Applied Numerical Mathematics最新文献

英文中文

Structure-preserving numerical methods for phase-field surfactant models based on the supplemental variable method (SVM) 基于补充变量法的相场表面活性剂模型保结构数值方法

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-10-14 DOI: 10.1016/j.apnum.2025.10.007

Mengchun Yuan, Qi Li

Numerous efficient numerical algorithms have been developed for phase-field surfactant models, including the convex splitting method, the scalar auxiliary variable (SAV) approach, the invariant energy quadratization (IEQ) approach, and the new Lagrange multiplier method. In this paper, we introduce novel numerical schemes based on the supplementary variable method for solving and simulating the binary fluid phase-field surfactant model of the Cahn-Hilliard type. The key innovation of our method is the introduction of an auxiliary variable, which reformulates the original problem into a constrained optimization framework. This reformulation offers several significant advantages over existing approaches. Firstly, our schemes require solving only a few Poisson-type systems with constant coefficient matrices, significantly reducing computational costs. Secondly, the proposed schemes preserve mass conservation and adhere to the original energy dissipation law at the discrete level, in contrast to the SAV and IEQ approaches, which adhere to modified dissipation laws. Additionally, we rigorously establish the energy stability of the schemes. Extensive 2D and 3D numerical experiments confirm the accuracy and efficiency of the proposed schemes.

许多有效的相场表面活性剂模型的数值算法已经被开发出来，包括凸分裂法、标量辅助变量法、不变能量二次化法和新的拉格朗日乘子法。本文介绍了基于补充变量法求解和模拟Cahn-Hilliard型二元流体相场表面活性剂模型的新数值格式。该方法的关键创新在于引入了一个辅助变量，将原问题重新表述为一个约束优化框架。与现有方法相比，这种重新表述提供了几个显著的优势。首先，我们的方案只需要求解几个常系数矩阵的泊松型系统，大大降低了计算成本。其次，与SAV和IEQ方法遵循修正的耗散规律相比，所提出的方案保持了质量守恒，并在离散水平上遵循原始的能量耗散规律。此外，我们严格地建立了方案的能量稳定性。大量的二维和三维数值实验验证了所提方案的准确性和有效性。

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引用次数: 0

Parameter choice strategies for regularized least squares approximation of noisy continuous functions on the unit circle 单位圆上带噪声连续函数正则化最小二乘逼近的参数选择策略

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-10-10 DOI: 10.1016/j.apnum.2025.10.005

Congpei An , Mou Cai

This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific cases. With the aid of the de la Vallée-Poussin approximation, we derive a uniform error bound and a concrete

L_{2}

error bound. These error estimates demonstrate the effectiveness of Tikhonov regularization in the denoising process. A new regularity condition for the selection of regularization parameters is proposed. We investigate three strategies for choosing regularization parameters: Morozov’s discrepancy principle, the L-curve, and generalized cross-validation, by explicitly combining these error bounds of the approximating trigonometric polynomial. We show that Morozov’s discrepancy principle satisfies the proposed regularity condition, while the other two methods do not. Finally, numerical examples are provided to illustrate how the aforementioned methodologies, when applied with well-chosen parameters, can significantly improve the quality of approximation.

本文探讨了利用单位圆上的三角多项式将Tikhonov正则化纳入最小二乘近似方案。这种方法包括插值和超插值作为具体案例。利用de la vall - poussin近似，导出了统一的误差界和具体的L2误差界。这些误差估计证明了吉洪诺夫正则化在去噪过程中的有效性。提出了一种新的正则化参数选择的正则性条件。我们研究了三种选择正则化参数的策略：Morozov的差异原理、l曲线和广义交叉验证，通过显式地组合这些近似三角多项式的误差界限。结果表明，Morozov的差异原理满足所提出的正则性条件，而其他两种方法则不满足。最后，提供了数值实例来说明上述方法如何在选择良好的参数时应用，可以显着提高近似的质量。

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引用次数: 0

Plug-and-play superiorization 即插即用superiorization

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-10-10 DOI: 10.1016/j.apnum.2025.10.002

Jon Henshaw , Aviv Gibali , Thomas Humphries

The superiorization methodology (SM) is an optimization heuristic in which an iterative algorithm, which aims to solve a particular problem, is “superiorized” to promote solutions that are improved with respect to some secondary criterion. This superiorization is achieved by perturbing iterates of the algorithm in nonascending directions of a prescribed function that penalizes undesirable characteristics in the solution; the solution produced by the superiorized algorithm should therefore be improved with respect to the value of this function. In this paper, we broaden the SM to allow for the perturbations to be introduced by an arbitrary procedure instead, using a plug-and-play approach. This allows for operations such as image denoisers or deep neural networks, which have applications to a broad class of problems, to be incorporated within the superiorization methodology. As proof of concept, we perform numerical simulations involving low-dose and sparse-view computed tomography image reconstruction, comparing the plug-and-play approach to two conventionally superiorized algorithms, as well as a post-processing approach. The plug-and-play approach provides comparable or better image quality in most cases, while also providing advantages in terms of computing time, and data fidelity of the solutions.

