中国科学院数学与系统科学研究院期刊网

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  • Zewen Wang, Shufang Qiu, Shuang Yu, Bin Wu, Wen Zhang
    Journal of Computational Mathematics. 2023, 41(2): 173-190. https://doi.org/10.4208/jcm.2107-m2020-0133
    CSCD(1)
    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.
  • Haishen Dai, Qiumei Huang, Cheng Wang
    Journal of Computational Mathematics. 2023, 41(3): 370-394. https://doi.org/10.4208/jcm.2107-m2021-0051
    CSCD(1)
    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 L2 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(k3 + hr) (ETD3- Padé) or O(k4 + hr) (ETD4-Padé) in the L2 norm, respectively. Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.
  • Huifang Zhou, Zhiqiang Sheng, Guangwei Yuan
    Journal of Computational Mathematics. 2023, 41(3): 345-369. https://doi.org/10.4208/jcm.2107-m2020-0266
    CSCD(1)
    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.
  • Xiaolin Li
    Journal of Computational Mathematics. 2023, 41(3): 502-524. https://doi.org/10.4208/jcm.2201-m2021-0361
    CSCD(1)
    Numerical integration poses greater challenges in Galerkin meshless methods than finite element methods owing to the non-polynomial feature of meshless shape functions. The reproducing kernel gradient smoothing integration (RKGSI) is one of the optimal numerical integration techniques in Galerkin meshless methods with minimum integration points. In this paper, properties, quadrature rules and the effect of the RKGSI on meshless methods are analyzed. The existence, uniqueness and error estimates of the solution of Galerkin meshless methods under numerical integration with the RKGSI are established. A procedure on how to choose quadrature rules to recover the optimal convergence rate is presented.
  • Yuan Li, Xuewei Cui
    Journal of Computational Mathematics. 2023, 41(2): 211-223. https://doi.org/10.4208/jcm.2107-m2020-0243
    CSCD(1)
    This paper aims to study a second-order semi-implicit BDF finite element scheme for the Kuramoto-Tsuzuki equations in two dimensional and three dimensional spaces. The proposed scheme is stable and the nonlinear term is linearized by the extrapolation technique. Moreover, we prove that the error estimate in L2-norm is unconditionally optimal which means that there has not any restriction on the time step and the mesh size. Finally, numerical results are displayed to illustrate our theoretical analysis
  • Xianmin Xu
    Journal of Computational Mathematics. 2023, 41(2): 191-210. https://doi.org/10.4208/jcm.2107-m2020-0227
    CSCD(1)
    By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.
  • Yayun Fu, Wenjun Cai, Yushun Wang
    Journal of Computational Mathematics. 2023, 41(5): 797-816. https://doi.org/10.4208/jcm.2111-m2020-0177
    The main objective of this paper is to present an efficient structure-preserving scheme, which is based on the idea of the scalar auxiliary variable approach, for solving the twodimensional space-fractional nonlinear Schrödinger equation. First, we reformulate the equation as an canonical Hamiltonian system, and obtain a new equivalent system via introducing a scalar variable. Then, we construct a semi-discrete energy-preserving scheme by using the Fourier pseudo-spectral method to discretize the equivalent system in space direction. After that, applying the Crank-Nicolson method on the temporal direction gives a linearly-implicit scheme in the fully-discrete version. As expected, the proposed scheme can preserve the energy exactly and more efficient in the sense that only decoupled equations with constant coefficients need to be solved at each time step. Finally, numerical experiments are provided to demonstrate the efficiency and conservation of the scheme.
  • Yuhuan Yuan, Huazhong Tang
    Journal of Computational Mathematics. 2023, 41(2): 305-324. https://doi.org/10.4208/jcm.2201-m2020-0288
    CSCD(1)
    This paper continues to study the explicit two-stage fourth-order accurate time discretizations [5, 7]. By introducing variable weights, we propose a class of more general explicit one-step two-stage time discretizations, which are different from the existing methods, e.g. the Euler methods, Runge-Kutta methods, and multistage multiderivative methods etc. We study the absolute stability, the stability interval, and the intersection between the imaginary axis and the absolute stability region. Our results show that our two-stage time discretizations can be fourth-order accurate conditionally, the absolute stability region of the proposed methods with some special choices of the variable weights can be larger than that of the classical explicit fourth-or fifth-order Runge-Kutta method, and the interval of absolute stability can be almost twice as much as the latter. Several numerical experiments are carried out to demonstrate the performance and accuracy as well as the stability of our proposed methods.
