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KNOT PLACEMENT FOR B-SPLINE CURVE APPROXIMATION VIA l ∞,1-NORM AND DIFFERENTIAL EVOLUTION ALGORITHM Jiaqi Luo, Hongmei Kang, Zhouwang Yang Journal of Computational Mathematics    2022, 40 (4): 589-606.   DOI: 10.4208/jcm.2012-m2020-0203 Abstract （61）      PDF Knowledge map    In this paper, we consider the knot placement problem in B-spline curve approximation. A novel two-stage framework is proposed for addressing this problem. In the first step, the l ∞,1-norm model is introduced for the sparse selection of candidate knots from an initial knot vector. By this step, the knot number is determined. In the second step, knot positions are formulated into a nonlinear optimization problem and optimized by a global optimization algorithm — the differential evolution algorithm (DE). The candidate knots selected in the first step are served for initial values of the DE algorithm. Since the candidate knots provide a good guess of knot positions, the DE algorithm can quickly converge. One advantage of the proposed algorithm is that the knot number and knot positions are determined automatically. Compared with the current existing algorithms, the proposed algorithm finds approximations with smaller fitting error when the knot number is fixed in advance. Furthermore, the proposed algorithm is robust to noisy data and can handle with few data points. We illustrate with some examples and applications. Reference | Related Articles | Metrics Cited: CSCD(1)

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TWO-GRID ALGORITHM OF H 1-GALERKIN MIXED FINITE ELEMENT METHODS FOR SEMILINEAR PARABOLIC INTEGRO-DIFFERENTIAL EQUATIONS Tianliang Hou, Chunmei Liu, Chunlei Dai, Luoping Chen, Yin Yang Journal of Computational Mathematics    2022, 40 (5): 667-685.   DOI: 10.4208/jcm.2101-m2019-0159 Abstract （51）      PDF Knowledge map    In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by H 1-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice h = H 2. Reference | Related Articles | Metrics Cited: CSCD(1)

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A FINITE VOLUME METHOD PRESERVING MAXIMUM PRINCIPLE FOR THE CONJUGATE HEAT TRANSFER PROBLEMS WITH GENERAL INTERFACE CONDITIONS Huifang Zhou, Zhiqiang Sheng, Guangwei Yuan Journal of Computational Mathematics    2023, 41 (3): 345-369.   DOI: 10.4208/jcm.2107-m2020-0266 Abstract （39）      PDF Knowledge map    In this paper, we present a unified finite volume method preserving discrete maximum principle (DMP) for the conjugate heat transfer problems with general interface conditions. We prove the existence of the numerical solution and the DMP-preserving property. Numerical experiments show that the nonlinear iteration numbers of the scheme in [24] increase rapidly when the interfacial coefficients decrease to zero. In contrast, the nonlinear iteration numbers of the unified scheme do not increase when the interfacial coefficients decrease to zero, which reveals that the unified scheme is more robust than the scheme in [24]. The accuracy and DMP-preserving property of the scheme are also verified in the numerical experiments. Reference | Related Articles | Metrics Cited: CSCD(1)

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EXPONENTIAL TIME DIFFERENCING-PADé FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY Haishen Dai, Qiumei Huang, Cheng Wang Journal of Computational Mathematics    2023, 41 (3): 370-394.   DOI: 10.4208/jcm.2107-m2021-0051 Abstract （33）      PDF Knowledge map    In this paper, ETD3-Padé and ETD4-Padé Galerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions. An ETD-based RK is used for time integration of the corresponding equation. To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator, the Padé approach is used for such an exponential operator approximation, which in turn leads to the corresponding ETD-Padé schemes. An unconditional L 2 numerical stability is proved for the proposed numerical schemes, under a global Lipshitz continuity assumption. In addition, optimal rate error estimates are provided, which gives the convergence order of O( k 3 + hr) (ETD3- Padé) or O( k 4 + hr) (ETD4-Padé) in the L 2 norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes. Reference | Related Articles | Metrics Cited: CSCD(1)

