The aim of the paper is to derive minimization algorithms based on the Nesterov accelerated gradient flow [Y. Nesterov, Gradient methods for minimizing composite objective function. Core discussion...
{"title":"Adaptation and assessement of projected Nesterov accelerated gradient flow to compute stationary states of nonlinear Schrödinger equations","authors":"Xavier Antoine, Chorouq Bentayaa, Jérémie Gaidamour","doi":"10.1080/00207160.2023.2294688","DOIUrl":"https://doi.org/10.1080/00207160.2023.2294688","url":null,"abstract":"The aim of the paper is to derive minimization algorithms based on the Nesterov accelerated gradient flow [Y. Nesterov, Gradient methods for minimizing composite objective function. Core discussion...","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138715985","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-11DOI: 10.1080/00207160.2023.2294432
Bin Wang, Xianfa Hu, Xinyuan Wu
We note a fact that stiff systems or differential equations that have highly oscillatory solutions cannot be solved efficiently using conventional methods. In this paper, we study two new classes o...
{"title":"Two new classes of exponential Runge–Kutta integrators for efficiently solving stiff systems or highly oscillatory problems","authors":"Bin Wang, Xianfa Hu, Xinyuan Wu","doi":"10.1080/00207160.2023.2294432","DOIUrl":"https://doi.org/10.1080/00207160.2023.2294432","url":null,"abstract":"We note a fact that stiff systems or differential equations that have highly oscillatory solutions cannot be solved efficiently using conventional methods. In this paper, we study two new classes o...","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":null,"pages":null},"PeriodicalIF":1.8,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138572567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-10DOI: 10.1080/00207160.2023.2279511
M. Yousuf, Shahzad Sarwar
AbstractA computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.Keywords: Inhomogeneous parabolic PDEsReal pole rational approximationComputationally fastfractional distributed order PDEsRiesz derivativeDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.
{"title":"A fast third order algorithm for two dimensional inhomogeneous fractional parabolic partial differential equations","authors":"M. Yousuf, Shahzad Sarwar","doi":"10.1080/00207160.2023.2279511","DOIUrl":"https://doi.org/10.1080/00207160.2023.2279511","url":null,"abstract":"AbstractA computationally fast third order numerical algorithm is developed for inhomogeneous parabolic partial differential equations. The algorithm is based on a third order method developed by using a rational approximation with single Gaussian quadrature pole to avoid complex arithmetic and to achieve high efficiency and accuracy. Difficulties with computational efficiency and accuracy are addressed using partial fraction decomposition technique. Third order accuracy and convergence of the method is proved analytically and verified numerically. Several classical as well as more challenging fractional and distributed order inhomogeneous problems are considered to perform numerical experiments. Computational efficiency of the method is demonstrated through central processing unit (CPU) time and is given in the convergence tables.Keywords: Inhomogeneous parabolic PDEsReal pole rational approximationComputationally fastfractional distributed order PDEsRiesz derivativeDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136450","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1080/00207160.2023.2279006
Shuaikang Wang, Yongbin Ge, Tingfu Ma
AbstractFirst, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.Keywords: Three-dimensional nonlinear wave equationNonlinear difference schemeSixth-order accuracyLinearized difference schemeRichardson extrapolationDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThis work is partially supported by National Natural Science Foundation of China (12161067), Natural Science Foundation of Ningxia (2022AAC02023, 2022AAC03313), the Key Research and Development Program of Ningxia (2021YCZX0036, 2021BEB04053), the Scientific Research Program in Higher Institution of Ningxia (NGY2020110), National Youth Top-notch Talent Support Program of Ningxia.Data AvailabilityThe data used to support the findings of this study are available from the corresponding author upon request. Conflicts of InterestThe authors declare no conflict of interest.