  • Datong Zhou, Jing Chen, Hao Wu, Dinghui Yang, Lingyun Qiu
    Journal of Computational Mathematics. 2023, 41(3): 437-458. https://doi.org/10.4208/jcm.2109-m2021-0045
    CSCD(1)
    In this paper, we apply the Wasserstein-Fisher-Rao (WFR) metric from the unbalanced optimal transport theory to the earthquake location problem. Compared with the quadratic Wasserstein (W2) metric from the classical optimal transport theory, the advantage of this method is that it retains the important amplitude information as a new constraint, which avoids the problem of the degeneration of the optimization objective function near the real earthquake hypocenter and origin time. As a result, the deviation of the global minimum of the optimization objective function based on the WFR metric from the true solution can be much smaller than the results based on the W2 metric when there exists strong data noise. Thus, we develop an accurate earthquake location method under strong data noise. Many numerical experiments verify our conclusions.
  • Yonghui Bo, Wenjun Cai, Yushun Wang
    Journal of Computational Mathematics. 2023, 41(3): 395-414. https://doi.org/10.4208/jcm.2108-m2021-0076
    CSCD(1)
    In this paper, we systematically construct two classes of structure-preserving schemes with arbitrary order of accuracy for canonical Hamiltonian systems. The one class is the symplectic scheme, which contains two new families of parameterized symplectic schemes that are derived by basing on the generating function method and the symmetric composition method, respectively. Each member in these schemes is symplectic for any fixed parameter. A more general form of generating functions is introduced, which generalizes the three classical generating functions that are widely used to construct symplectic algorithms. The other class is a novel family of energy and quadratic invariants preserving schemes, which is devised by adjusting the parameter in parameterized symplectic schemes to guarantee energy conservation at each time step. The existence of the solutions of these schemes is verified. Numerical experiments demonstrate the theoretical analysis and conservation of the proposed schemes.
  • Lexing Ying
    Journal of Computational Mathematics. 2023, 41(3): 542-550. https://doi.org/10.4208/jcm.2211-m2022-0172
    CSCD(1)
    This note introduces a method for sampling Ising models with mixed boundary conditions. As an application of annealed importance sampling and the Swendsen-Wang algorithm, the method adopts a sequence of intermediate distributions that keeps the temperature fixed but turns on the boundary condition gradually. The numerical results show that the variance of the sample weights is relatively small.
  • Yanping Chen, Xinliang Liu, Jiaoyan Zeng, Lei Zhang
    Journal of Computational Mathematics. 2023, 41(5): 841-865. https://doi.org/10.4208/jcm.2112-m2021-0123
    This paper concerns the convex optimal control problem governed by multiscale elliptic equations with arbitrarily rough L coefficients, which has not only complex coupling between nonseparable scales and nonlinearity, but also important applications in composite materials and geophysics. We use one of the recently developed numerical homogenization techniques, the so-called Rough Polyharmonic Splines (RPS) and its generalization (GRPS) for the efficient resolution of the elliptic operator on the coarse scale. Those methods have optimal convergence rate which do not rely on the regularity of the coefficients nor the concepts of scale-separation or periodicity. As the iterative solution of the nonlinearly coupled OCP-OPT formulation for the optimal control problem requires solving the corresponding (state and co-state) multiscale elliptic equations many times with different right hand sides, numerical homogenization approach only requires one-time pre-computation on the fine scale and the following iterations can be done with computational cost proportional to coarse degrees of freedom. Numerical experiments are presented to validate the theoretical analysis.
