中国科学院数学与系统科学研究院期刊网
JCM

15 November 2023, Volume 41 Issue 6
    

  • Select all
    |
  • AN INDEFINITE-PROXIMAL-BASED STRICTLY CONTRACTIVE PEACEMAN-RACHFORD SPLITTING METHOD
    Yan Gu, Bo Jiang, Deren Han
    Journal of Computational Mathematics. 2023, 41(6): 1017-1040. https://doi.org/10.4208/jcm.2112-m2020-0023
    Abstract ( ) Download PDF   Knowledge map   Save
    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.
  • THE NONCONFORMING CROUZEIX-RAVIART ELEMENT APPROXIMATION AND TWO-GRID DISCRETIZATIONS FOR THE ELASTIC EIGENVALUE PROBLEM
    Hai Bi, Xuqing Zhang, Yidu Yang
    Journal of Computational Mathematics. 2023, 41(6): 1041-1063. https://doi.org/10.4208/jcm.2201-m2020-0128
    Abstract ( ) Download PDF   Knowledge map   Save
    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.
  • HIGH ORDER FINITE DIFFERENCE HERMITE WENO FAST SWEEPING METHODS FOR STATIC HAMILTON-JACOBI EQUATIONS
    Yupeng Ren, Yulong Xing, Jianxian Qiu
    Journal of Computational Mathematics. 2023, 41(6): 1064-1092. https://doi.org/10.4208/jcm.2112-m2020-0283
    Abstract ( ) Download PDF   Knowledge map   Save
    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.
  • A FAST FREE MEMORY METHOD FOR AN EFFICIENT COMPUTATION OF CONVOLUTION KERNELS
    Matthieu Aussal, Marc Bakry
    Journal of Computational Mathematics. 2023, 41(6): 1093-1116. https://doi.org/10.4208/jcm.2202-m2021-0324
    Abstract ( ) Download PDF   Knowledge map   Save
    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.
  • IMPLICIT DETERMINANT METHOD FOR SOLVING AN HERMITIAN EIGENVALUE OPTIMIZATION PROBLEM
    Siru Gong, Yangfeng Su
    Journal of Computational Mathematics. 2023, 41(6): 1117-1136. https://doi.org/10.4208/jcm.2203-m2020-0303
    Abstract ( ) Download PDF   Knowledge map   Save
    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.
  • STABLE AND ROBUST RECOVERY OF APPROXIMATELY k-SPARSE SIGNALS WITH PARTIAL SUPPORT INFORMATION IN NOISE SETTINGS VIA WEIGHTED lp (0 < p ≤ 1) MINIMIZATION
    Biao Du, Anhua Wan
    Journal of Computational Mathematics. 2023, 41(6): 1137-1170. https://doi.org/10.4208/jcm.2207-m2022-0058
    Abstract ( ) Download PDF   Knowledge map   Save
    In the existing work, the recovery of strictly k-sparse signals with partial support information was derived in the l2 bounded noise setting. In this paper, the recovery of approximately k-sparse signals with partial support information in two noise settings is investigated via weighted lp (0 < p ≤ 1) minimization method. The restricted isometry constant (RIC) condition δtk < $\frac{1}{{p{\eta ^{\frac{2}{p} - 1}} + 1}}$ on the measurement matrix for some t ∈ [1+ $\frac{{2 - p}}{{2 + p}}$σ, 2] is proved to be sufficient to guarantee the stable and robust recovery of signals under sparsity defect in noisy cases. Herein, σ ∈ [0, 1] is a parameter related to the prior support information of the original signal, and η ≥ 0 is determined by p, t and σ. The new results not only improve the recent work in [17], but also include the optimal results by weighted l1 minimization or by standard lp minimization as special cases.
  • IMPULSE NOISE REMOVAL BY L1 WEIGHTED NUCLEAR NORM MINIMIZATION
    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
    Abstract ( ) Download PDF   Knowledge map   Save
    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.
