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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
    Abstract57)      PDF (534KB)(68)      
    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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    A FAST COMPACT DIFFERENCE METHOD FOR TWO-DIMENSIONAL NONLINEAR SPACE-FRACTIONAL COMPLEX GINZBURG-LANDAU EQUATIONS
    Lu Zhang, Qifeng Zhang, Hai-wei Sun
    Journal of Computational Mathematics    2021, 39 (5): 708-732.   DOI: 10.4208/jcm.2005-m2020-0029
    Abstract53)      PDF (368KB)(61)      
    This paper focuses on a fast and high-order finite difference method for two-dimensional space-fractional complex Ginzburg-Landau equations. We firstly establish a three-level finite difference scheme for the time variable followed by the linearized technique of the nonlinear term. Then the fourth-order compact finite difference method is employed to discretize the spatial variables. Hence the accuracy of the discretization is $\mathcal{O}$( τ 2 + $h_1^4$ + $h_2^4$) in L 2-norm, where τ is the temporal step-size, both h 1 and h 2 denote spatial mesh sizes in x- and y- directions, respectively. The rigorous theoretical analysis, including the uniqueness, the almost unconditional stability, and the convergence, is studied via the energy argument. Practically, the discretized system holds the block Toeplitz structure. Therefore, the coefficient Toeplitz-like matrix only requires $\mathcal{O}$( M 1 M 2) memory storage, and the matrix-vector multiplication can be carried out in $\mathcal{O}$( M 1 M 2(log M 1 + log M 2)) computational complexity by the fast Fourier transformation, where M 1 and M 2 denote the numbers of the spatial grids in two different directions. In order to solve the resulting Toeplitz-like system quickly, an efficient preconditioner with the Krylov subspace method is proposed to speed up the iteration rate. Numerical results are given to demonstrate the well performance of the proposed method.
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    NUMERICAL ANALYSIS OF A NONLINEAR SINGULARLY PERTURBED DELAY VOLTERRA INTEGRO-DIFFERENTIAL EQUATION ON AN ADAPTIVE GRID
    Libin Liu, Yanping Chen, Ying Liang
    Journal of Computational Mathematics    2022, 40 (2): 258-274.   DOI: 10.4208/jcm.2008-m2020-0063
    Abstract41)      PDF (290KB)(35)      
    In this paper, we study a nonlinear first-order singularly perturbed Volterra integrodifferential equation with delay. This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived. Based on the a priori error bound and mesh equidistribution principle, we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter. The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Furthermore, we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations. Numerical results are provided to demonstrate the effectiveness of our presented monitor function. Meanwhile, it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.
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    ANALYSIS ON A NUMERICAL SCHEME WITH SECOND-ORDER TIME ACCURACY FOR NONLINEAR DIFFUSION EQUATIONS
    Xia Cui, Guangwei Yuan, Fei Zhao
    Journal of Computational Mathematics    2021, 39 (5): 777-800.   DOI: 10.4208/jcm.2007-m2020-0058
    Abstract35)      PDF (276KB)(44)      
    A nonlinear fully implicit finite difference scheme with second-order time evolution for nonlinear diffusion problem is studied. The scheme is constructed with two-layer coupled discretization (TLCD) at each time step. It does not stir numerical oscillation, while permits large time step length, and produces more accurate numerical solutions than the other two well-known second-order time evolution nonlinear schemes, the Crank-Nicolson (CN) scheme and the backward difference formula second-order (BDF2) scheme. By developing a new reasoning technique, we overcome the difficulties caused by the coupled nonlinear discrete diffusion operators at different time layers, and prove rigorously the TLCD scheme is uniquely solvable, unconditionally stable, and has second-order convergence in both space and time. Numerical tests verify the theoretical results, and illustrate its superiority over the CN and BDF2 schemes.
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    A CELL-CENTERED ALE METHOD WITH HLLC-2D RIEMANN SOLVER IN 2D CYLINDRICAL GEOMETRY
    Jian Ren, Zhijun Shen, Wei Yan, Guangwei Yuan
    Journal of Computational Mathematics    2021, 39 (5): 666-692.   DOI: 10.4208/jcm.2005-m2019-0173
    Abstract30)      PDF (3267KB)(36)      
    This paper presents a second-order direct arbitrary Lagrangian Eulerian (ALE) method for compressible flow in two-dimensional cylindrical geometry. This algorithm has half-face fluxes and a nodal velocity solver, which can ensure the compatibility between edge fluxes and the nodal flow intrinsically. In two-dimensional cylindrical geometry, the control volume scheme and the area-weighted scheme are used respectively, which are distinguished by the discretizations for the source term in the momentum equation. The two-dimensional second-order extensions of these schemes are constructed by employing the monotone upwind scheme of conservation law (MUSCL) on unstructured meshes. Numerical results are provided to assess the robustness and accuracy of these new schemes.