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ANALYSIS OF A MULTI-TERM VARIABLE-ORDER TIME-FRACTIONAL DIFFUSION EQUATION AND ITS GALERKIN FINITE ELEMENT APPROXIMATION Huan Liu, Xiangcheng Zheng, Hongfei Fu Journal of Computational Mathematics    2022, 40 (5): 814-834.   DOI: 10.4208/jcm.2102-m2020-0211 Abstract （31）      PDF Knowledge map    In this paper, we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation, and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution. We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t = 0. More precisely, we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0, T ]) in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness, otherwise the solution exhibits the same singular behavior like its constant-order counterpart. Based on these regularity results, we prove optimalorder convergence rate of the Galerkin finite element scheme. Furthermore, we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives. Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme. Reference | Related Articles | Metrics Cited: CSCD(1)

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STRONG CONVERGENCE OF THE EULER-MARUYAMA METHOD FOR A CLASS OF STOCHASTIC VOLTERRA INTEGRAL EQUATIONS Wei Zhang Journal of Computational Mathematics    2022, 40 (4): 607-623.   DOI: 10.4208/jcm.2101-m2020-0070 Abstract （28）      PDF Knowledge map    In this paper, we consider the Euler-Maruyama method for a class of stochastic Volterra integral equations (SVIEs). It is known that the strong convergence order of the Euler-Maruyama method is $\frac{1}{2}$. However, the strong superconvergence order 1 can be obtained for a class of SVIEs if the kernels σ i( t, t) = 0 for i = 1 and 2; otherwise, the strong convergence order is $\frac{1}{2}$ . Moreover, the theoretical results are illustrated by some numerical examples. Reference | Related Articles | Metrics Cited: CSCD(1)

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A DISCRETIZING LEVENBERG-MARQUARDT SCHEME FOR SOLVING NONLIEAR ILL-POSED INTEGRAL EQUATIONS Rong Zhang, Hongqi Yang Journal of Computational Mathematics    2022, 40 (5): 686-710.   DOI: 10.4208/jcm.2101-m2020-0218 Abstract （27）      PDF Knowledge map    To reduce the computational cost, we propose a regularizing modified LevenbergMarquardt scheme via multiscale Galerkin method for solving nonlinear ill-posed problems. Convergence results for the regularizing modified Levenberg-Marquardt scheme for the solution of nonlinear ill-posed problems have been proved. Based on these results, we propose a modified heuristic parameter choice rule to terminate the regularizing modified Levenberg-Marquardt scheme. By imposing certain conditions on the noise, we derive optimal convergence rates on the approximate solution under special source conditions. Numerical results are presented to illustrate the performance of the regularizing modified Levenberg-Marquardt scheme under the modified heuristic parameter choice. Reference | Related Articles | Metrics Cited: CSCD(1)

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PENALTY-FACTOR-FREE STABILIZED NONCONFORMING FINITE ELEMENTS FOR SOLVING STATIONARY NAVIER-STOKES EQUATIONS Linshuang He, Minfu Feng, Qiang Ma Journal of Computational Mathematics    2022, 40 (5): 728-755.   DOI: 10.4208/jcm.2101-m2020-0156 Abstract （26）      PDF Knowledge map    Two nonconforming penalty methods for the two-dimensional stationary Navier-Stokes equations are studied in this paper. These methods are based on the weakly continuous P 1 vector fields and the locally divergence-free (LDF) finite elements, which respectively penalize local divergence and are discontinuous across edges. These methods have no penalty factors and avoid solving the saddle-point problems. The existence and uniqueness of the velocity solution are proved, and the optimal error estimates of the energy norms and L 2-norms are obtained. Moreover, we propose unified pressure recovery algorithms and prove the optimal error estimates of L 2-norm for pressure. We design a unified iterative method for numerical experiments to verify the correctness of the theoretical analysis. Reference | Related Articles | Metrics Cited: CSCD(1)