  • Jing Chen, Zhaojie Zhou, Huanzhen Chen, Hong Wang
    Journal of Computational Mathematics. 2023, 41(5): 817-840. https://doi.org/10.4208/jcm.2112-m2021-0204
    In this article, we propose a new finite element space Λh for the expanded mixed finite element method (EMFEM) for second-order elliptic problems to guarantee its computing capability and reduce the computation cost. The new finite element space Λh is designed in such a way that the strong requirement VhΛh in [9] is weakened to {vhVh; divvh=0} ⊂ Λh so that it needs fewer degrees of freedom than its classical counterpart. Furthermore, the new Λh coupled with the Raviart-Thomas space satisfies the inf-sup condition, which is crucial to the computation of mixed methods for its close relation to the behavior of the smallest nonzero eigenvalue of the stiff matrix, and thus the existence, uniqueness and optimal approximate capability of the EMFEM solution are proved for rectangular partitions in $\mathbb{R}^d$, d=2, 3 and for triangular partitions in $\mathbb{R}^2$. Also, the solvability of the EMFEM for triangular partition in $\mathbb{R}^3$ can be directly proved without the inf-sup condition. Numerical experiments are conducted to confirm these theoretical findings.
  • Zhiyun Yu, Dongyang Shi, Huiqing Zhu
    Journal of Computational Mathematics. 2023, 41(4): 569-587. https://doi.org/10.4208/jcm.2107-m2021-0114
    A low order nonconforming mixed finite element method (FEM) is established for the fully coupled non-stationary incompressible magnetohydrodynamics (MHD) problem in a bounded domain in 3D. The lowest order finite elements on tetrahedra or hexahedra are chosen to approximate the pressure, the velocity field and the magnetic field, in which the hydrodynamic unknowns are approximated by inf-sup stable finite element pairs and the magnetic field by H1(?)-conforming finite elements, respectively. The existence and uniqueness of the approximate solutions are shown. Optimal order error estimates of L2(H1)-norm for the velocity field, L2(L2)-norm for the pressure and the broken L2(H1)-norm for the magnetic field are derived.
  • Faouzi Triki, Tao Yin
    Journal of Computational Mathematics. 2023, 41(3): 483-501. https://doi.org/10.4208/jcm.2111-m2021-0093
    CSCD(1)
    This paper concerns the reconstruction of a scalar coefficient of a second-order elliptic equation in divergence form posed on a bounded domain from internal data. This problem finds applications in multi-wave imaging, greedy methods to approximate parameterdependent elliptic problems, and image treatment with partial differential equations. We first show that the inverse problem for smooth coefficients can be rewritten as a linear transport equation. Assuming that the coefficient is known near the boundary, we study the well-posedness of associated transport equation as well as its numerical resolution using discontinuous Galerkin method. We propose a regularized transport equation that allow us to derive rigorous convergence rates of the numerical method in terms of the order of the polynomial approximation as well as the regularization parameter. We finally provide numerical examples for the inversion assuming a lower regularity of the coefficient, and using synthetic data.
  • Victor Churchill
    Journal of Computational Mathematics. 2023, 41(2): 246-262. https://doi.org/10.4208/jcm.2110-m2021-0157
    CSCD(1)
    This paper presents an application of the sparse Bayesian learning (SBL) algorithm to linear inverse problems with a high order total variation (HOTV) sparsity prior. For the problem of sparse signal recovery, SBL often produces more accurate estimates than maximum a posteriori estimates, including those that use l1 regularization. Moreover, rather than a single signal estimate, SBL yields a full posterior density estimate which can be used for uncertainty quantification. However, SBL is only immediately applicable to problems having a direct sparsity prior, or to those that can be formed via synthesis. This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis, and then utilizes SBL. This expands the class of problems available to Bayesian learning to include, e.g., inverse problems dealing with the recovery of piecewise smooth functions or signals from data. Numerical examples are provided to demonstrate how this new technique is effectively employed.
  • Hong-lin Liao, Tao Tang, Tao Zhou
    Journal of Computational Mathematics. 2023, 41(2): 325-344. https://doi.org/10.4208/jcm.2207-m2022-0020
    CSCD(1)
    This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see, e.g., [SIAM J. Numer. Anal., 58:2294-2314] and [Math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the L2 norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical results.