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    A GREEDY ALGORITHM FOR SPARSE PRECISION MATRIX APPROXIMATION
    Didi Lv, Xiaoqun Zhang
    Journal of Computational Mathematics    2021, 39 (5): 693-707.   DOI: 10.4208/jcm.2005-m2019-0151
    Abstract28)      PDF (204KB)(24)      
    Precision matrix estimation is an important problem in statistical data analysis. This paper proposes a sparse precision matrix estimation approach, based on CLIME estimator and an efficient algorithm GISS ρ that was originally proposed for l 1 sparse signal recovery in compressed sensing. The asymptotic convergence rate for sparse precision matrix estimation is analyzed with respect to the new stopping criteria of the proposed GISS ρ algorithm. Finally, numerical comparison of GISS ρ with other sparse recovery algorithms, such as ADMM and HTP in three settings of precision matrix estimation is provided and the numerical results show the advantages of the proposed algorithm.
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    MODIFIED ALTERNATING POSITIVE SEMIDEFINITE SPLITTING PRECONDITIONER FOR TIME-HARMONIC EDDY CURRENT MODELS
    Yifen Ke, Changfeng Ma
    Journal of Computational Mathematics    2021, 39 (5): 733-754.   DOI: 10.4208/jcm.2006-m2020-0037
    Abstract28)      PDF (607KB)(27)      
    In this paper, we consider a modified alternating positive semidefinite splitting preconditioner for solving the saddle point problems arising from the finite element discretization of the hybrid formulation of the time-harmonic eddy current model. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are studied for both simple and general topology. Numerical results demonstrate the effectiveness of the proposed preconditioner when it is used to accelerate the convergence rate of Krylov subspace methods such as GMRES.
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    ELLIPTIC RECONSTRUCTION AND A POSTERIORI ERROR ESTIMATES FOR FULLY DISCRETE SEMILINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS
    Ram Manohar, Rajen Kumar Sinha
    Journal of Computational Mathematics    2022, 40 (2): 147-176.   DOI: 10.4208/jcm.2009-m2019-0194
    Abstract27)      PDF (521KB)(39)      
    This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the L (0, T; L 2(Ω))-norm. Finally, a numerical experiment is performed to illustrate the performance of the derived estimators.
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    CONVERGENCE AND MEAN-SQUARE STABILITY OF EXPONENTIAL EULER METHOD FOR SEMI-LINEAR STOCHASTIC DELAY INTEGRO-DIFFERENTIAL EQUATIONS
    Haiyan Yuan
    Journal of Computational Mathematics    2022, 40 (2): 177-204.   DOI: 10.4208/jcm.2010-m2019-0200
    Abstract27)      PDF (272KB)(34)      
    In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order 1/2 and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.
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    STOCHASTIC TRUST-REGION METHODS WITH TRUST-REGION RADIUS DEPENDING ON PROBABILISTIC MODELS
    Xiaoyu Wang, Ya-xiang Yuan
    Journal of Computational Mathematics    2022, 40 (2): 294-334.   DOI: 10.4208/jcm.2012-m2020-0144
    Abstract27)      PDF (1165KB)(31)      
    We present a stochastic trust-region model-based framework in which its radius is related to the probabilistic models. Especially, we propose a specific algorithm termed STRME, in which the trust-region radius depends linearly on the gradient used to define the latest model. The complexity results of the STRME method in nonconvex, convex and strongly convex settings are presented, which match those of the existing algorithms based on probabilistic properties. In addition, several numerical experiments are carried out to reveal the benefits of the proposed methods compared to the existing stochastic trust-region methods and other relevant stochastic gradient methods.
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    A POSTERIORI ERROR ESTIMATES FOR A MODIFIED WEAK GALERKIN FINITE ELEMENT APPROXIMATION OF SECOND ORDER ELLIPTIC PROBLEMS WITH DG NORM
    Yuping Zeng, Feng Wang, Zhifeng Weng, Hanzhang Hu
    Journal of Computational Mathematics    2021, 39 (5): 755-776.   DOI: 10.4208/jcm.2006-m2019-0010
    Abstract22)      PDF (531KB)(28)      
    In this paper, we derive a residual based a posteriori error estimator for a modified weak Galerkin formulation of second order elliptic problems. We prove that the error estimator used for interior penalty discontinuous Galerkin methods still gives both upper and lower bounds for the modified weak Galerkin method, though they have essentially different bilinear forms. More precisely, we prove its reliability and efficiency for the actual error measured in the standard DG norm. We further provide an improved a priori error estimate under minimal regularity assumptions on the exact solution. Numerical results are presented to verify the theoretical analysis.