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STABILIZED NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY ON RECTANGULAR OR CUBIC MESHES Bei Zhang, Jikun Zhao, Minghao Li, Hongru Chen Journal of Computational Mathematics    2022, 40 (6): 865-881.   DOI: 10.4208/jcm.2103-m2020-0143 Abstract （26）      PDF Knowledge map    Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity on rectangular and cubic meshes. Two kinds of penalty terms are introduced in the stabilized mixed formulation, which are the jump penalty term for the displacement and the divergence penalty term for the stress. We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress, where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation. The stabilized mixed method is locking-free. The optimal convergence order is derived in the L 2-norm for stress and in the broken H 1-norm and L 2-norm for displacement. A numerical test is carried out to verify the optimal convergence of the stabilized method. Reference | Related Articles | Metrics Cited: CSCD(1)

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BOUNDARY INTEGRAL EQUATIONS FOR ISOTROPIC LINEAR ELASTICITY Benjamin Stamm, Shuyang Xiang Journal of Computational Mathematics    2022, 40 (6): 835-864.   DOI: 10.4208/jcm.2103-m2019-0031 Abstract （26）      PDF Knowledge map    This articles first investigates boundary integral operators for the three-dimensional isotropic linear elasticity of a biphasic model with piecewise constant Lamé coefficients in the form of a bounded domain of arbitrary shape surrounded by a background material. In the simple case of a spherical inclusion, the vector spherical harmonics consist of eigenfunctions of the single and double layer boundary operators and we provide their spectra. Further, in the case of many spherical inclusions with isotropic materials, each with its own set of Lamé parameters, we propose an integral equation and a subsequent Galerkin discretization using the vector spherical harmonics and apply the discretization to several numerical test cases. Reference | Related Articles | Metrics Cited: CSCD(1)

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AN L ∞ SECOND ORDER CARTESIAN METHOD FOR 3D ANISOTROPIC INTERFACE PROBLEMS Baiying Dong, Xiufeng Feng, Zhilin Li Journal of Computational Mathematics    2022, 40 (6): 882-912.   DOI: 10.4208/jcm.2103-m2020-0107 Abstract （25）      PDF Knowledge map    A second order accurate method in the infinity norm is proposed for general three dimensional anisotropic elliptic interface problems in which the solution and its derivatives, the coefficients, and source terms all can have finite jumps across one or several arbitrary smooth interfaces. The method is based on the 2D finite element-finite difference (FEFD) method but with substantial differences in method derivation, implementation, and convergence analysis. One of challenges is to derive 3D interface relations since there is no invariance anymore under coordinate system transforms for the partial differential equations and the jump conditions. A finite element discretization whose coefficient matrix is a symmetric semi-positive definite is used away from the interface; and the maximum preserving finite difference discretization whose coefficient matrix part is an M-matrix is constructed at irregular elements where the interface cuts through. We aim to get a sharp interface method that can have second order accuracy in the point-wise norm. We show the convergence analysis by splitting errors into several parts. Nontrivial numerical examples are presented to confirm the convergence analysis. Reference | Related Articles | Metrics Cited: CSCD(1)

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A TWO-GRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWO-TERM MIXED SUB-DIFFUSION AND DIFFUSION WAVE EQUATIONS Yanping Chen, Qiling Gu, Qingfeng Li, Yunqing Huang Journal of Computational Mathematics    2022, 40 (6): 936-954.   DOI: 10.4208/jcm.2104-m2020-0332 Abstract （25）      PDF Knowledge map    In this paper, we develop a two-grid method (TGM) based on the FEM for 2D nonlinear time fractional two-term mixed sub-diffusion and diffusion wave equations. A two-grid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order α ∈ (1, 2) and α 1 ∈ (0, 1). Numerical stability and optimal error estimate O( h r+1 + H 2r+2 + τ min-3-α,2-α1}) in L 2-norm are presented for two-grid scheme, where t, H and h are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm. Reference | Related Articles | Metrics Cited: CSCD(1)

