting from the spatially varying weight. Motivated by these results, this paper introduces a new technique to even further reduce computational cost by eliminating the requirement to first approximate the underlying image in order to construct the weights. Eliminating this preprocessing step moreover reduces the amount of information lost from the data, so that our method is more accurate. Numerical examples provided in the paper verify these benefits.
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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
    Abstract12)      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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    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
    Abstract11)      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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    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
    Abstract10)      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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    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
    Abstract10)      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
    Abstract10)      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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    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
    Abstract10)      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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    EFFICIENT AND ACCURATE CHEBYSHEV DUAL-PETROV-GALERKIN METHODS FOR ODD-ORDER DIFFERENTIAL EQUATIONS
    Xuhong Yu, Lusha Jin, Zhongqing Wang
    Journal of Computational Mathematics    2021, 39 (1): 43-62.   DOI: 10.4208/jcm.1907-m2018-0285
    Abstract8)      PDF      
    Efficient and accurate Chebyshev dual-Petrov-Galerkin methods for solving first-order equation, third-order equation, third-order KdV equation and fifth-order Kawahara equation are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions are expanded as an infinite and truncated Fourier-like series, respectively. Numerical experiments illustrate the effectiveness of the suggested approaches.
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    SUPERCONVERGENCE ANALYSIS OF LOW ORDER NONCONFORMING MIXED FINITE ELEMENT METHODS FOR TIME-DEPENDENT NAVIER-STOKES EQUATIONS
    Huaijun Yang, Dongyang Shi, Qian Liu
    Journal of Computational Mathematics    2021, 39 (1): 63-80.   DOI: 10.4208/jcm.1907-m2018-0263
    Abstract8)      PDF      
    In this paper, the superconvergence properties of the time-dependent Navier-Stokes equations are investigated by a low order nonconforming mixed finite element method (MFEM). In terms of the integral identity technique, the superclose error estimates for both the velocity in broken H 1-norm and the pressure in L 2-norm are first obtained, which play a key role to bound the numerical solution in L -norm. Then the corresponding global superconvergence results are derived through a suitable interpolation postprocessing approach. Finally, some numerical results are provided to demonstrated the theoretical analysis.
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    A MIXED VIRTUAL ELEMENT METHOD FOR THE BOUSSINESQ PROBLEM ON POLYGONAL MESHES
    Gabriel N. Gatica, Mauricio Munar, Filánder A. Sequeira
    Journal of Computational Mathematics    2021, 39 (3): 392-427.   DOI: 10.4208/jcm.2001-m2019-0187
    Abstract7)      PDF      
    In this work we introduce and analyze a mixed virtual element method (mixed-VEM) for the two-dimensional stationary Boussinesq problem. The continuous formulation is based on the introduction of a pseudostress tensor depending nonlinearly on the velocity, which allows to obtain an equivalent model in which the main unknowns are given by the aforementioned pseudostress tensor, the velocity and the temperature, whereas the pressure is computed via a postprocessing formula. In addition, an augmented approach together with a fixed point strategy is used to analyze the well-posedness of the resulting continuous formulation. Regarding the discrete problem, we follow the approach employed in a previous work dealing with the Navier-Stokes equations, and couple it with a VEM for the convection-diffusion equation modelling the temperature. More precisely, we use a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of ${\Bbb H}$(div) and H 1, respectively, whereas a VEM is proposed to approximate the temperature on a virtual element subspace of H 1. In this way, we make use of the L 2-orthogonal projectors onto suitable polynomial spaces, which allows the explicit integration of the terms that appear in the bilinear and trilinear forms involved in the scheme for the fluid equations. On the other hand, in order to manipulate the bilinear form associated to the heat equations, we define a suitable projector onto a space of polynomials to deal with the fact that the diffusion tensor, which represents the thermal conductivity, is variable. Next, the corresponding solvability analysis is performed using again appropriate fixed-point arguments. Further, Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure. The corresponding rates of convergence are also established. Finally, several numerical examples illustrating the performance of the mixed-VEM scheme and confirming these theoretical rates are presented.
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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
    Abstract6)      PDF (442KB)(10)      
    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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    A HYBRID EXPLICIT-IMPLICIT SCHEME FOR THE TIME-DEPENDENT WIGNER EQUATION
    Haiyan Jiang, Tiao Lu, Xu Yin
    Journal of Computational Mathematics    2021, 39 (1): 22-42.   DOI: 10.4208/jcm.1906-m2018-0081
    Abstract5)      PDF      
    This paper designs a hybrid scheme based on finite difference methods and a spectral method for the time-dependent Wigner equation, and gives the error analysis for the full discretization of its initial value problem. An explicit-implicit time-splitting scheme is used for time integration and the second-order upwind finite difference scheme is used to discretize the advection term. The consistence error and the stability of the full discretization are analyzed. A Fourier spectral method is used to approximate the pseudo-differential operator term and the corresponding error is studied in detail. The final convergence result shows clearly how the regularity of the solution affects the convergence order of the proposed scheme. Numerical results are presented for confirming the sharpness of the analysis. The scattering effects of a Gaussian wave packet tunneling through a Gaussian potential barrier are investigated. The evolution of the density function shows that a larger portion of the wave is reflected when the height and the width of the barrier increase.
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    DISCONTINUOUS GALERKIN METHODS AND THEIR ADAPTIVITY FOR THE TEMPERED FRACTIONAL (CONVECTION) DIFFUSION EQUATIONS
    Xudong Wang, Weihua Deng
    Journal of Computational Mathematics    2020, 38 (6): 839-867.   DOI: 10.4208/jcm.1906-m2019-0040
    Abstract5)      PDF (416KB)(5)      
    This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations. The DG schemes with interior penalty for the diffusion term and numerical flux for the convection term are used to solve the equations, and the detailed stability and convergence analyses are provided. Based on the derived posteriori error estimates, the local error indicator is designed. The theoretical results and the effectiveness of the adaptive DG methods are, respectively, verified and displayed by the extensive numerical experiments. The strategy of designing adaptive schemes presented in this paper works for the general PDEs with fractional operators.
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    A MULTIDIMENSIONAL FILTER SQP ALGORITHM FOR NONLINEAR PROGRAMMING
    Wenjuan Xue, Weiai Liu
    Journal of Computational Mathematics    2020, 38 (5): 683-704.   DOI: 10.4208/jcm.1903-m2018-0072
    Abstract5)      PDF (305KB)(5)      
    We propose a multidimensional filter SQP algorithm. The multidimensional filter technique proposed by Gould et al.[SIAM J. Optim., 2005] is extended to solve constrained optimization problems. In our proposed algorithm, the constraints are partitioned into several parts, and the entry of our filter consists of these different parts. Not only the criteria for accepting a trial step would be relaxed, but the individual behavior of each part of constraints is considered. One feature is that the undesirable link between the objective function and the constraint violation in the filter acceptance criteria disappears. The other is that feasibility restoration phases are unnecessary because a consistent quadratic programming subproblem is used. We prove that our algorithm is globally convergent to KKT points under the constant positive generators (CPG) condition which is weaker than the well-known Mangasarian-Fromovitz constraint qualification (MFCQ) and the constant positive linear dependence (CPLD). Numerical results are presented to show the efficiency of the algorithm.
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