优越化方法（SM）是一种优化启发式方法，其中旨在解决特定问题的迭代算法被“优越化”，以促进解决方案在某些次要准则方面得到改进。这种优越性是通过在规定函数的非升序方向上扰动算法的迭代来实现的，该算法会惩罚解中不希望出现的特征；因此，优越算法产生的解相对于该函数的值应该得到改进。在本文中，我们扩大了SM，以允许通过任意程序引入扰动，而不是使用即插即用的方法。这使得诸如图像去噪器或深度神经网络等应用于广泛问题的操作被纳入上级化方法。作为概念验证，我们进行了涉及低剂量和稀疏视图计算机断层扫描图像重建的数值模拟，将即插即用方法与两种传统的高级算法以及后处理方法进行了比较。即插即用方法在大多数情况下提供了相当或更好的图像质量，同时在计算时间和解决方案的数据保真度方面也具有优势。

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引用次数: 0

Numerical approximation of constrained optimal control problems in delayed systems using an enhanced Rayleigh-Ritz algorithm 基于增强瑞利-里兹算法的时滞系统约束最优控制问题数值逼近

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-10-10 DOI: 10.1016/j.apnum.2025.10.004

Behzad Kafash

In this study, a modified Rayleigh-Ritz method is presented for the solution of optimal control problems governed by time-delayed dynamical systems, considering both constrained and unconstrained control and state variables. In this approach, the control or state variables are approximated using shifted Chebyshev polynomials with unknown coefficients. The proposed modified Rayleigh-Ritz method transforms the constrained optimal control problems governed by time-delayed dynamical systems into an optimization problem with constraints. Furthermore, a computational algorithm is developed for implementing the proposed method, and its convergence is proven analytically. To evaluate the efficiency and accuracy of the proposed algorithm, several numerical examples are presented. These include the single-input/single-output system as a case study with control and final state constraints, and an optimal control problem of the harmonic oscillator under different scenarios, which involve constraints on state and control variables.

本文提出了一种改进的Rayleigh-Ritz方法，用于求解时滞动力系统的最优控制问题，同时考虑了有约束控制和无约束控制以及状态变量。在这种方法中，控制变量或状态变量使用带未知系数的移位切比雪夫多项式逼近。提出的改进瑞利-里兹方法将时滞动力系统的约束最优控制问题转化为带约束的优化问题。在此基础上，提出了实现该方法的计算算法，并对其收敛性进行了分析证明。为了评估该算法的效率和准确性，给出了几个数值算例。其中包括以控制和最终状态约束为例的单输入/单输出系统，以及涉及状态约束和控制变量约束的不同情况下谐振子的最优控制问题。

引用次数: 0

A feasible interior-point method with full-Newton step for P*(κ)-weighted linear complementarity problem via the algebraically equivalent transformation 基于代数等价变换的P*(κ)-加权线性互补问题可行的全牛顿步内点法

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-10-10 DOI: 10.1016/j.apnum.2025.10.003

Xiaoni Chi , Lin Gan , Zhuoran Gao , Jein-Shan Chen

This paper investigates feasible interior-point method (IPM) with full-Newton step for

P_{*} (κ)

-weighted linear complementarity problem (WLCP). In particular, by applying the algebraically equivalent transformation (AET) for linear optimization, we obtain the new search directions by solving the perturbed Newton system. The AET of the Newton system is based on the kernel function

φ (t) = t - \sqrt{t}

, which is used for solving WLCP for the first time. At each iteration, our algorithm takes only full-Newton steps. Therefore, no line-searches are needed to update the iterates. We show the strict feasibility of the full-Newton step and the polynomial iteration complexity of our algorithm under suitable assumptions. Some numerical experiments demonstrate the effectiveness of the proposed algorithm.