  • Huijun Fan, Yanmin Zhao, Fenling Wang, Yanhua Shi, Fawang Liu
    Journal of Computational Mathematics. 2023, 41(3): 459-482. https://doi.org/10.4208/jcm.2110-m2021-0180
    CSCD(1)
    By employing EQ1rot nonconforming finite element, the numerical approximation is presented for multi-term time-fractional mixed sub-diffusion and diffusion-wave equation on anisotropic meshes. Comparing with the multi-term time-fractional sub-diffusion equation or diffusion-wave equation, the mixed case contains a special time-space coupled derivative, which leads to many difficulties in numerical analysis. Firstly, a fully discrete scheme is established by using nonconforming finite element method (FEM) in spatial direction and L1 approximation coupled with Crank-Nicolson (L1-CN) scheme in temporal direction. Furthermore, the fully discrete scheme is proved to be unconditional stable. Besides, convergence and superclose results are derived by using the properties of EQ1rot nonconforming finite element. What's more, the global superconvergence is obtained via the interpolation postprocessing technique. Finally, several numerical results are provided to demonstrate the theoretical analysis on anisotropic meshes.
  • Dongyang Shi, Houchao Zhang
    Journal of Computational Mathematics. 2023, 41(2): 224-245. https://doi.org/10.4208/jcm.2108-m2020-0324
    CSCD(1)
    The focus of this paper is on a linearized backward differential formula (BDF) scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations (KGSEs) with damping mechanism. Optimal error estimates and superconvergence results are proved without any time-step restriction condition for the proposed scheme. The proof consists of three ingredients. First, a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong norms. Second, optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear terms. Third, by virtue of the relationship between the Ritz projection and the interpolation, as well as a so-called "lifting" technique, the superconvergence behavior of order O(h2 + τ2) in H1-norm for the original variables are deduced. Finally, a numerical experiment is conducted to confirm our theoretical analysis. Here, h is the spatial subdivision parameter, and τ is the time step.
  • Yongxia Hao, Ting Li
    Journal of Computational Mathematics. 2023, 41(4): 551-568. https://doi.org/10.4208/jcm.2106-m2021-0050
    In this paper, we present a method for generating Bézier surfaces from the boundary information based on a general second order functional and a third order functional associated with the triharmonic equation. By solving simple linear equations, the internal control points of the resulting Bézier surface can be obtained as linear combinations of the given boundary control points. This is a generalization of previous works on Plateau-Bézier problem, harmonic, biharmonic and quasi-harmonic Bézier surfaces. Some representative examples show the effectiveness of the presented method.
  • Qiang Han, Shaolin Ji
    Journal of Computational Mathematics. 2023, 41(2): 287-304. https://doi.org/10.4208/jcm.2112-m2019-0289
    CSCD(1)
    In this paper, a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations (BSDEs). A necessary and sufficient condition is given to judge the $\mathbb{L}$ 2-stability of our numerical schemes. This stochastic linear two-step method possesses a family of 3-order convergence schemes in the sense of strong stability. The coefficients in the numerical methods are inferred based on the constraints of strong stability and n-order accuracy (n∈$\mathbb{N}$ +). Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.
  • Pengcong Mu, Weiying Zheng
    Journal of Computational Mathematics. 2023, 41(5): 909-932. https://doi.org/10.4208/jcm.2206-m2021-0353
    In this paper, we propose a positivity-preserving finite element method for solving the three-dimensional quantum drift-diffusion model. The model consists of five nonlinear elliptic equations, and two of them describe quantum corrections for quasi-Fermi levels. We propose an interpolated-exponential finite element (IEFE) method for solving the two quantum-correction equations. The IEFE method always yields positive carrier densities and preserves the positivity of second-order differential operators in the Newton linearization of quantum-correction equations. Moreover, we solve the two continuity equations with the edge-averaged finite element (EAFE) method to reduce numerical oscillations of quasi-Fermi levels. The Poisson equation of electrical potential is solved with standard Lagrangian finite elements. We prove the existence of solution to the nonlinear discrete problem by using a fixed-point iteration and solving the minimum problem of a new discrete functional. A Newton method is proposed to solve the nonlinear discrete problem. Numerical experiments for a three-dimensional nano-scale FinFET device show that the Newton method is robust for source-to-gate bias voltages up to 9V and source-to-drain bias voltages up to 10V.
