中国科学院数学与系统科学研究院期刊网

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  • 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.
  • 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.
  • Haijin Wang, Anping Xu, Qi Tao
    Journal of Computational Mathematics. 2024, 42(1): 1-23. https://doi.org/10.4208/jcm.2202-m2021-0290
    In this paper, we first present the optimal error estimates of the semi-discrete ultra-weak discontinuous Galerkin method for solving one-dimensional linear convection-diffusion equations. Then, coupling with a kind of Runge-Kutta type implicit-explicit time discretization which treats the convection term explicitly and the diffusion term implicitly, we analyze the stability and error estimates of the corresponding fully discrete schemes. The fully discrete schemes are proved to be stable if the time-step ττ0, where τ0 is a constant independent of the mesh-size h. Furthermore, by the aid of a special projection and a careful estimate for the convection term, the optimal error estimate is also obtained for the third order fully discrete scheme. Numerical experiments are displayed to verify the theoretical results.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • Gengen Zhang, Chunmei Su
    Journal of Computational Mathematics. 2024, 42(1): 289-312. https://doi.org/10.4208/jcm.2204-m2022-0001
    In this paper, we consider a uniformly accurate compact finite difference method to solve the quantum Zakharov system (QZS) with a dimensionless parameter 0 < ε ≤ 1, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., when 0 < ε ? 1, the solution of QZS propagates rapidly oscillatory initial layers in time, and this brings significant difficulties in devising numerical algorithm and establishing their error estimates, especially as 0 < ε ? 1. The solvability, the mass and energy conservation laws of the scheme are also discussed. Based on the cut-off technique and energy method, we rigorously analyze two independent error estimates for the well-prepared and ill-prepared initial data, respectively, which are uniform in both time and space for ε ∈ (0, 1] and optimal at the fourth order in space. Numerical results are reported to verify the error behavior.
  • Shounian Deng, Chen Fei, Weiyin Fei, Xuerong Mao
    Journal of Computational Mathematics. 2024, 42(1): 178-216. https://doi.org/10.4208/jcm.2204-m2021-0270
    This work is concerned with the convergence and stability of the truncated EulerMaruyama (EM) method for super-linear stochastic differential delay equations (SDDEs) with time-variable delay and Poisson jumps. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present two types of the truncated EM method for such jump-diffusion SDDEs with time-variable delay, which is proposed to be approximated by the value taken at the nearest grid points on the left of the delayed argument. The first type is proved to have a strong convergence order which is arbitrarily close to 1/2 in mean-square sense, under the Khasminskii-type, global monotonicity with U function and polynomial growth conditions. The second type is convergent in q-th (q < 2) moment under the local Lipschitz plus generalized Khasminskii-type conditions. In addition, we show that the partially truncated EM method preserves the mean-square and H stabilities of the true solutions. Lastly, we carry out some numerical experiments to support the theoretical results.
  • 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.
  • 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.
  • 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.
  • Xiaoya Zhai
    Journal of Computational Mathematics. 2024, 42(1): 134-155. https://doi.org/10.4208/jcm.2209-m2021-0358
    Topology optimization (TO) has developed rapidly recently. However, topology optimization with stress constraints still faces many challenges due to its highly non-linear properties which will cause inefficient computation, iterative oscillation, and convergence guarantee problems. At the same time, isogeometric analysis (IGA) is accepted by more and more researchers, and it has become one important tool in the field of topology optimization because of its high fidelity. In this paper, we focus on topology optimization with stress constraints based on isogeometric analysis to improve computation efficiency and stability. A new hybrid solver combining the alternating direction method of multipliers and the method of moving asymptotes (ADMM-MMA) is proposed to solve this problem. We first generate an initial feasible point by alternating direction method of multipliers (ADMM) in virtue of the rapid initial descent property. After that, we adopt the method of moving asymptotes (MMA) to get the final results. Several benchmark examples are used to verify the proposed method, and the results show its feasibility and effectiveness.
  • 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.
