中国科学院数学与系统科学研究院期刊网

14 December 2024, Volume 45 Issue 4
    

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  • Zhang Bo, Sheng Hailong, Yang Chao
    Journal on Numerica Methods and Computer Applications. 2024, 45(4): 301-313. https://doi.org/10.12288/szjs.s2024-0949
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    In recent years, the researches on employing artificial neural networks to solve forward and inverse problems involving partial differential equations have developed rapidly. In solving the forward problems, Penalty-Free Neural Network-2 (PFNN-2) method can accurately approximate the initial and essential boundary conditions of the problem, relax the smoothness requirement about the solution, and achieve satisfactory solution accuracy (Sheng and Yang, CiCP, 2022)[1]. In this paper, we extend PFNN-2 to the parameter inversion problem of partial differential equation by combining its characteristics. To achieve this goal, a data-driven loss term is introduced on the basis of the original PFNN-2 loss function, and an adaptive strategy for the corresponding balance coefficient is designed. In numerical experiments, taking inversions of parameters in Burgers equation and convection-diffusion equation as examples, the proposed inversion method is tested, validating its feasibility. This study extends the application scope of the PFNN-2 method.
  • Huang Weijia, Huang Zhongyi, Yang Wenli
    Journal on Numerica Methods and Computer Applications. 2024, 45(4): 314-335. https://doi.org/10.12288/szjs.s2024-0965
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    In this paper, we propose a new image denoising model using the dual Lipschitz norm from optimal transport (OT) and the total variation minimization. We show the relations of this model to the $G$-TV model proposed by Yves Meyer to decomposition an image into a cartoon component, and a component representing the texture or noise. The proposed model is solved by minimizing a convex functional alternately in two variables. We design the numerical algorithm based on the Primal-Dual Hybrid Gradient (PDHG) algorithm for the Wasserstein-1 distance and the projection algorithm for the ROF model, and we establish the convergence analysis of the proposed algorithm. The existence of a minimizer of the proposed model is proved. Numerical examples demonstrate the distinct features of the proposed model compared with the traditional models such as the ROF model, and show the effectiveness of the proposed numerical method.
  • Zhang Linjie, Cui Haitao, Li Yongjie, Guo Yaqian, Lv Zhiyi
    Journal on Numerica Methods and Computer Applications. 2024, 45(4): 336-353. https://doi.org/10.12288/szjs.s2024-0936
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    Vertex dynamics model is a mathematical model based on energy and forces, which is widely used to simulate biological processes such as cell division, cell migration, cell death and cell shape change during morphogenesis. The application of vertex dynamics model plays an important role in contribution to the understanding of biological phenomena and their underlying mechanisms. Current vertex dynamics models focus on the simulation of a tissue composed of a single type of cells. However, multiple type of tissues and cells are often involved in real biological processes. For example, the dorsal closure process during embryonic development of the model organism Drosophila melanogaster, which is accomplished by the collaboration of amnioserosa tissues consisting of flat squamous cells and epidermal tissues consisting of columnar epithelial cells. In this paper, the dorsal closure process of Drosophila embryos is successfully simulated by a vertex dynamics model, in which the above two types of cells are included, and the input data is generated by using real time-lapse images. The simulation result is very close to the experimental observation data. The construction of this model provides a new research perspective to further understand the biophysical mechanism of dorsal closure in Drosophila embryos and other related morphogenesis processes.
  • Zhou Fengying, Zhang Jiakun, Huang Yingjie
    Journal on Numerica Methods and Computer Applications. 2024, 45(4): 354-372. https://doi.org/10.12288/szjs.s2024-0938
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    A numerical method for solving variable-order time fractional differential equations is developed by using two-dimensional fractional-order Legendre wavelets (FOLWs). In the sense of Riemann-Liouville (R-L) variable fractional-order integral, the variable fractional-order integral formulas of FOLWs are derived by means of unit step function and regularized $\beta$ function. Based on the generalized fractional-order Taylor expansion, the error estimation of two-dimensional FOLWs expansion is studied. The variable-order time fractional differential equation is discretized into a system of algebraic equation by using the collocation method. The resulted linear and nonlinear system are solved by Gauss elimination method and Picard iterative method, respectively. The effectiveness, applicability and accuracy of the proposed method are verified by several numerical examples.
  • Bao Tiantian, Feng Xiufang
    Journal on Numerica Methods and Computer Applications. 2024, 45(4): 373-386. https://doi.org/10.12288/szjs.s2024-0939
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    In this paper, a new high-order accuracy hybrid compact finite-difference scheme for solving the two-dimensional Helmholtz equation is developed using a hybrid compact finite-difference methods. To address the inefficiency of serial algorithm in solving the Helmholtz equation with large wave numbers, we propose a parallel high-order hybrid compact finite-difference algorithm based on the MPI environment on Linux cluster systems. Truncation error analysis shows that the proposed scheme has sixth-order accuracy. Numerical experimental results show that the proposed method can achieve the theoretical sixth-order accuracy in solving Helmholtz equation problems with variable wave numbers and large wave numbers. In addition, the parallel algorithm designed in this paper exhibits good parallel speedup, which can effectively improve computational efficiency.