  • Baoli Yin, Yang Liu, Hong Li, Zhimin Zhang
    Journal of Computational Mathematics. 2023, 41(5): 980-1002. https://doi.org/10.4208/jcm.2210-m2021-0257
    A simple criterion is studied for the first time for identifying the discrete energy dissipation of the Crank-Nicolson scheme for Maxwell's equations in a Cole-Cole dispersive medium. Several numerical formulas that approximate the time fractional derivatives are investigated based on this criterion, including the L1 formula, the fractional BDF-2, and the shifted fractional trapezoidal rule (SFTR). Detailed error analysis is provided within the framework of time domain mixed finite element methods for smooth solutions. The convergence results and discrete energy dissipation law are confirmed by numerical tests. For nonsmooth solutions, the method SFTR can still maintain the optimal convergence order at a positive time on uniform meshes. Authors believe this is the first appearance that a second-order time-stepping method can restore the optimal convergence rate for Maxwell's equations in a Cole-Cole dispersive medium regardless of the initial singularity of the solution.
  • Yanfang Zhang
    Journal of Computational Mathematics. 2023, 41(3): 415-436. https://doi.org/10.4208/jcm.2109-m2020-0099
    CSCD(1)
    In this paper, we consider the generalized Nash equilibrium with shared constraints in the stochastic environment, and we call it the stochastic generalized Nash equilibrium. The stochastic variational inequalities are employed to solve this kind of problems, and the expected residual minimization model and the conditional value-at-risk formulations defined by the residual function for the stochastic variational inequalities are discussed. We show the risk for different kinds of solutions for the stochastic generalized Nash equilibrium by the conditional value-at-risk formulations. The properties of the stochastic quadratic generalized Nash equilibrium are shown. The smoothing approximations for the expected residual minimization formulation and the conditional value-at-risk formulation are employed. Moreover, we establish the gradient consistency for the measurable smoothing functions and the integrable functions under some suitable conditions, and we also analyze the properties of the formulations. Numerical results for the applications arising from the electricity market model illustrate that the solutions for the stochastic generalized Nash equilibrium given by the ERM model have good properties, such as robustness, low risk and so on.
  • Xiaoqiang Yan, Xu Qian, Hong Zhang, Songhe Song, Xiujun Cheng
    Journal of Computational Mathematics. 2023, 41(4): 643-662. https://doi.org/10.4208/jcm.2109-m2021-0020
    Block boundary value methods (BBVMs) are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation (DDAESP). It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP. Besides, whenever the classic Lipschitz conditions are satisfied, the extended BBVMs are preconsistent and pth order consistent. Moreover, through some numerical examples, the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.
  • Wenbo Li, Jicheng Li, Xuenian Liu
    Journal of Computational Mathematics. 2023, 41(5): 866-878. https://doi.org/10.4208/jcm.2201-m2019-0145
    In this paper, we develop an active set identification technique. By means of the active set technique, we present an active set adaptive monotone projected Barzilai-Borwein method (ASAMPBB) for solving nonnegative matrix factorization (NMF) based on the alternating nonnegative least squares framework, in which the Barzilai-Borwein (BB) step sizes can be adaptively picked to get meaningful convergence rate improvements. To get optimal step size, we take into account of the curvature information. In addition, the larger step size technique is exploited to accelerate convergence of the proposed method. The global convergence of the proposed method is analysed under mild assumption. Finally, the results of the numerical experiments on both synthetic and real-world datasets show that the proposed method is effective.
  • Tianqi Wu, Shing-Tung Yau
    Journal of Computational Mathematics. 2023, 41(5): 879-908. https://doi.org/10.4208/jcm.2206-m2020-0251
    We use a narrow-band approach to compute harmonic maps and conformal maps for surfaces embedded in the Euclidean 3-space, using point cloud data only. Given a surface, or a point cloud approximation, we simply use the standard cubic lattice to approximate its ∈-neighborhood. Then the harmonic map of the surface can be approximated by discrete harmonic maps on lattices. The conformal map, or the surface uniformization, is achieved by minimizing the Dirichlet energy of the harmonic map while deforming the target surface of constant curvature. We propose algorithms and numerical examples for closed surfaces and topological disks. To the best of the authors' knowledge, our approach provides the first meshless method for computing harmonic maps and uniformizations of higher genus surfaces.