  • Xiaojing Dong, Yinnian He
    Journal of Computational Mathematics. 2024, 42(1): 49-70. https://doi.org/10.4208/jcm.2201-m2021-0140
    By combination of iteration methods with the partition of unity method (PUM), some finite element parallel algorithms for the stationary incompressible magnetohydrodynamics (MHD) with different physical parameters are presented and analyzed. These algorithms are highly efficient. At first, a global solution is obtained on a coarse grid for all approaches by one of the iteration methods. By parallelized residual schemes, local corrected solutions are calculated on finer meshes with overlapping sub-domains. The subdomains can be achieved flexibly by a class of PUM. The proposed algorithm is proved to be uniformly stable and convergent. Finally, one numerical example is presented to confirm the theoretical findings.
  • Jauny Prajapati, Debdas Ghosh, Ashutosh Upadhayay
    Journal of Computational Mathematics. 2024, 42(1): 24-48. https://doi.org/10.4208/jcm.2204-m2021-0241
    This paper proposes an interior-point technique for detecting the nondominated points of multi-objective optimization problems using the direction-based cone method. Cone method decomposes the multi-objective optimization problems into a set of single-objective optimization problems. For this set of problems, parametric perturbed KKT conditions are derived. Subsequently, an interior point technique is developed to solve the parametric perturbed KKT conditions. A differentiable merit function is also proposed whose stationary point satisfies the KKT conditions. Under some mild assumptions, the proposed algorithm is shown to be globally convergent. Numerical results of unconstrained and constrained multi-objective optimization test problems are presented. Also, three performance metrics (modified generational distance, hypervolume, inverted generational distance) are used on some test problems to investigate the efficiency of the proposed algorithm. We also compare the results of the proposed algorithm with the results of some other existing popular methods.
  • 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.
  • Shipeng Mao, Jiaao Sun, Wendong Xue
    Journal of Computational Mathematics. 2024, 42(1): 71-110. https://doi.org/10.4208/jcm.2201-m2021-0315
    In this paper, we consider the initial-boundary value problem (IBVP) for the micropolar Naviers-Stokes equations (MNSE) and analyze a first order fully discrete mixed finite element scheme. We first establish some regularity results for the solution of MNSE, which seem to be not available in the literature. Next, we study a semi-implicit time-discrete scheme for the MNSE and prove L2-H1 error estimates for the time discrete solution. Furthermore, certain regularity results for the time discrete solution are establishes rigorously. Based on these regularity results, we prove the unconditional L2-H1 error estimates for the finite element solution of MNSE. Finally, some numerical examples are carried out to demonstrate both accuracy and efficiency of the fully discrete finite element scheme.
  • Xu Yang, Weidong Zhao
    Journal of Computational Mathematics. 2024, 42(1): 248-270. https://doi.org/10.4208/jcm.2206-m2021-0354
    In this paper, we study the strong convergence of a jump-adapted implicit Milstein method for a class of jump-diffusion stochastic differential equations with non-globally Lipschitz drift coefficients. Compared with the regular methods, the jump-adapted methods can significantly reduce the complexity of higher order methods, which makes them easily implementable for scenario simulation. However, due to the fact that jump-adapted time discretization is path dependent and the stepsize is not uniform, this makes the numerical analysis of jump-adapted methods much more involved, especially in the non-globally Lipschitz setting. We provide a rigorous strong convergence analysis of the considered jump-adapted implicit Milstein method by developing some novel analysis techniques and optimal rate with order one is also successfully recovered. Numerical experiments are carried out to verify the theoretical findings.
  • Lina Wang, Qian Tong, Lijun Yi, Mingzhu Zhang
    Journal of Computational Mathematics. 2024, 42(1): 217-247. https://doi.org/10.4208/jcm.2203-m2021-0244
    We propose and analyze a single-interval Legendre-Gauss-Radau (LGR) spectral collocation method for nonlinear second-order initial value problems of ordinary differential equations. We design an efficient iterative algorithm and prove spectral convergence for the single-interval LGR collocation method. For more effective implementation, we propose a multi-interval LGR spectral collocation scheme, which provides us great flexibility with respect to the local time steps and local approximation degrees. Moreover, we combine the multi-interval LGR collocation method in time with the Legendre-Gauss-Lobatto collocation method in space to obtain a space-time spectral collocation approximation for nonlinear second-order evolution equations. Numerical results show that the proposed methods have high accuracy and excellent long-time stability. Numerical comparison between our methods and several commonly used methods are also provided.