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PRIMAL-DUAL PATH-FOLLOWING METHODS AND THE TRUST-REGION UPDATING STRATEGY FOR LINEAR PROGRAMMING WITH NOISY DATA Xinlong Luo, Yiyan Yao Journal of Computational Mathematics    2022, 40 (5): 756-776.   DOI: 10.4208/jcm.2101-m2020-0173 Abstract （25）      PDF Knowledge map    In this article, we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem. For the rank-deficient problem with the small noisy data, we also give the preprocessing method based on the QR decomposition with column pivoting. Then, we prove the global convergence of the new method when the initial point is strictly primal-dual feasible. Finally, for some rankdeficient problems with or without the small noisy data from the NETLIB collection, we compare it with other two popular interior-point methods, i.e. the subroutine pathfollow.m and the built-in subroutine linprog.m of the MATLAB environment. Numerical results show that the new method is more robust than the other two methods for the rank-deficient problem with the small noise data. Reference | Related Articles | Metrics Cited: CSCD(1)

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A NEW HYBRIDIZED MIXED WEAK GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC PROBLEMS* Abdelhamid Zaghdani, Sayed Sayari, Miled EL Hajji Journal of Computational Mathematics    2022, 40 (4): 499-516.   DOI: 10.4208/jcm.2011-m2019-0142 Abstract （24）      PDF Knowledge map    In this paper, a new hybridized mixed formulation of weak Galerkin method is studied for a second order elliptic problem. This method is designed by approximate some operators with discontinuous piecewise polynomials in a shape regular finite element partition. Some discrete inequalities are presented on discontinuous spaces and optimal order error estimations are established. Some numerical results are reported to show super convergence and confirm the theory of the mixed weak Galerkin method. Reference | Related Articles | Metrics Cited: CSCD(1)

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STABLE BOUNDARY CONDITIONS AND DISCRETIZATION FOR P N EQUATIONS Jonas Bünger, Neeraj Sarna, Manuel Torrilhon Journal of Computational Mathematics    2022, 40 (6): 977-1003.   DOI: 10.4208/jcm.2104-m2019-0231 Abstract （24）      PDF Knowledge map    A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the energy transported into the volume by particles entering the volume over time. In this paper, we present boundary conditions (BCs) for the spherical harmonic ( P N) approximation, which ensure that this fundamental energy bound is satisfied by the P N approximation. Our BCs are compatible with the characteristic waves of P N equations and determine the incoming waves uniquely. Both, energy bound and compatibility, are shown on abstract formulations of P N equations and BCs to isolate the necessary structures and properties. The BCs are derived from a Marshak type formulation of BC and base on a non-classical even/odd-classification of spherical harmonic functions and a stabilization step, which is similar to the truncation of the series expansion in the P N method. We show that summation by parts (SBP) finite differences on staggered grids in space and the method of simultaneous approximation terms (SAT) allows to maintain the energy bound also on the semi-discrete level. Reference | Related Articles | Metrics Cited: CSCD(1)

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AN EXPLICIT MULTISTEP SCHEME FOR MEAN-FIELD FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS Yabing Sun, Jie Yang, Weidong Zhao, Tao Zhou Journal of Computational Mathematics    2022, 40 (4): 517-540.   DOI: 10.4208/jcm.2011-m2019-0205 Abstract （23）      PDF Knowledge map    This is one of our series works on numerical methods for mean-field forward backward stochastic differential equations (MFBSDEs). In this work, we propose an explicit multistep scheme for MFBSDEs which is easy to implement, and is of high order rate of convergence. Rigorous error estimates of the proposed multistep scheme are presented. Numerical experiments are carried out to show the efficiency and accuracy of the proposed scheme. Reference | Related Articles | Metrics Cited: CSCD(1)