{"title":"Sixth-order Finite Difference Schemes for Nonlinear Wave Equations with Variable Coefficients in Three Dimensions","authors":"Shuaikang Wang, Yongbin Ge, Tingfu Ma","doi":"10.1080/00207160.2023.2279006","DOIUrl":"https://doi.org/10.1080/00207160.2023.2279006","url":null,"abstract":"AbstractFirst, a nonlinear difference scheme is proposed to solve the three-dimensional (3D) nonlinear wave equation by combining the correction technique of truncation error remainder in time and a sixth-order finite difference operator in space, resulting in fourth-order accuracy in time and sixth-order accuracy in space. Then, the Richardson extrapolation method is applied to improve the temporal accuracy from the fourth-order to the sixth-order. To enhance computational efficiency, a linearized difference scheme is obtained by linear interpolation based on the nonlinear scheme. In addition, the stability of the linearized scheme is proved. Finally, the accuracy, stability and efficiency of the two proposed schemes are tested numerically.Keywords: Three-dimensional nonlinear wave equationNonlinear difference schemeSixth-order accuracyLinearized difference schemeRichardson extrapolationDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThis work is partially supported by National Natural Science Foundation of China (12161067), Natural Science Foundation of Ningxia (2022AAC02023, 2022AAC03313), the Key Research and Development Program of Ningxia (2021YCZX0036, 2021BEB04053), the Scientific Research Program in Higher Institution of Ningxia (NGY2020110), National Youth Top-notch Talent Support Program of Ningxia.Data AvailabilityThe data used to support the findings of this study are available from the corresponding author upon request. Conflicts of InterestThe authors declare no conflict of interest.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135371205","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-23DOI: 10.1080/00207160.2023.2274278
Hongling Shi, Minghui Song, Mingzhu Liu
AbstractThis paper constructs a modified partially truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift and diffusion coefficients grow superlinearly. We divide the coefficients of SDEPCAs into global Lipschitz continuous and superlinearly growing parts. Our method only truncates the superlinear terms of the coefficients to overcome the potential explosions caused by the nonlinearities of the coefficients. The strong convergence theory of this method is established and the 1/2 convergence rate is presented. Furthermore, an explicit scheme is developed to preserve the mean square exponential stability of underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results.Keywords: Modified partially truncated EM methodstochastic differential equations with piecewise continuous argumentsstrong convergenceconvergence ratemean square exponential stabilityDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThis work is supported by the NSF of PR China (No. 12071101 and No. 11671113).
{"title":"Convergence and stability of modified partially truncated Euler-Maruyama method for stochastic differential equations with piecewise continuous arguments","authors":"Hongling Shi, Minghui Song, Mingzhu Liu","doi":"10.1080/00207160.2023.2274278","DOIUrl":"https://doi.org/10.1080/00207160.2023.2274278","url":null,"abstract":"AbstractThis paper constructs a modified partially truncated Euler-Maruyama (EM) method for stochastic differential equations with piecewise continuous arguments (SDEPCAs), where the drift and diffusion coefficients grow superlinearly. We divide the coefficients of SDEPCAs into global Lipschitz continuous and superlinearly growing parts. Our method only truncates the superlinear terms of the coefficients to overcome the potential explosions caused by the nonlinearities of the coefficients. The strong convergence theory of this method is established and the 1/2 convergence rate is presented. Furthermore, an explicit scheme is developed to preserve the mean square exponential stability of underlying SDEPCAs. Several numerical experiments are offered to illustrate the theoretical results.Keywords: Modified partially truncated EM methodstochastic differential equations with piecewise continuous argumentsstrong convergenceconvergence ratemean square exponential stabilityDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThis work is supported by the NSF of PR China (No. 12071101 and No. 11671113).","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135368481","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-22DOI: 10.1080/00207160.2023.2274277
Zhaolin Yan, Jianfang Gao
AbstractThe purpose of this paper is to study oscillation and non-oscillation of Runge-Kutta methods for linear mixed type impulsive differential equations with piecewise constant arguments. The conditions for oscillation and non-oscillation of numerical solutions are obtained. Also conditions under which Runge-Kutta methods can preserve the oscillation and non-oscillation of linear mixed type impulsive differential equations with piecewise constant arguments are obtained. Moreover, the interpolation function of numerical solutions is introduced and the properties of the interpolation function is discussed. It turns out that the zeros of the interpolation function converge to ones of the analytic solution with the same order of accuracy as that of the corresponding Runge-Kutta method. To confirm the theoretical results, the numerical examples are given.Keywords: oscillationnumerical solutionRunge-Kutta methodsimpulsive delay differential equationspiecewise constant argumentsDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.
{"title":"Numerical oscillation and non-oscillation analysis of the mixed type impulsive differential equation with piecewise constant arguments","authors":"Zhaolin Yan, Jianfang Gao","doi":"10.1080/00207160.2023.2274277","DOIUrl":"https://doi.org/10.1080/00207160.2023.2274277","url":null,"abstract":"AbstractThe purpose of this paper is to study oscillation and non-oscillation of Runge-Kutta methods for linear mixed type impulsive differential equations with piecewise constant arguments. The conditions for oscillation and non-oscillation of numerical solutions are obtained. Also conditions under which Runge-Kutta methods can preserve the oscillation and non-oscillation of linear mixed type impulsive differential equations with piecewise constant arguments are obtained. Moreover, the interpolation function of numerical solutions is introduced and the properties of the interpolation function is discussed. It turns out that the zeros of the interpolation function converge to ones of the analytic solution with the same order of accuracy as that of the corresponding Runge-Kutta method. To confirm the theoretical results, the numerical examples are given.Keywords: oscillationnumerical solutionRunge-Kutta methodsimpulsive delay differential equationspiecewise constant argumentsDisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135461885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-18DOI: 10.1080/00207160.2023.2272589
Fariba Bakrani Balani, Masoud Hajarian
AbstractWe introduce a new block preconditioner for the solution of weighted Toeplitz regularized least-squares problems written in augmented system form. The proposed preconditioner is obtained based on the new splitting of coefficient matrix which results in an unconditionally convergent stationary iterative method. Spectral analysis of the preconditioned matrix is investigated. In particular, we show that the preconditioned matrix has a very nice eigenvalue distribution which can lead to fast convergence of the preconditioned Krylov subspace methods such as GMRES. Numerical experiments are reported to demonstrate the performance of preconditioner used with (flexible) GMRES method in the solution of augmented system form of weighted Toeplitz regularized least-squares problems.Keywords: PreconditioningSplittingLeast-squares problemsWeighted Toeplitz matricesAMS classification 2010:: 65F1065F50DisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe authors express their thanks to the referees for the comments and constructive suggestions, which were valuable in improving the quality of the manuscript.