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    UNCONDITIONALLY OPTIMAL ERROR ESTIMATES OF THE BILINEAR-CONSTANT SCHEME FOR TIME-DEPENDENT NAVIER-STOKES EQUATIONS
    Huaijun Yang, Dongyang Shi
    Journal of Computational Mathematics    2022, 40 (1): 127-146.   DOI: 10.4208/jcm.2007-m2020-0164
    Abstract18)      PDF (9635KB)(10)      
    In this paper, the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme. The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally, while the previous works require certain time-step restrictions. The analysis is based on an iterated timediscrete system, with which the error function is split into a temporal error and a spatial error. The τ-independent ( τ is the time stepsize) error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis, which implies that the numerical solution in L -norm is bounded. Thus optimal error estimates can be obtained in a traditional way. Numerical results are provided to confirm the theoretical analysis.
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    ADAPTIVE AND OPTIMAL POINT-WISE ESTIMATIONS FOR DENSITIES IN GARCH-TYPE MODEL BY WAVELETS
    Cong Wu, Jinru Wang, Xiaochen Zeng
    Journal of Computational Mathematics    2022, 40 (1): 108-126.   DOI: 10.4208/jcm.2007-m2020-0109
    Abstract17)      PDF (221KB)(4)      
    This paper considers adaptive point-wise estimations of density functions in GARCHtype model under the local Hölder condition by wavelet methods. A point-wise lower bound estimation of that model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate, which means that the convergence rate coincides with the lower bound. The non-linear wavelet estimator is introduced for adaptivity, although it is nearly-optimal. However, the non-linear wavelet one depends on an upper bound of the smoothness index of unknown functions, we finally discuss a data driven version without any assumptions on the estimated functions.
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    THEORETICAL ANALYSES ON DISCRETE FORMULAE OF DIRECTIONAL DIFFERENTIALS IN THE FINITE POINT METHOD
    Guixia Lv, Longjun Shen
    Journal of Computational Mathematics    2022, 40 (1): 1-25.   DOI: 10.4208/jcm.2005-m2019-0304
    Abstract17)      PDF (263KB)(11)      
    For the five-point discrete formulae of directional derivatives in the finite point method, overcoming the challenge resulted from scattered point sets and making full use of the explicit expressions and accuracy of the formulae, this paper obtains a number of theoretical results:(1) a concise expression with definite meaning of the complicated directional difference coefficient matrix is presented, which characterizes the correlation between coefficients and the connection between coefficients and scattered geometric characteristics; (2) various expressions of the discriminant function for the solvability of numerical differentials along with the estimation of its lower bound are given, which are the bases for selecting neighboring points and making analysis; (3) the estimations of combinatorial elements and of each element in the directional difference coefficient matrix are put out, which exclude the existence of singularity. Finally, the theoretical analysis results are verified by numerical calculations.
    The results of this paper have strong regularity, which lay the foundation for further research on the finite point method for solving partial differential equations.
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    DEEP RELU NETWORKS OVERCOME THE CURSE OF DIMENSIONALITY FOR GENERALIZED BANDLIMITED FUNCTIONS
    Hadrien Montanelli, Haizhao Yang, Qiang Du
    Journal of Computational Mathematics    2021, 39 (6): 801-815.   DOI: 10.4208/jcm.2007-m2019-0239
    Abstract17)      PDF (380KB)(4)      
    We prove a theorem concerning the approximation of generalized bandlimited multivariate functions by deep ReLU networks for which the curse of the dimensionality is overcome. Our theorem is based on a result by Maurey and on the ability of deep ReLU networks to approximate Chebyshev polynomials and analytic functions efficiently.
