14 October 2025, Volume 47 Issue 4

    

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  STRONG ERROR ANALYSIS OF JUMP-ADAPTED SPLIT-STEP BACKWARD EULER METHOD FOR NONLINEAR JUMP-DIFFUSION PROBLEMS

  Yang Xu, Chen Qing, Zhao Weidong

  Mathematica Numerica Sinica. 2025, 47(4): 561-575. https://doi.org/10.12286/jssx.j2024-1241

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  Based on jump-adapted time partition, this paper proposes a jump-adapted split-step backward Euler numerical approximation method for solving a class of nonlinear jump-diffusion problems. Under non-global Lipschitz conditions, by overcoming the main difficulties caused by strong nonlinear coefficients and random time partition in numerical analysis, we establish strong error estimates for the proposed numerical method, and obtain the optimal mean square convergence order. Finally, numerical experiments are provided to validate the theoretical results.
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  A STEP-FREE PRIMAL-DUAL ALGORITHM FOR SOLVING BILINEAR SADDLE POINT PROBLEM

  Xu Long, Chang Xiaokai

  Mathematica Numerica Sinica. 2025, 47(4): 576-590. https://doi.org/10.12286/jssx.j2024-1258

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  The primal-dual algorithm (PDA) is a full-splitting method that simultaneously obtain solutions to both primal and dual problems, which is a classical approach to solve bilinear saddle point problems. However in the existing PDA, the step size depends on the spectral norm of linear transform or can be estimated by linesearch, which is often overly conservative or requires extra computations of proximal operators or linear transform. In this paper, we present a splitting preconditioned PDA (SP-PDA), by adding proximal terms to the Lagrangian function and introducing a preconditioning strategy by solving linear matrix inversion problem. The proposed method has free step sizes and involves matrix decomposition only once, thus the computational can be burdened for solving linear inverse problems. We establish global iterative convergence and derive an $\mathcal{O}(1/N)$ ergodic convergence rate measured by function value residuals and constraint violations. Finally, numerical experiments on LASSO and matrix game problems demonstrate the efficient of SP-PDA.
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  PROXIMAL PENALTY METHOD FOR SPARSE OPTIMIZATION WITH COMPLEMENTARITY CONSTRAINTS

  Wang Luyao, Li Gaoxi, Lv Yibing

  Mathematica Numerica Sinica. 2025, 47(4): 591-604. https://doi.org/10.12286/jssx.j2024-1259

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  We consider how to solve a class of mathematical program with complementarity constraints (MPCC) where the objective function is a cardinality function. For tackling the cardinality function, we use capped-$ \ell_1 $ function to transform it to a difference-of-convex function, and give a continuous approximation. Then, a proximal penalty method is proposed for finding a weak directional (d)-stationary point of the continuous approximation, which is stronger than Clarke stationary point. The proposed algorithm is a novel combination of penalty method and non-monotonic proximal gradient method. We prove that our algorithm converges to a weak d-stationary point of MPCC under MPCC linear independence constraint qualification. The numerical results demonstrate the effectiveness of the proposed method.
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  THE DAI-LIAO CONJUGATE GRADIENT METHOD WITH STRONG CONVERGENCE AND ITS APPLICATION IN IMAGE RESTORATION AND MACHINE LEARNING

  Jiang Xianzhen, Sun Guoqing, Jian Jinbao

  Mathematica Numerica Sinica. 2025, 47(4): 605-623. https://doi.org/10.12286/jssx.j2024-1266

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  The conjugate gradient method is one of the most effective methods to solve large-scale optimization problems. In this paper, three sets of Dai-Liao conjugate condition parameters are provided, with truncated Dai-Liao conjugate parameters, and a restart procedure is set in the search direction. Thus a new Dai-Liao conjugate gradient algorithm is proposed. The search direction generated by the new algorithm satisfies the sufficient descent condition at each iteration without depending on any line search condition. Under the usual assumptions and the weak Wolfe line search condition, the algorithm is strongly convergent. Finally, the new algorithm is applied to solve large-scale unconstrained optimization, image restorations and machine learning. Numerical results show that the proposed algorithm is effective.
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  GOLDEN RATIO TYPE DOUGLAS-RACHFORD SPLITTING METHOD FOR SOLVING STRUCTURED INVERSE VARIATIONAL INEQUALITY PROBLEMS

  Jiang Yaning, Cai Xingju, Han Deren

  Mathematica Numerica Sinica. 2025, 47(4): 624-642. https://doi.org/10.12286/jssx.j2024-1271

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  This paper designs a golden ratio type Douglas-Rachford (DR) splitting method for a class of structured inverse variational inequality problems. The proposed method is based on an inexact customized DR splitting method, effectively integrating the golden ratio convex combination coefficients with a strategy for dynamically adjusting the step size parameter. Under general assumption conditions, we prove the global convergence of the new method and further establish the sublinear convergence rate results of the new method. In addition, we apply the new method to solve actual spatial price equilibrium control problems, and the relevant numerical experimental results also verify the effectiveness and superiority of the new method.
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  AN EXTRAGRADIENT METHOD FOR GENERALIZED DC PROGRAMMING

  Mao Ying, Wang Qun

  Mathematica Numerica Sinica. 2025, 47(4): 643-658. https://doi.org/10.12286/jssx.j2024-1274

