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Preserving invariant domains and strong approximation of stochastic differential equations 随机微分方程的保不变域与强逼近
IF 2.4 2区 数学
Applied Numerical Mathematics Pub Date : 2026-03-01 Epub Date: 2025-11-11 DOI: 10.1016/j.apnum.2025.11.003
Utku Erdoğan , Gabriel Lord
{"title":"Preserving invariant domains and strong approximation of stochastic differential equations","authors":"Utku Erdoğan ,&nbsp;Gabriel Lord","doi":"10.1016/j.apnum.2025.11.003","DOIUrl":"10.1016/j.apnum.2025.11.003","url":null,"abstract":"<div><div>In this paper, we develop numerical methods for solving stochastic differential equations with solutions that evolve within a hypercube <em>D</em> in <span><math><msup><mi>R</mi><mi>d</mi></msup></math></span>. Our approach is based on a convex combination of two numerical flows, both of which are constructed from positivity preserving methods. The strong convergence of the Euler version of the method is proven to be of order <span><math><mstyle><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></math></span>, and numerical examples are provided to demonstrate that, in some cases, first-order convergence is observed in practice. We compare the Euler and Milstein versions of these new methods to existing domain preservation methods in the literature and observe our methods are robust, more widely applicable and that, in most cases, have a smaller error constant.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"221 ","pages":"Pages 30-45"},"PeriodicalIF":2.4,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145616684","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-09-19 DOI: 10.1016/j.apnum.2025.09.008
Mahmoud A. Zaky
{"title":"A linearized two-dimensional Galerkin-L1 spectral method with diagonalization for time-fractional diffusion equations with delay","authors":"Mahmoud A. Zaky","doi":"10.1016/j.apnum.2025.09.008","DOIUrl":"10.1016/j.apnum.2025.09.008","url":null,"abstract":"<div><div>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 <span><math><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>ϑ</mi><mo>)</mo></mrow></math></span>-order accuracy in time, as well as validating the robustness and efficiency of the proposed method across different fractional orders and delay parameters.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 1-12"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145265082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-10 DOI: 10.1016/j.apnum.2025.10.005
Congpei An , Mou Cai
{"title":"Parameter choice strategies for regularized least squares approximation of noisy continuous functions on the unit circle","authors":"Congpei An ,&nbsp;Mou Cai","doi":"10.1016/j.apnum.2025.10.005","DOIUrl":"10.1016/j.apnum.2025.10.005","url":null,"abstract":"<div><div>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 <span><math><msub><mi>L</mi><mn>2</mn></msub></math></span> 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.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 84-103"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Plug-and-play superiorization 即插即用superiorization
IF 2.4 2区 数学
Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-10 DOI: 10.1016/j.apnum.2025.10.002
Jon Henshaw , Aviv Gibali , Thomas Humphries
{"title":"Plug-and-play superiorization","authors":"Jon Henshaw ,&nbsp;Aviv Gibali ,&nbsp;Thomas Humphries","doi":"10.1016/j.apnum.2025.10.002","DOIUrl":"10.1016/j.apnum.2025.10.002","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 29-43"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A unified preconditioned minimal residual (PMR) algorithm for matrix problems: Linear systems, multiple right-hand sides linear systems, least squares problems, inversion and pseudo-inversion with application to color image encryption 矩阵问题的统一预条件最小残差（PMR）算法：线性系统，多重右手边线性系统，最小二乘问题，反演和伪反演及其在彩色图像加密中的应用
IF 2.4 2区 数学
Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-22 DOI: 10.1016/j.apnum.2025.10.012
Akbar Shirilord, Mehdi Dehghan
{"title":"A unified preconditioned minimal residual (PMR) algorithm for matrix problems: Linear systems, multiple right-hand sides linear systems, least squares problems, inversion and pseudo-inversion with application to color image encryption","authors":"Akbar Shirilord,&nbsp;Mehdi Dehghan","doi":"10.1016/j.apnum.2025.10.012","DOIUrl":"10.1016/j.apnum.2025.10.012","url":null,"abstract":"<div><div>In this article, we introduce a preconditioned minimal residual (PMR) algorithm designed to address a wide range of matrix equations and linear systems. We illustrate the efficacy of this algorithm through several numerical examples, including the solution of matrix equations. Notably, we tackle various significant problems such as the minimization of Frobenius norms, least squares optimization, and the computation of the Moore-Penrose pseudo-inverse. Convergence analysis shows that it converges without any constraints and for any initial guess, although this algorithm is more efficient when the matrices are sparse. To validate the effectiveness of our proposed iterative algorithm, we offer various numerical examples by large matrices. As an application of the matrix equation, we explore a method for encrypting and decrypting color images.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 216-245"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145413757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Directional gradient and curvature approximation via Legendre quadrature in unconstrained optimization 无约束优化中基于勒让德正交的方向梯度和曲率逼近
IF 2.4 2区 数学
Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-11-03 DOI: 10.1016/j.apnum.2025.10.018
Stefan R. Panic