  • Chunxiao Liu, Shengfeng Zhu
    Journal of Computational Mathematics. 2023, 41(5): 956-979. https://doi.org/10.4208/jcm.2208-m2020-0142
    Shape gradient flows are widely used in numerical shape optimization algorithms. We investigate the accuracy and effectiveness of approximate shape gradients flows for shape optimization of elliptic problems. We present convergence analysis with a priori error estimates for finite element approximations of shape gradient flows associated with a distributed or boundary expression of Eulerian derivative. Numerical examples are presented to verify theory and show that using the volume expression is effective for shape optimization with Dirichlet and Neumann boundary conditions.
  • Ming-Jun Lai, Jiaxin Xie, Zhiqiang Xu
    Journal of Computational Mathematics. 2023, 41(4): 741-770. https://doi.org/10.4208/jcm.2201-m2021-0130
    Graph sparsification is to approximate an arbitrary graph by a sparse graph and is useful in many applications, such as simplification of social networks, least squares problems, and numerical solution of symmetric positive definite linear systems. In this paper, inspired by the well-known sparse signal recovery algorithm called orthogonal matching pursuit (OMP), we introduce a deterministic, greedy edge selection algorithm, which is called the universal greedy approach (UGA) for the graph sparsification problem. For a general spectral sparsification problem, e.g., the positive subset selection problem from a set of m vectors in $\mathbb{R}{^n}$, we propose a nonnegative UGA algorithm which needs O(mn2 + n3/∈2) time to find a $\frac{{1 + \in/\beta }}{{1-\in/\beta }}$-spectral sparsifier with positive coefficients with sparsity at most $\left[{\frac{n}{{{ \in ^2}}}} \right]$, where β is the ratio between the smallest length and largest length of the vectors. The convergence of the nonnegative UGA algorithm is established. For the graph sparsification problem, another UGA algorithm is proposed which can output a $\frac{{1 + O/(\in)}}{{1-O/(\in)}}$-spectral sparsifier with $\left[{\frac{n}{{{ \in ^2}}}} \right]$ edges in O(m+n2/∈2) time from a graph with m edges and n vertices under some mild assumptions. This is a linear time algorithm in terms of the number of edges that the community of graph sparsification is looking for. The best result in the literature to the knowledge of the authors is the existence of a deterministic algorithm which is almost linear, i.e. O(m1+o(1)) for some o(1)=O($\frac{{{{(\log \log (m))}^{2/3}}}}{{{{\log }^{1/3}}(m)}}$). Finally, extensive experimental results, including applications to graph clustering and least squares regression, show the effectiveness of proposed approaches.
  • Wei Yang, Xin Liu, Bin He, Yunqing Huang
    Journal of Computational Mathematics. 2023, 41(2): 263-286. https://doi.org/10.4208/jcm.2112-m2020-0330
    CSCD(1)
    In this paper, we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell’s equations. We propose two a posteriori error estimators, one is the recovery-type estimator, and the other is the residual-type estimator. We first propose the curl-recovery method for the staggered discontinuous Galerkin method (SDGM), and based on the super-convergence result of the postprocessed solution, an asymptotically exact error estimator is constructed. The residual-type a posteriori error estimator is also proposed, and it’s reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell’s equations. The efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.
  • Yan Gu, Bo Jiang, Deren Han
    Journal of Computational Mathematics. 2023, 41(6): 1017-1040. https://doi.org/10.4208/jcm.2112-m2020-0023
    The Peaceman-Rachford splitting method is efficient for minimizing a convex optimization problem with a separable objective function and linear constraints. However, its convergence was not guaranteed without extra requirements. He et al. (SIAM J. Optim. 24: 1011 - 1040, 2014) proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor. In this paper, we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors. Besides, motivated by the recent advances on the ADMM type method with indefinite proximal terms, we employ the indefinite proximal term in the strictly contractive Peaceman-Rachford splitting method. We show that the proposed indefinite-proximal strictly contractive Peaceman-Rachford splitting method is convergent and also prove the o(1/t) convergence rate in the nonergodic sense. The numerical tests on the l1 regularized least square problem demonstrate the efficiency of the proposed method.