{"title":"A new block preconditioner for weighted Toeplitz regularized least-squares problems","authors":"Fariba Bakrani Balani, Masoud Hajarian","doi":"10.1080/00207160.2023.2272589","DOIUrl":"https://doi.org/10.1080/00207160.2023.2272589","url":null,"abstract":"AbstractWe introduce a new block preconditioner for the solution of weighted Toeplitz regularized least-squares problems written in augmented system form. The proposed preconditioner is obtained based on the new splitting of coefficient matrix which results in an unconditionally convergent stationary iterative method. Spectral analysis of the preconditioned matrix is investigated. In particular, we show that the preconditioned matrix has a very nice eigenvalue distribution which can lead to fast convergence of the preconditioned Krylov subspace methods such as GMRES. Numerical experiments are reported to demonstrate the performance of preconditioner used with (flexible) GMRES method in the solution of augmented system form of weighted Toeplitz regularized least-squares problems.Keywords: PreconditioningSplittingLeast-squares problemsWeighted Toeplitz matricesAMS classification 2010:: 65F1065F50DisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe authors express their thanks to the referees for the comments and constructive suggestions, which were valuable in improving the quality of the manuscript.","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135889289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-10DOI: 10.1080/00207160.2023.2269430
Peng Wang, Detong Zhu
AbstractIn this paper, we propose a single timescale stochastic quasi-Newton method for solving the stochastic optimization problems. The objective function of the problem is a composition of two smooth functions and their derivatives are not available. The algorithm sets to approximate sequences to estimate the gradient of the composite objective function and the inner function. The matrix correction parameters are given in BFGS update form for avoiding the assumption that Hessian matrix of objective is positive definite. We show the global convergence of the algorithm. The algorithm achieves the complexity O(ϵ−1) to find an ϵ−approximate stationary point and ensure that the expectation of the squared norm of the gradient is smaller than the given accuracy tolerance ϵ. The numerical results of nonconvex binary classification problem using the support vector machine and a multicall classification problem using neural networks are reported to show the effectiveness of the algorithm.Keywords: stochastic optimizationquasi-Newton methodBFGS update techniquemachine learning2010: 49M3765K0590C3090C56DisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe author thanks the support of National Natural Science Foundation (11371253) and Hainan Natural Science Foundation (120MS029).
{"title":"A single timescale stochastic quasi-Newton method for stochastic optimization","authors":"Peng Wang, Detong Zhu","doi":"10.1080/00207160.2023.2269430","DOIUrl":"https://doi.org/10.1080/00207160.2023.2269430","url":null,"abstract":"AbstractIn this paper, we propose a single timescale stochastic quasi-Newton method for solving the stochastic optimization problems. The objective function of the problem is a composition of two smooth functions and their derivatives are not available. The algorithm sets to approximate sequences to estimate the gradient of the composite objective function and the inner function. The matrix correction parameters are given in BFGS update form for avoiding the assumption that Hessian matrix of objective is positive definite. We show the global convergence of the algorithm. The algorithm achieves the complexity O(ϵ−1) to find an ϵ−approximate stationary point and ensure that the expectation of the squared norm of the gradient is smaller than the given accuracy tolerance ϵ. The numerical results of nonconvex binary classification problem using the support vector machine and a multicall classification problem using neural networks are reported to show the effectiveness of the algorithm.Keywords: stochastic optimizationquasi-Newton methodBFGS update techniquemachine learning2010: 49M3765K0590C3090C56DisclaimerAs a service to authors and researchers we are providing this version of an accepted manuscript (AM). Copyediting, typesetting, and review of the resulting proofs will be undertaken on this manuscript before final publication of the Version of Record (VoR). During production and pre-press, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal relate to these versions also. AcknowledgmentsThe author thanks the support of National Natural Science Foundation (11371253) and Hainan Natural Science Foundation (120MS029).","PeriodicalId":13911,"journal":{"name":"International Journal of Computer Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136293592","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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