Journal of Computational and Applied Mathematics最新文献_第4页

New error estimates for the conjugate gradient method 共轭梯度法的新误差估计

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-12 DOI: 10.1016/j.cam.2024.116357

Hanan Almutairi , Gérard Meurant , Lothar Reichel , Miodrag M. Spalević

{"title":"New error estimates for the conjugate gradient method","authors":"Hanan Almutairi , Gérard Meurant , Lothar Reichel , Miodrag M. Spalević","doi":"10.1016/j.cam.2024.116357","DOIUrl":"10.1016/j.cam.2024.116357","url":null,"abstract":"<div><div>The conjugate gradient method is the default iterative method for the solution of linear systems of equations with a large symmetric positive definite matrix <span><math><mi>A</mi></math></span>. The development of techniques for estimating the norm of the error in iterates computed by this method has received considerable attention. Available methods for bracketing the <span><math><mi>A</mi></math></span>-norm of the error evaluate pairs of Gauss and Gauss–Radau quadrature rules to determine lower and upper bounds. The latter rule requires a user to allocate a node (the Radau node) between the origin and the smallest eigenvalue of the system matrix. The determination of such a node generally demands further computations to estimate the location of the smallest eigenvalue; see, e.g., Golub and Meurant (1997), Golub and Meurant (2010), Golub and Strakoš (1994), Meurant (1997), Meurant (1999). An approach that avoids the need to know a lower bound for the smallest eigenvalue is to replace the Gauss–Radau quadrature rule by an anti-Gauss rule as described by Calvetti et al. (2000). However, this approach may sometimes yield inaccurate error norm estimates. This paper proposes the use of pairs of Gauss and associated optimal averaged Gauss quadrature rules to estimate the <span><math><mi>A</mi></math></span>-norm of the error in iterates determined by the conjugate gradient method.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116357"},"PeriodicalIF":2.1,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661496","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 solutions for second-order neutral volterra integro-differential equations: Stability analysis and finite difference method 二阶中性伏特拉积分微分方程的数值解法：稳定性分析和有限差分法

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-10 DOI: 10.1016/j.cam.2024.116371

Burcu Fedakar , Ilhame Amirali , Muhammet Enes Durmaz , Gabil M. Amiraliyev

引用次数: 0

Well-posedness of a class of evolutionary variational–hemivariational inequalities in contact mechanics 接触力学中一类进化变分-半变分不等式的好求解性

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-10 DOI: 10.1016/j.cam.2024.116366

Wei Xu , Weimin Han , Ting Li , Ziping Huang

引用次数: 0

A robust parameterized enhanced shift-splitting preconditioner for three-by-three block saddle point problems 三乘三块鞍点问题的鲁棒参数化增强移位分割预处理器

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-09 DOI: 10.1016/j.cam.2024.116358

Sk. Safique Ahmad, Pinki Khatun

引用次数: 0

Two-grid finite element methods for space-fractional nonlinear Schrödinger equations 空间分数非线性薛定谔方程的双网格有限元方法

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-09 DOI: 10.1016/j.cam.2024.116370

Yanping Chen , Hanzhang Hu

引用次数: 0

A study of the Orr–Sommerfeld and induction equations by Galerkin and Petrov–Galerkin spectral methods utilizing Chebyshev polynomials 利用切比雪夫多项式的 Galerkin 和 Petrov-Galerkin 频谱法研究 Orr-Sommerfeld 和感应方程

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-09 DOI: 10.1016/j.cam.2024.116374

Anna Piterskaya, Mikael Mortensen

{"title":"A study of the Orr–Sommerfeld and induction equations by Galerkin and Petrov–Galerkin spectral methods utilizing Chebyshev polynomials","authors":"Anna Piterskaya, Mikael Mortensen","doi":"10.1016/j.cam.2024.116374","DOIUrl":"10.1016/j.cam.2024.116374","url":null,"abstract":"<div><div>The article discusses two spectral methods, namely the Galerkin and the Petrov–Galerkin methods, for linear stability analysis of magneto-hydrodynamic (MHD) equations describing the flow of an electrically conducting fluid in the presence of a tangential magnetic field. The stability and spectral accuracy of both methods have been compared by examining the most unstable eigensolution of the Orr–Sommerfeld (OS) and induction equations. The Petrov–Galerkin spectral method (PGSM) used in this work has been developed by choosing function spaces and basis functions that always lead to banded coefficient matrices. The Galerkin spectral method (GSM), on the contrary, leads to dense matrices when Chebyshev polynomials are utilized in a weighted inner product space. We have found that both the GSM and the PGSM can produce results with minimal round-off errors, as confirmed by computing the most unstable eigenvalue of the OS equations (Re <span><math><mrow><mo>=</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>4</mn></mrow></msup></mrow></math></span>) to 14 decimal places of accuracy in double precision. We show that with properly scaled basis functions the GSM leads to coefficient matrices with bounded condition numbers, both for the OS equation and for the coupled OS and induction equations. This allows to achieve accurate results with double precision for any number of <span><math><mi>N</mi></math></span> for both the GSM and the PGSM. The analysis of the different behavior of the condition numbers suggests that the proposed two methods, based on the Chebyshev polynomials, can become a useful computer-based tool that is capable of finding a numerical solution to both the hydrodynamic and the MHD equations at very high Reynolds numbers.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116374"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661492","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Generic preconditioning based on the inverse with respect to the semi-inner product for iterative solvers for radial basis function interpolation 基于半内积逆的通用预处理，用于径向基函数插值的迭代求解器