  • Xiuhui Guo, Lulu Tian, Yang Yang, Hui Guo
    Journal of Computational Mathematics. 2023, 41(4): 623-642. https://doi.org/10.4208/jcm.2108-m2021-0143
    In this paper, we apply local discontinuous Galerkin (LDG) methods for pattern formation dynamical model in polymerizing actin flocks. There are two main difficulties in designing effective numerical solvers. First of all, the density function is non-negative, and zero is an unstable equilibrium solution. Therefore, negative density values may yield blow-up solutions. To obtain positive numerical approximations, we apply the positivitypreserving (PP) techniques. Secondly, the model may contain stiff source. The most commonly used time integration for the PP technique is the strong-stability-preserving Runge-Kutta method. However, for problems with stiff source, such time discretizations may require strictly limited time step sizes, leading to large computational cost. Moreover, the stiff source any trigger spurious filament polarization, leading to wrong numerical approximations on coarse meshes. In this paper, we combine the PP LDG methods with the semi-implicit Runge-Kutta methods. Numerical experiments demonstrate that the proposed method can yield accurate numerical approximations with relatively large time steps.
  • Yidan Geng, Minghui Song, Mingzhu Liu
    Journal of Computational Mathematics. 2023, 41(4): 663-682. https://doi.org/10.4208/jcm.2109-m2021-0116
    In this paper, we consider the stochastic differential equations with piecewise continuous arguments (SDEPCAs) in which the drift coefficient satisfies the generalized one-sided Lipschitz condition and the diffusion coefficient satisfies the linear growth condition. Since the delay term t-[t] of SDEPCAs is not continuous and differentiable, the variable substitution method is not suitable. To overcome this difficulty, we adopt new techniques to prove the boundedness of the exact solution and the numerical solution. It is proved that the truncated Euler-Maruyama method is strongly convergent to SDEPCAs in the sense of Lq(q ≥ 2). We obtain the convergence order with some additional conditions. An example is presented to illustrate the analytical theory.
  • Ziang Chen, Andre Milzarek, Zaiwen Wen
    Journal of Computational Mathematics. 2023, 41(4): 683-716. https://doi.org/10.4208/jcm.2110-m2020-0317
    We propose a trust-region type method for a class of nonsmooth nonconvex optimization problems where the objective function is a summation of a (probably nonconvex) smooth function and a (probably nonsmooth) convex function. The model function of our trust-region subproblem is always quadratic and the linear term of the model is generated using abstract descent directions. Therefore, the trust-region subproblems can be easily constructed as well as efficiently solved by cheap and standard methods. When the accuracy of the model function at the solution of the subproblem is not sufficient, we add a safeguard on the stepsizes for improving the accuracy. For a class of functions that can be "truncated", an additional truncation step is defined and a stepsize modification strategy is designed. The overall scheme converges globally and we establish fast local convergence under suitable assumptions. In particular, using a connection with a smooth Riemannian trust-region method, we prove local quadratic convergence for partly smooth functions under a strict complementary condition. Preliminary numerical results on a family of ${\ell _1}$-optimization problems are reported and demonstrate the efficiency of our approach.
  • Weizhu Bao, Quan Zhao
    Journal of Computational Mathematics. 2023, 41(4): 771-796. https://doi.org/10.4208/jcm.2205-m2021-0237
    We propose an accurate and energy-stable parametric finite element method for solving the sharp-interface continuum model of solid-state dewetting in three-dimensional space. The model describes the motion of the film/vapor interface with contact line migration and is governed by the surface diffusion equation with proper boundary conditions at the contact line. We present a weak formulation for the problem, in which the contact angle condition is weakly enforced. By using piecewise linear elements in space and backward Euler method in time, we then discretize the formulation to obtain a parametric finite element approximation, where the interface and its contact line are evolved simultaneously. The resulting numerical method is shown to be well-posed and unconditionally energystable. Furthermore, the numerical method is generalized to the case of anisotropic surface energies in the Riemannian metric form. Numerical results are reported to show the convergence and efficiency of the proposed numerical method as well as the anisotropic effects on the morphological evolution of thin films in solid-state dewetting.