  • A STOCHASTIC NEWTON METHOD FOR NONLINEAR EQUATIONS
    Jiani Wang, Xiao Wang, Liwei Zhang
    Journal of Computational Mathematics. 2023, 41(6): 1192-1221. https://doi.org/10.4208/jcm.2112-m2021-0072
    Abstract ( ) Download PDF   Knowledge map   Save
    In this paper, we study a stochastic Newton method for nonlinear equations, whose exact function information is difficult to obtain while only stochastic approximations are available. At each iteration of the proposed algorithm, an inexact Newton step is first computed based on stochastic zeroth- and first-order oracles. To encourage the possible reduction of the optimality error, we then take the unit step size if it is acceptable by an inexact Armijo line search condition. Otherwise, a small step size will be taken to help induce desired good properties. Then we investigate convergence properties of the proposed algorithm and obtain the almost sure global convergence under certain conditions. We also explore the computational complexities to find an approximate solution in terms of calls to stochastic zeroth- and first-order oracles, when the proposed algorithm returns a randomly chosen output. Furthermore, we analyze the local convergence properties of the algorithm and establish the local convergence rate in high probability. At last we present preliminary numerical tests and the results demonstrate the promising performances of the proposed algorithm.
  • A MULTISCALE PROJECTION METHOD FOR SOLVING NONLINEAR INTEGRAL EQUATIONS UNDER THE LIPSCHITZ CONDITION
    Linxiu Fan, Xingjun Luo, Rong Zhang, Chunmei Zeng, Suhua Yang
    Journal of Computational Mathematics. 2023, 41(6): 1222-1245. https://doi.org/10.4208/jcm.2202-m2021-0206
    Abstract ( ) Download PDF   Knowledge map   Save
    We propose a multiscale projection method for the numerical solution of the irtatively regularized Gauss-Newton method of nonlinear integral equations. An a posteriori rule is suggested to choose the stopping index of iteration and the rates of convergence are also derived under the Lipschitz condition. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.
  • A MODIFIED WEAK GALERKIN FINITE ELEMENT METHOD FOR SINGULARLY PERTURBED PARABOLIC CONVECTION-DIFFUSION-REACTION PROBLEMS
    Suayip Toprakseven, Fuzheng Gao
    Journal of Computational Mathematics. 2023, 41(6): 1246-1280. https://doi.org/10.4208/jcm.2203-m2021-0031
    Abstract ( ) Download PDF   Knowledge map   Save
    In this work, a modified weak Galerkin finite element method is proposed for solving second order linear parabolic singularly perturbed convection-diffusion equations. The key feature of the proposed method is to replace the classical gradient and divergence operators by the modified weak gradient and modified divergence operators, respectively. We apply the backward finite difference method in time and the modified weak Galerkin finite element method in space on uniform mesh. The stability analyses are presented for both semi-discrete and fully-discrete modified weak Galerkin finite element methods. Optimal order of convergences are obtained in suitable norms. We have achieved the same accuracy with the weak Galerkin method while the degrees of freedom are reduced in our method. Various numerical examples are presented to support the theoretical results. It is theoretically and numerically shown that the method is quite stable.
  • A DEEP LEARNING BASED DISCONTINUOUS GALERKIN METHOD FOR HYPERBOLIC EQUATIONS WITH DISCONTINUOUS SOLUTIONS AND RANDOM UNCERTAINTIES
    Jingrun Chen, Shi Jin, Liyao Lyu
    Journal of Computational Mathematics. 2023, 41(6): 1281-1304. https://doi.org/10.4208/jcm.2205-m2021-0277
    Abstract ( ) Download PDF   Knowledge map   Save
    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.
  • ALIKHANOV LINEARIZED GALERKIN FINITE ELEMENT METHODS FOR NONLINEAR TIME-FRACTIONAL SCHRÖDINGER EQUATIONS
    Hongyu Qin, Fengyan Wu, Boya Zhou
    Journal of Computational Mathematics. 2023, 41(6): 1305-1324. https://doi.org/10.4208/jcm.2112-m2021-0113
    Abstract ( ) Download PDF   Knowledge map   Save
    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.