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-09 DOI: 10.1016/j.cam.2024.116355

Dirk Martin , Gundolf Haase , Günter Offner

{"title":"Generic preconditioning based on the inverse with respect to the semi-inner product for iterative solvers for radial basis function interpolation","authors":"Dirk Martin , Gundolf Haase , Günter Offner","doi":"10.1016/j.cam.2024.116355","DOIUrl":"10.1016/j.cam.2024.116355","url":null,"abstract":"<div><div>The inverse matrix of radial basis function (RBF) interpolation systems can be stated concisely in terms of an inverse with respect to the semi-inner product induced by the interpolation kernel. Based on this representation, a separation of the solution process is justified and consequently splitting methods and an orthogonal projection method based on the semi-inner norm induced by the RBF are established. The requirements for preconditioning operators are derived and exemplary domain decomposition method preconditioning operators are presented. The introduced representation using the inverse with respect to the semi-inner product clarifies the coherence with well-known concepts from numerical linear algebra. The generic formulation of the preconditioned orthogonal projection method and the requirements for suitable preconditioners serve as building blocks to create solvers tailored for the specific assets of available hardware. Exemplary, design variants of the established subspace projection method and the respective preconditioners are tested on replicable data up to <span><math><msup><mrow><mn>2</mn></mrow><mrow><mn>19</mn></mrow></msup></math></span> interpolation centers.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116355"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661489","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

Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh–Stokes equations 半线性时分数雷利-斯托克斯方程的时序二阶两网格有限元法

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-09 DOI: 10.1016/j.cam.2024.116375

Zhijun Tan , Yunhua Zeng

{"title":"Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh–Stokes equations","authors":"Zhijun Tan , Yunhua Zeng","doi":"10.1016/j.cam.2024.116375","DOIUrl":"10.1016/j.cam.2024.116375","url":null,"abstract":"<div><div>In this paper, we have developed a temporal second-order two-grid FEM to solve the semilinear time-fractional Rayleigh–Stokes equations. The proposed two-grid FEM uses the L2-<span><math><msub><mrow><mn>1</mn></mrow><mrow><mi>σ</mi></mrow></msub></math></span> scheme and second order scheme to approximate the Caputo fractional derivative and the time first-order derivative in temporal direction and the standard FEM in spatial direction. The <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-norm and <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-norm stability and error estimates for the standard finite element solution and the two-grid solution are derived. The results shown that as long as the mesh sizes satisfy <span><math><mrow><mi>H</mi><mo>=</mo><msup><mrow><mi>h</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math></span> and <span><math><mrow><mi>H</mi><mo>=</mo><msup><mrow><mi>h</mi></mrow><mrow><mfrac><mrow><mi>r</mi></mrow><mrow><mn>2</mn><mi>r</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></msup></mrow></math></span> respectively, the two-grid algorithm can achieve asymptotically optimal approximation. Furthermore, the non-uniform L2-<span><math><msub><mrow><mn>1</mn></mrow><mrow><mi>σ</mi></mrow></msub></math></span> scheme was applied for temporal discretization to handle the weak singularity of the solution. Finally, the theoretical findings were confirmed by numerical results, and the effectiveness of the two-grid algorithm was demonstrated.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116375"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660934","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

An inexact semismooth Newton SAA-based algorithm for stochastic nonsmooth SOC complementarity problems with application to a stochastic power flow programming problem 基于非精确半光滑牛顿 SAA 算法的随机非光滑 SOC 互补问题，并应用于随机电力流编程问题

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-09 DOI: 10.1016/j.cam.2024.116361

Pin-Bo Chen , Gui-Hua Lin , Zhen-Ping Yang

引用次数: 0

Improving the accuracy and consistency of the energy quadratization method with an energy-optimized technique 用能量优化技术提高能量四分法的准确性和一致性

IF 2.1 2区数学

Journal of Computational and Applied Mathematics Pub Date : 2024-11-09 DOI: 10.1016/j.cam.2024.116368

Xiaoqing Meng , Aijie Cheng , Zhengguang Liu

{"title":"Improving the accuracy and consistency of the energy quadratization method with an energy-optimized technique","authors":"Xiaoqing Meng , Aijie Cheng , Zhengguang Liu","doi":"10.1016/j.cam.2024.116368","DOIUrl":"10.1016/j.cam.2024.116368","url":null,"abstract":"<div><div>We propose an energy-optimized invariant energy quadratization method to solve the gradient flow models in this paper, which requires only one linear energy-optimized step to correct the auxiliary variables on each time step. In addition to inheriting the benefits of the baseline and relaxed invariant energy quadratization method, our approach has several other advantages. Firstly, in the process of correcting auxiliary variables, we can directly solve linear programming problem by the energy-optimized technique, which greatly simplifies the nonlinear optimization problem in the previous relaxed invariant energy quadratization method. Secondly, we construct new linear unconditionally energy stable schemes by applying backward differentiation formulas and Crank–Nicolson formula, so that the accuracy in time can reach the first- and second-order. Thirdly, comparing with relaxation technique, the modified energy obtained by energy-optimized technique is closer to the original energy, and the accuracy and consistency of the numerical solutions can be improved. Ample numerical examples have been presented to demonstrate the accuracy, efficiency and energy stability of the proposed schemes.</div></div>","PeriodicalId":50226,"journal":{"name":"Journal of Computational and Applied Mathematics","volume":"459 ","pages":"Article 116368"},"PeriodicalIF":2.1,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142661491","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