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    A FINITE ELEMENT ALGORITHM FOR NEMATIC LIQUID CRYSTAL FLOW BASED ON THE GAUGE-UZAWA METHOD
    Pengzhan Huang, Yinnian He, Ting Li
    Journal of Computational Mathematics    2022, 40 (1): 26-43.   DOI: 10.4208/jcm.2005-m2020-0010
    Abstract15)      PDF (194KB)(12)      
    In this paper, we present a finite element algorithm for the time-dependent nematic liquid crystal flow based on the Gauge-Uzawa method. This algorithm combines the Gauge and Uzawa methods within a finite element variational formulation, which is a fully discrete projection type algorithm, whereas many projection methods have been studied without space discretization. Besides, error estimates for velocity and molecular orientation of the nematic liquid crystal flow are shown. Finally, numerical results are given to show that the presented algorithm is reliable and confirm the theoretical analysis.
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    METRICALLY REGULAR MAPPING AND ITS UTILIZATION TO CONVERGENCE ANALYSIS OF A RESTRICTED INEXACT NEWTON-TYPE METHOD
    Mohammed Harunor Rashid
    Journal of Computational Mathematics    2022, 40 (1): 44-69.   DOI: 10.4208/jcm.2005-m2019-0019
    Abstract14)      PDF (275KB)(5)      
    In the present paper, we study the restricted inexact Newton-type method for solving the generalized equation 0 ∈ f( x)+ F ( x), where X and Y are Banach spaces, f: XY is a Fréchet differentiable function and F: XY is a set-valued mapping with closed graph. We establish the convergence criteria of the restricted inexact Newton-type method, which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Fréchet derivative of f. Indeed, we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Fréchet derivative of f is continuous and Lipschitz continuous as well as f + F is metrically regular. An application of this method to variational inequality is given. In addition, a numerical experiment is given which illustrates the theoretical result.
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    CONSTRUCTION OF CUBATURE FORMULAS VIA BIVARIATE QUADRATIC SPLINE SPACES OVER NON-UNIFORM TYPE-2 TRIANGULATION
    Jiang Qian, Xiquan Shi, Jinming Wu, Dianxuan Gong
    Journal of Computational Mathematics    2022, 40 (2): 205-230.   DOI: 10.4208/jcm.2008-m2020-0077
    Abstract14)      PDF (232KB)(21)      
    In this paper, matrix representations of the best spline quasi-interpolating operator over triangular sub-domains in S 2 1mn (2)), and coefficients of splines in terms of B-net are calculated firstly. Moreover, by means of coefficients in terms of B-net, computation of bivariate numerical cubature over triangular sub-domains with respect to variables x and y is transferred into summation of coefficients of splines in terms of B-net. Thus concise bivariate cubature formulas are constructed over rectangular sub-domain. Furthermore, by means of module of continuity and max-norms, error estimates for cubature formulas are derived over both sub-domains and the domain.
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    THE RANDOM BATCH METHOD FOR N-BODY QUANTUM DYNAMICS
    François Golse, Shi Jin, Thierry Paul
    Journal of Computational Mathematics    2021, 39 (6): 808-922.   DOI: 10.4208/jcm.2107-m2020-0306
    Abstract13)      PDF (545KB)(0)      
    This paper discusses a numerical method for computing the evolution of large interacting system of quantum particles. The idea of the random batch method is to replace the total interaction of each particle with the N - 1 other particles by the interaction with p << N particles chosen at random at each time step, multiplied by ( N - 1)/p. This reduces the computational cost of computing the interaction potential per time step from O( N 2) to O( N). For simplicity, we consider only in this work the case p=1-in other words, we assume that N is even, and that at each time step, the N particles are organized in N/2 pairs, with a random reshuffling of the pairs at the beginning of each time step. We obtain a convergence estimate for the Wigner transform of the single-particle reduced density matrix of the particle system at time t that is both uniform in N > 1 and independent of the Planck constant ħ. The key idea is to use a new type of distance on the set of quantum states that is reminiscent of the Wasserstein distance of exponent 1 (or Monge-Kantorovich-Rubinstein distance) on the set of Borel probability measures on R d used in the context of optimal transport.
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    CONVERGENCE OF THE WEIGHTED NONLOCAL LAPLACIAN ON RANDOM POINT CLOUD
    Zuoqiang Shi, Bao Wang
    Journal of Computational Mathematics    2021, 39 (6): 865-879.   DOI: 10.4208/jcm.2104-m2020-0309
    Abstract13)      PDF (206KB)(7)      
    We analyze the convergence of the weighted nonlocal Laplacian (WNLL) on the high dimensional randomly distributed point cloud. Our analysis reveals the importance of the scaling weight, μ~| P|/| S|with| P|and| S|being the number of entire and labeled data, respectively, in WNLL. The established result gives a theoretical foundation of the WNLL for high dimensional data interpolation.
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