{"title":"Directional gradient and curvature approximation via Legendre quadrature in unconstrained optimization","authors":"Stefan R. Panic","doi":"10.1016/j.apnum.2025.10.018","DOIUrl":"10.1016/j.apnum.2025.10.018","url":null,"abstract":"<div><div>A novel quasi-Newton method for solving systems of nonlinear equations by leveraging directional approximations of both the gradient and curvature via Legendre-Gauss quadrature has been proposed. The method reformulates the root-finding problem <span><math><mrow><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></math></span> as the minimization of a scalar merit function <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mrow><mo>∥</mo><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∥</mo></mrow><mn>2</mn></msup></mrow></math></span>, and approximates second-order information using three-node orthogonal polynomial integration along search directions. A rank-1 approximation of the Jacobian action is constructed without requiring explicit derivative information. The resulting scheme features a scalar curvature parameter <span><math><msub><mi>γ</mi><mi>k</mi></msub></math></span> that dynamically controls the step size, enabling stable updates through an inexact Armijo-type line search. The method remains numerically stable across problems without requiring explicit Jacobian evaluations or storage. We establish global convergence under mild assumptions and explore quasi-Newton properties under additional curvature conditions. Extensive numerical experiments demonstrate competitive accuracy, robustness, and reduced iteration counts compared to existing diagonal quasi-Newton methods.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 294-309"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145463216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A structure-preserving variational time stepping scheme for Cahn-Hilliard equation with dynamic boundary conditions 具有动态边界条件的Cahn-Hilliard方程的保结构变分时间步进格式
IF 2.4 2区 数学
Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-11-08 DOI: 10.1016/j.apnum.2025.11.001
Changlun Ye , Hai Bi , Liangkun Xu , Xianbing Luo
{"title":"A structure-preserving variational time stepping scheme for Cahn-Hilliard equation with dynamic boundary conditions","authors":"Changlun Ye ,&nbsp;Hai Bi ,&nbsp;Liangkun Xu ,&nbsp;Xianbing Luo","doi":"10.1016/j.apnum.2025.11.001","DOIUrl":"10.1016/j.apnum.2025.11.001","url":null,"abstract":"<div><div>In this paper, for the Cahn-Hilliard equation with dynamic boundary conditions, we establish a variational time stepping numerical scheme integrated with finite element methods. This scheme is a structure-preserving scheme, which effectively maintains the inherent physical properties of the continuous model including mass conservation and energy dissipation. We demonstrate the existence of discrete solutions without restrictions on the discretization parameters, and establish the uniqueness under mild conditions. Finally, we present ample numerical results which validate our theoretical findings and demonstrate that our numerical scheme can achieve second-order convergence in time. We also apply our scheme to the KLS (proposed by P. Knopf, K.F. Lam, and J. Stange) and KLLM (proposed by P. Knopf, K. F. Lam, C. Liu, and S. Metzger) models, two other Cahn-Hilliard models with dynamic boundaries, and verify that the solutions of KLS model converge to the solutions of KLLM model numerically.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 346-372"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145576255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Optimal error estimates and stability of a local discontinuous Galerkin method for the stochastic two-dimensional KdV equation 随机二维KdV方程局部不连续Galerkin方法的最优误差估计和稳定性
IF 2.4 2区 数学
Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-14 DOI: 10.1016/j.apnum.2025.10.008
Xuewei Liu , Zhenyu Wang , Xiaohua Ding , Shao-Liang Zhang
{"title":"Optimal error estimates and stability of a local discontinuous Galerkin method for the stochastic two-dimensional KdV equation","authors":"Xuewei Liu ,&nbsp;Zhenyu Wang ,&nbsp;Xiaohua Ding ,&nbsp;Shao-Liang Zhang","doi":"10.1016/j.apnum.2025.10.008","DOIUrl":"10.1016/j.apnum.2025.10.008","url":null,"abstract":"<div><div>The stochastic two-dimensional KdV equation arises as a mathematical model for shallow water wave dynamics in physical systems. To efficiently handle the equation’s high-order spatial derivatives and stochastic terms, a local discontinuous Galerkin method is proposed. The method is proved to be <span><math><msup><mi>L</mi><mn>2</mn></msup></math></span>-stable and to achieve the optimal mean-square convergence rate of order <span><math><mrow><mi>N</mi><mo>+</mo><mn>1</mn></mrow></math></span> when degree-<span><math><mi>N</mi></math></span> polynomials are used. For temporal discretization, the implicit midpoint method is applied, and the restarted Generalized Minimum Residual method is employed to solve the resulting linear systems in two-dimensional simulations. Numerical experiments demonstrate optimal convergence rates and confirm both the theoretical analysis and the effectiveness of the method.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 310-328"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145526261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-10 DOI: 10.1016/j.apnum.2025.10.004
Behzad Kafash
{"title":"Numerical approximation of constrained optimal control problems in delayed systems using an enhanced Rayleigh-Ritz algorithm","authors":"Behzad Kafash","doi":"10.1016/j.apnum.2025.10.004","DOIUrl":"10.1016/j.apnum.2025.10.004","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 104-122"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323038","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Applied Numerical Mathematics Pub Date : 2026-02-01 Epub Date: 2025-10-09 DOI: 10.1016/j.apnum.2025.10.006
Feng Shao , Hu Shao , Bin Wu , Haijun Wang , Pengjie Liu , Meixing Liu
{"title":"A conjugate gradient algorithmic framework for unconstrained optimization with applications: Convergence and rate analyses","authors":"Feng Shao ,&nbsp;Hu Shao ,&nbsp;Bin Wu ,&nbsp;Haijun Wang ,&nbsp;Pengjie Liu ,&nbsp;Meixing Liu","doi":"10.1016/j.apnum.2025.10.006","DOIUrl":"10.1016/j.apnum.2025.10.006","url":null,"abstract":"<div><div>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.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"220 ","pages":"Pages 13-28"},"PeriodicalIF":2.4,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145323041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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