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WAVEFORM RELAXATION METHODS FOR LIE-GROUP EQUATIONS* Yaolin Jiang, Zhen Miao, Yi Lu Journal of Computational Mathematics    2022, 40 (4): 649-666.   DOI: 10.4208/jcm.2101-m2020-0214 Abstract （21）      PDF Knowledge map    In this paper, we derive and analyse waveform relaxation (WR) methods for solving differential equations evolving on a Lie-group. We present both continuous-time and discrete-time WR methods and study their convergence properties. In the discrete-time case, the novel methods are constructed by combining WR methods with Runge-KuttaMunthe-Kaas (RK-MK) methods. The obtained methods have both advantages of WR methods and RK-MK methods, which simplify the computation by decoupling strategy and preserve the numerical solution of Lie-group equations on a manifold. Three numerical experiments are given to illustrate the feasibility of the new WR methods. Reference | Related Articles | Metrics Cited: CSCD(1)

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REQUIRED NUMBER OF ITERATIONS FOR SPARSE SIGNAL RECOVERY VIA ORTHOGONAL LEAST SQUARES Haifeng Li, Jing Zhang, Jinming Wen, Dongfang Li Journal of Computational Mathematics    2023, 41 (1): 1-17.   DOI: 10.4208/jcm.2104-m2020-0093 Abstract （21）      PDF Knowledge map    In countless applications, we need to reconstruct a K-sparse signal x ∈ R n from noisy measurements y= Φx+ v, where Φ∈ R m×n is a sensing matrix and v ∈ R m is a noise vector. Orthogonal least squares (OLS), which selects at each step the column that results in the most significant decrease in the residual power, is one of the most popular sparse recovery algorithms. In this paper, we investigate the number of iterations required for recovering x with the OLS algorithm. We show that OLS provides a stable reconstruction of all K-sparse signals x in [2.8 K] iterations provided that Φ satisfies the restricted isometry property (RIP). Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013. Reference | Related Articles | Metrics Cited: CSCD(1)

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EXPONENTIAL TIKHONOV REGULARIZATION METHOD FOR SOLVING AN INVERSE SOURCE PROBLEM OF TIME FRACTIONAL DIFFUSION EQUATION Zewen Wang, Shufang Qiu, Shuang Yu, Bin Wu, Wen Zhang Journal of Computational Mathematics    2023, 41 (2): 173-190.   DOI: 10.4208/jcm.2107-m2020-0133 Abstract （19）      PDF Knowledge map    In this paper, we mainly study an inverse source problem of time fractional diffusion equation in a bounded domain with an over-specified terminal condition at a fixed time. A novel regularization method, which we call the exponential Tikhonov regularization method with a parameter γ, is proposed to solve the inverse source problem, and the corresponding convergence analysis is given under a-priori and a-posteriori regularization parameter choice rules. When γ is less than or equal to zero, the optimal convergence rate can be achieved and it is independent of the value of γ. However, when γ is great than zero, the optimal convergence rate depends on the value of γ which is related to the regularity of the unknown source. Finally, numerical experiments are conducted for showing the effectiveness of the proposed exponential regularization method. Reference | Related Articles | Metrics Cited: CSCD(1)

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RECONSTRUCTION OF SPARSE POLYNOMIALS VIA QUASI-ORTHOGONAL MATCHING PURSUIT METHOD Renzhong Feng, Aitong Huang, Ming-Jun Lai, Zhaiming Shen Journal of Computational Mathematics    2023, 41 (1): 18-38.   DOI: 10.4208/jcm.2104-m2020-0250 Abstract （18）      PDF Knowledge map    In this paper, we propose a Quasi-Orthogonal Matching Pursuit (QOMP) algorithm for constructing a sparse approximation of functions in terms of expansion by orthonormal polynomials. For the two kinds of sampled data, data with noises and without noises, we apply the mutual coherence of measurement matrix to establish the convergence of the QOMP algorithm which can reconstruct s-sparse Legendre polynomials, Chebyshev polynomials and trigonometric polynomials in s step iterations. The results are also extended to general bounded orthogonal system including tensor product of these three univariate orthogonal polynomials. Finally, numerical experiments will be presented to verify the e ectiveness of the QOMP method. Reference | Related Articles | Metrics Cited: CSCD(1)