研究了P*(κ)加权线性互补问题的全牛顿步可行内点法。特别地，我们利用代数等价变换（AET）进行线性优化，通过求解扰动牛顿系统得到新的搜索方向。牛顿系统的AET基于核函数φ(t)=t−t，首次用于求解WLCP。在每次迭代中，我们的算法只需要一整牛顿步。因此，更新迭代时不需要行搜索。在适当的假设条件下，证明了算法的多项式迭代复杂度和全牛顿步的严格可行性。数值实验证明了该算法的有效性。

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引用次数: 0

A conjugate gradient algorithmic framework for unconstrained optimization with applications: Convergence and rate analyses 无约束优化的共轭梯度算法框架及其应用：收敛性和速率分析

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-10-09 DOI: 10.1016/j.apnum.2025.10.006

Feng Shao , Hu Shao , Bin Wu , Haijun Wang , Pengjie Liu , Meixing Liu

In this paper, we aim to develop a general conjugate gradient (CG) algorithmic framework for solving unconstrained optimization problems. Additionally, we employ different hybrid techniques to derive two hybrid conjugate parameters, which are then integrated into the algorithmic framework to develop two effective hybrid CG methods. Under common assumptions, we establish the global convergence of the proposed framework without any convexity assumption. Furthermore, we reveal the convergence rate of the framework under the uniformly convex condition. Preliminary numerical experiment results, including applications to unconstrained optimization and image restoration problems, are presented to explicitly illustrate the performance of the proposed methods in comparison with several existing methods.

在本文中，我们的目标是建立一个通用的共轭梯度（CG）算法框架来解决无约束优化问题。此外，我们采用不同的混合技术来推导两个混合共轭参数，然后将其集成到算法框架中，以开发两种有效的混合CG方法。在一般假设下，我们建立了该框架的全局收敛性，而不需要任何凸性假设。进一步给出了该框架在一致凸条件下的收敛速度。初步的数值实验结果，包括在无约束优化和图像恢复问题上的应用，明确地说明了所提方法与几种现有方法的性能。

引用次数: 0

Error estimate for the Cahn-Hilliard equation by the SAV-Runge-Kutta scheme 用SAV-Runge-Kutta格式估计Cahn-Hilliard方程的误差

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-10-09 DOI: 10.1016/j.apnum.2025.10.001

Lizhen Chen , Xiaozhuang Ma , Guohui Zhang , Jing Zhang

In this paper, we introduce a Scalar Auxiliary Variable (SAV) approach combined with the Runge-Kutta method for solving the Cahn-Hilliard equation. Within the framework of the SAV method, this Runge-Kutta method is particularly well-suited for obtaining numerical solutions that preserve key structural properties of the system, including energy conservation and momentum conservation. In the SAV-Runge-Kutta method, through a comparison of approximate solutions derived at various time points, the error for each step can be rigorously estimated. This approach thereby guarantees the stability and accuracy of the entire solution process. Finally, to illustrate the effectiveness and precision of our proposed method, we present several numerical examples. These examples demonstrate the capability of the SAV-Runge-Kutta method to accurately capture the intricate dynamics of the Cahn-Hilliard equation while maintaining energy conservation and momentum conservation.

本文将标量辅助变量法与龙格-库塔法相结合，引入求解Cahn-Hilliard方程的方法。在SAV方法的框架内，这种龙格-库塔方法特别适合于获得保持系统关键结构性质的数值解，包括能量守恒和动量守恒。在SAV-Runge-Kutta方法中，通过比较不同时间点的近似解，可以严格估计每一步的误差。因此，这种方法保证了整个解决过程的稳定性和准确性。最后，为了说明所提出方法的有效性和精度，给出了几个数值算例。这些例子证明了SAV-Runge-Kutta方法能够准确地捕捉Cahn-Hilliard方程的复杂动力学，同时保持能量守恒和动量守恒。

引用次数: 0

A linearized two-dimensional Galerkin-L1 spectral method with diagonalization for time-fractional diffusion equations with delay 具有对角化的线性化二维Galerkin-L1谱法求解时间分数阶时滞扩散方程

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-09-19 DOI: 10.1016/j.apnum.2025.09.008

Mahmoud A. Zaky

In this paper, we construct and analyze a linearized L1–Galerkin Legendre spectral method for solving two-dimensional time-fractional diffusion equations with delay. The approach combines the L1 temporal discretization of the Caputo derivative with a Legendre spectral approximation in space, while nonlinear source terms are efficiently handled through a linearization strategy. To further enhance computational performance, we employ matrix diagonalization approach to solve the resulting algebraic systems in numerical implementation. Rigorous stability and convergence analyses are carried out using discrete fractional Grönwall and fractional Halanay inequalities, establishing unconditional stability and spectral error estimates. Numerical experiments confirm the theoretical predictions, demonstrating spectral accuracy in space and

(2 - ϑ)

-order accuracy in time, as well as validating the robustness and efficiency of the proposed method across different fractional orders and delay parameters.