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  This paper introduces an extended extragradient algorithm to solve a class of generalized nonsmooth DC problems. We establish the global convergence of the proposed algorithm under appropriate conditions. Our algorithm efficiently exploits the DC structure, and some numerical results demonstrate that it works better than the classical DCA algorithm.
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  CHEBYSHEV SPECTRAL METHOD FOR SOLVING THE SCHRÖDINGER EQUATION UNDER TRANSMISSION BOUNDARY CONDITIONS

  Han Yu, Jiang Haiyan, Lu Tiao

  Mathematica Numerica Sinica. 2025, 47(4): 659-676. https://doi.org/10.12286/jssx.j2024-1276

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  This article designs a Chebyshev-Galerkin spectral method based on a linear combination of Chebyshev polynomials as the basis functions to solve the Schrödinger equation with transparent boundary conditions. The paper rigorously analyzes the convergence of the spectral method. Through the design of numerical experiments, it verifies the high-order convergence of this algorithm. We also compare it with the traditional finite difference method, highlighting the advantages of this algorithm. For potential energy functions of single and double barriers, the quantum tunneling and resonance tunneling phenomena are simulated by calculating the transmission rate variation curves. The algorithm is then applied to simulate the current-voltage characteristics of the resonant tunneling diode, successfully reproducing the negative resistance characteristic of the resonant tunneling diode.
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  PRECONDITIONED CONJUGATE GRADIENT METHOD ARISING FROM MULTIPLICATIVE HALF-QUADRATIC IMAGE RESTORATION

  Zhao Peipei, Huang Yumei

  Mathematica Numerica Sinica. 2025, 47(4): 677-695. https://doi.org/10.12286/jssx.j2025-1280

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  Image restoration is to estimate the clean image from the degraded image, it is a highly ill-posed inverse problem. Regularized methods can mitigate the ill-posedness, which can usually be achieved by minimizing a cost function consisting of a data-fidelity term and a regularization term. In this paper, we consider the multiplicative half-quadratic regularized method for the image restoration problem and employ the Newton method to solve the model. At each Newton iteration step, a linear system of equations with symmetric positive definite coefficient matrix arises. In order to solve the linear system efficiently, we propose a linear Taylor approximation preconditioner for the Schur complement inverse matrix, based on the block triangular decomposition of the coefficient matrix, and the preconditioned conjugate gradient method is applied to solve the linear system. Spectral analysis of the preconditioned matrix reveals that the proposed preconditioner yields a relatively clustered eigenvalue distribution, with some eigenvalues exactly equal to one. Numerical experiments demonstrate that the proposed preconditioner significantly reduces both the number of iterations and the computational time compared to existing methods when solving the system using PCG.
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  EXPONENTIAL STABILITY OF NUMERICAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS—THE NECESSITY OF FULLY IMPLICIT METHODS

  Liu Kai, Zhu Quanxin

  Mathematica Numerica Sinica. 2025, 47(4): 696-713. https://doi.org/10.12286/jssx.j2025-1282

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  This paper investigates the exponential stability of numerical solutions for stochastic differential equations (SDEs) and explores the necessity of fully implicit methods. Centered around two counterexamples, the limitations of Euler-type methods (including the stochastic theta method and the truncated Euler method) are discussed. Based on the theory of exponential martingales, the almost sure exponential stability conditions of the zero solution for SDEs are improved. It is then proven that the fully implicit Milstein method performs well for these two counterexamples. Numerical experiments validate the conclusions. Specifically, there exist SDEs for which, when considering exponential stability, commonly used Eulertype methods (such as the stochastic theta method and the truncated Euler method) are not applicable, whereas the fully implicit Milstein method remains effective. Thus, fully implicit schemes are essential in the study of exponential stability for numerical solutions of SDEs.
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  SOLVING LARGE-WAVENUMBER HELMHOLTZ EQUATIONS WITH FREQUENCY-ENHANCED HIGH-ORDER RELU-KAN

  Zhang Haoran, Ji Xia, Hu Donghao

  Mathematica Numerica Sinica. 2025, 47(4): 714-742. https://doi.org/10.12286/jssx.j2025-1316

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  Solving large-wavenumber Helmholtz equations with traditional numerical methods faces an inherent trade-off between computational accuracy and efficiency. This paper proposes Frequency-Enhanced High-order ReLU-KAN (FE-HRKAN). It introduces a learnable adaptive frequency modulation mechanism into the existing High-order ReLU-KAN (HRKAN) framework, expanding the input features to a combination of the original variables and parameterized high-frequency oscillatory features. The paper proves HRKAN’s spectral limitations and demonstrates the extended high-frequency expressiveness of FE-HRKAN, ensuring that FE-HRKAN enhances the capability to represent high-frequency oscillations while maintaining the original performance of HRKAN. Experimental results show that in function approximation tasks, FE-HRKAN reduces the L₂ relative error for approximating high-frequency oscillatory functions by two orders of magnitude compared to the baseline HRKAN model, while also reducing the L₂ relative error for approximating non-oscillatory functions by 34%. In solving large-wavenumber Helmholtz equations, FE-HRKAN achieves L₂ relative errors on the order of 10^-3 to 10^-4 across wavenumbers ranging from 5 to 1000, reducing errors by 3 to 4 orders of magnitude compared to HRKAN in large-wavenumber scenarios.