  • 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.
  • Wenli Yang, Zhongyi Huang, Wei Zhu
    Journal of Computational Mathematics. 2024, 42(2): 313-336. https://doi.org/10.4208/jcm.2201-m2021-0287
    In this paper, we propose using the tailored finite point method (TFPM) to solve the resulting parabolic or elliptic equations when minimizing the Huber regularization based image super-resolution model using the augmented Lagrangian method (ALM). The Huber regularization based image super-resolution model can ameliorate the staircase for restored images. TFPM employs the method of weighted residuals with collocation technique, which helps get more accurate approximate solutions to the equations and reserve more details in restored images. We compare the new schemes with the Marquina-Osher model, the image super-resolution convolutional neural network (SRCNN) and the classical interpolation methods: bilinear interpolation, nearest-neighbor interpolation and bicubic interpolation. Numerical experiments are presented to demonstrate that with the new schemes the quality of the super-resolution images has been improved. Besides these, the existence of the minimizer of the Huber regularization based image super-resolution model and the convergence of the proposed algorithm are also established in this paper.
  • Siru Gong, Yangfeng Su
    Journal of Computational Mathematics. 2023, 41(6): 1117-1136. https://doi.org/10.4208/jcm.2203-m2020-0303
    Implicit determinant method is an effective method for some linear eigenvalue optimization problems since it solves linear systems of equations rather than eigenpairs. In this paper, we generalize the implicit determinant method to solve an Hermitian eigenvalue optimization problem for smooth case and non-smooth case. We prove that the implicit determinant method converges locally and quadratically. Numerical experiments confirm our theoretical results and illustrate the efficiency of implicit determinant method.
  • Yupeng Ren, Yulong Xing, Jianxian Qiu
    Journal of Computational Mathematics. 2023, 41(6): 1064-1092. https://doi.org/10.4208/jcm.2112-m2020-0283
    In this paper, we propose a novel Hermite weighted essentially non-oscillatory (HWENO) fast sweeping method to solve the static Hamilton-Jacobi equations efficiently. During the HWENO reconstruction procedure, the proposed method is built upon a new finite difference fifth order HWENO scheme involving one big stencil and two small stencils. However, one major novelty and difference from the traditional HWENO framework lies in the fact that, we do not need to introduce and solve any additional equations to update the derivatives of the unknown function φ. Instead, we use the current φ and the old spatial derivative of φ to update them. The traditional HWENO fast sweeping method is also introduced in this paper for comparison, where additional equations governing the spatial derivatives of φ are introduced. The novel HWENO fast sweeping methods are shown to yield great savings in computational time, which improves the computational efficiency of the traditional HWENO scheme. In addition, a hybrid strategy is also introduced to further reduce computational costs. Extensive numerical experiments are provided to validate the accuracy and efficiency of the proposed approaches.
  • Jingjun Zhao, Wenjiao Zhao, Yang Xu
    Journal of Computational Mathematics. 2024, 42(1): 156-177. https://doi.org/10.4208/jcm.2203-m2021-0233
    This paper deals with the numerical approximation for the time fractional diffusion problem with fractional dynamic boundary conditions. The well-posedness for the weak solutions is studied. A direct discontinuous Galerkin approach is used in spatial direction under the uniform meshes, together with a second-order Alikhanov scheme is utilized in temporal direction on the graded mesh, and then the fully discrete scheme is constructed. Furthermore, the stability and the error estimate for the full scheme are analyzed in detail. Numerical experiments are also given to illustrate the effectiveness of the proposed method.
  • Hongyu Qin, Fengyan Wu, Boya Zhou
    Journal of Computational Mathematics. 2023, 41(6): 1305-1324. https://doi.org/10.4208/jcm.2112-m2021-0113
    We present Alikhanov linearized Galerkin methods for solving the nonlinear time fractional Schrödinger equations. Unconditionally optimal estimates of the fully-discrete scheme are obtained by using the fractional time-spatial splitting argument. The convergence results indicate that the error estimates hold without any spatial-temporal stepsize restrictions. Numerical experiments are done to verify the theoretical results.