本文构造并分析了求解二维时间分数阶时滞扩散方程的线性化L1-Galerkin Legendre谱方法。该方法将卡普托导数的L1时间离散化与空间中的勒让德谱近似相结合，同时通过线性化策略有效地处理非线性源项。为了进一步提高计算性能，我们在数值实现中采用矩阵对角化方法来求解所得到的代数系统。使用离散分数阶Grönwall和分数阶Halanay不等式进行了严格的稳定性和收敛性分析，建立了无条件稳定性和谱误差估计。数值实验证实了理论预测，证明了空间上的频谱精度和时间上的（2−−）阶精度，并验证了该方法在不同分数阶和延迟参数下的鲁棒性和效率。

引用次数: 0

A Schur method for robust eigenvalue assignment of second-order singular systems 二阶奇异系统鲁棒特征值分配的Schur方法

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-09-17 DOI: 10.1016/j.apnum.2025.09.002

Kang Hu, Huiqing Xie

A new method for eigenvalue assignment problem of a second-order singular system is proposed via displacement-velocity-acceleration feedback. The real Schur form of a regular quadratic pencil and a robustness measurement for the closed-loop second-order system are introduced. On these grounds, a Schur method for robust eigenvalue assignment of a second-order singular system is proposed. The presented method also can be seen as an extension of the Schur method for eigenvalue assignment of a first-order system to a second-order system. However, our method directly deals with a second-order system and avoids to transform a second-order system into a first-order system. There are two open problems in the Schur method for eigenvalue assignment problem. One is how to determine the first Schur vector and another is how to order the eigenvalues in the Schur form. We provide the strategies for these two open problems, which are our main contributions. The efficiency of the proposed method is illustrated by some numerical examples.

提出了一种利用位移-速度-加速度反馈求解二阶奇异系统特征值分配问题的新方法。介绍了正则二次型铅笔的实Schur形式和闭环二阶系统的鲁棒性测量。在此基础上，提出了二阶奇异系统鲁棒特征值分配的Schur方法。所提出的方法也可以看作是一阶系统特征值分配的Schur方法到二阶系统的推广。然而，我们的方法直接处理二阶系统，避免了将二阶系统转化为一阶系统。特征值分配问题的Schur方法有两个开放问题。一个是如何确定第一个舒尔向量另一个是如何在舒尔形式中对特征值排序。我们为这两个开放问题提供了策略，这是我们的主要贡献。数值算例说明了该方法的有效性。

引用次数: 0

A Crank-Nicolson finite difference scheme for coupled nonlinear Schrödinger equations with saturable nonlinearity and nonlinear damping 具有饱和非线性和非线性阻尼的耦合非线性Schrödinger方程的Crank-Nicolson有限差分格式

IF 2.4 2区数学 Q1 MATHEMATICS, APPLIED

Applied Numerical Mathematics

Pub Date : 2025-09-15 DOI: 10.1016/j.apnum.2025.09.007

Anh Ha Le , Quan M. Nguyen

We propose a Crank-Nicolson finite difference scheme to simulate a 2D perturbed soliton interaction under the framework of coupled (2+1)D nonlinear Schrödinger equations with saturable nonlinearity and nonlinear damping. We rigorously demonstrate that the proposed numerical scheme achieves a second-order convergence rate in both the discrete

H_{0}^{1}

and

L^{2}

norms, relative to the time step and spatial mesh size. We establish the boundedness of discrete energies to prove the existence and uniqueness of the solutions derived from the Crank-Nicolson scheme. The validity of the analysis is confirmed through numerical simulations that apply to the corresponding coupled (2+1)D saturable nonlinear Schrödinger equations with damping terms.

我们提出了一种Crank-Nicolson有限差分格式来模拟具有可饱和非线性和非线性阻尼的（2+1）D耦合非线性Schrödinger方程框架下的二维摄动孤子相互作用。我们严格地证明了所提出的数值格式在离散的H01和L2范数下，相对于时间步长和空间网格尺寸，都达到了二阶收敛速率。建立了离散能量的有界性，证明了由Crank-Nicolson格式导出的解的存在唯一性。通过对相应的具有阻尼项的耦合（2+1）D饱和非线性Schrödinger方程的数值模拟，验证了分析的有效性。

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