  • Lexing Ying
    Journal of Computational Mathematics. 2023, 41(5): 1003-1016. https://doi.org/10.4208/jcm.2211-m2022-0186
    This note introduces the double flip move to accelerate the Swendsen-Wang algorithm for Ising models with mixed boundary conditions below the critical temperature. The double flip move consists of a geometric flip of the spin lattice followed by a spin value flip. Both symmetric and approximately symmetric models are considered. We prove the detailed balance of the double flip move and demonstrate its empirical efficiency in mixing.
  • Pascal Heid
    Journal of Computational Mathematics. 2023, 41(5): 933-955. https://doi.org/10.4208/jcm.2207-m2020-0302
    The purpose of this paper is to verify that the computational scheme from[Heid et al., Gradient flow finite element discretizations with energy-based adaptivity for the Gross-Pitaevskii equation, J. Comput. Phys. 436 (2021)] for the numerical approximation of the ground state of the Gross-Pitaevskii equation can equally be applied for the effective approximation of excited states of Schrödinger's equation. That procedure employs an adaptive interplay of a Sobolev gradient flow iteration and a novel local mesh refinement strategy, and yields a guaranteed energy decay in each step of the algorithm. The computational tests in the present work highlight that this strategy is indeed able to approximate excited states, with (almost) optimal convergence rate with respect to the number of degrees of freedom.
  • Yazid Dendani, Radouen Ghanem
    Journal of Computational Mathematics. 2023, 41(4): 717-740. https://doi.org/10.4208/jcm.2110-m2021-0131
    In this paper we deal with the convergence analysis of the finite element method for an elliptic penalized unilateral obstacle optimal control problem where the control and the obstacle coincide. Error estimates are established for both state and control variables. We apply a fixed point type iteration method to solve the discretized problem.
    To corroborate our error estimations and the efficiency of our algorithms, the convergence results and numerical experiments are illustrated by concrete examples.
  • Weina Wang, Nannan Tian, Chunlin Wu
    Journal of Computational Mathematics. 2023, 41(4): 588-622. https://doi.org/10.4208/jcm.2108-m2021-0057
    Two-phase image segmentation is a fundamental task to partition an image into foreground and background. In this paper, two types of nonconvex and nonsmooth regularization models are proposed for basic two-phase segmentation. They extend the convex regularization on the characteristic function on the image domain to the nonconvex case, which are able to better obtain piecewise constant regions with neat boundaries. By analyzing the proposed non-Lipschitz model, we combine the proximal alternating minimization framework with support shrinkage and linearization strategies to design our algorithm. This leads to two alternating strongly convex subproblems which can be easily solved. Similarly, we present an algorithm without support shrinkage operation for the nonconvex Lipschitz case. Using the Kurdyka-Lojasiewicz property of the objective function, we prove that the limit point of the generated sequence is a critical point of the original nonconvex nonsmooth problem. Numerical experiments and comparisons illustrate the effectiveness of our method in two-phase image segmentation.
  • Hai Bi, Xuqing Zhang, Yidu Yang
    Journal of Computational Mathematics. 2023, 41(6): 1041-1063. https://doi.org/10.4208/jcm.2201-m2020-0128
    In this paper, we extend the work of Brenner and Sung [Math. Comp. 59, 321–338 (1992)] and present a regularity estimate for the elastic equations in concave domains. Based on the regularity estimate we prove that the constants in the error estimates of the nonconforming Crouzeix-Raviart element approximations for the elastic equations/eigenvalue problem are independent of Lamé constant, which means the nonconforming Crouzeix-Raviart element approximations are locking-free. We also establish two kinds of two-grid discretization schemes for the elastic eigenvalue problem, and analyze that when the mesh sizes of coarse grid and fine grid satisfy some relationship, the resulting solutions can achieve the optimal accuracy. Numerical examples are provided to show the efficiency of two-grid schemes for the elastic eigenvalue problem.