  • Jian Lu, Yuting Ye, Yiqiu Dong, Xiaoxia Liu, Yuru Zou
    Journal of Computational Mathematics. 2023, 41(6): 1171-1191. https://doi.org/10.4208/jcm.2201-m2021-0183
    In recent years, the nuclear norm minimization (NNM) as a convex relaxation of the rank minimization has attracted great research interest. By assigning different weights to singular values, the weighted nuclear norm minimization (WNNM) has been utilized in many applications. However, most of the work on WNNM is combined with the l2-data-fidelity term, which is under additive Gaussian noise assumption. In this paper, we introduce the L1-WNNM model, which incorporates the l1-data-fidelity term and the regularization from WNNM. We apply the alternating direction method of multipliers (ADMM) to solve the non-convex minimization problem in this model. We exploit the low rank prior on the patch matrices extracted based on the image non-local self-similarity and apply the L1-WNNM model on patch matrices to restore the image corrupted by impulse noise. Numerical results show that our method can effectively remove impulse noise.
  • Ruihan Guo, Yan Xu
    Journal of Computational Mathematics. 2024, 42(1): 111-133. https://doi.org/10.4208/jcm.2202-m2021-0302
    In [20], a semi-implicit spectral deferred correction (SDC) method was proposed, which is efficient for highly nonlinear partial differential equations (PDEs). The semi-implicit SDC method in [20] is based on first-order time integration methods, which are corrected iteratively, with the order of accuracy increased by one for each additional iteration. In this paper, we will develop a class of semi-implicit SDC methods, which are based on second-order time integration methods and the order of accuracy are increased by two for each additional iteration. For spatial discretization, we employ the local discontinuous Galerkin (LDG) method to arrive at fully-discrete schemes, which are high-order accurate in both space and time. Numerical experiments are presented to demonstrate the accuracy, efficiency and robustness of the proposed semi-implicit SDC methods for solving complex nonlinear PDEs.
  • Matthieu Aussal, Marc Bakry
    Journal of Computational Mathematics. 2023, 41(6): 1093-1116. https://doi.org/10.4208/jcm.2202-m2021-0324
    We introduce the Fast Free Memory method (FFM), a new implementation of the Fast Multipole Method (FMM) for the evaluation of convolution products. The FFM aims at being easier to implement while maintaining a high level of performance, capable of handling industrially-sized problems. The FFM avoids the implementation of a recursive tree and is a kernel independent algorithm. We give the algorithm and the relevant complexity estimates. The quasi-linear complexity enables the evaluation of convolution products with up to one billion entries. We illustrate numerically the capacities of the FFM by solving Boundary Integral Equations problems featuring dozen of millions of unknowns. Our implementation is made freely available under the GPL 3.0 license within the Gypsilab framework.
  • Jingrun Chen, Shi Jin, Liyao Lyu
    Journal of Computational Mathematics. 2023, 41(6): 1281-1304. https://doi.org/10.4208/jcm.2205-m2021-0277
    We propose a deep learning based discontinuous Galerkin method (D2GM) to solve hyperbolic equations with discontinuous solutions and random uncertainties. The main computational challenges for such problems include discontinuities of the solutions and the curse of dimensionality due to uncertainties. Deep learning techniques have been favored for high-dimensional problems but face difficulties when the solution is not smooth, thus have so far been mainly used for viscous hyperbolic system that admits only smooth solutions. We alleviate this difficulty by setting up the loss function using discrete shock capturing schemes–the discontinous Galerkin method as an example–since the solutions are smooth in the discrete space. The convergence of D2GM is established via the Lax equivalence theorem kind of argument. The high-dimensional random space is handled by the Monte-Carlo method. Such a setup makes the D2GM approximate high-dimensional functions over the random space with satisfactory accuracy at reasonable cost. The D2GM is found numerically to be first-order and second-order accurate for (stochastic) linear conservation law with smooth solutions using piecewise constant and piecewise linear basis functions, respectively. Numerous examples are given to verify the efficiency and the robustness of D2GM with the dimensionality of random variables up to 200 for (stochastic) linear conservation law and (stochastic) Burgers’ equation.