Pub Date : 2024-06-10DOI: 10.1016/j.apnum.2024.06.006
Mengmeng Liu , Tao Guo , Mahmoud A. Zaky , Ahmed S. Hendy
An alternating direction implicit (ADI) scheme is proposed to study the numerical solution of a three-dimensional integrodifferential equation (IDE) with multi-term tempered singular kernels. Firstly, we employ the Crank-Nicolson method and the product integral (PI) rule on a uniform grid to approximate the temporal derivative and the multi-term tempered-type integral terms, thus establishing a second-order temporal discrete scheme. Then, a second-order finite difference method is used for spatial discretization and combined with the ADI technique to improve computational efficiency. Based on regularity conditions, the stability and convergence analysis of the ADI scheme is given by the energy argument. Finally, numerical examples confirm the results of the theoretical analysis and show that the method is effective.
本文提出了一种交替方向隐式(ADI)方案,用于研究具有多期回火奇异内核的三维积分微分方程(IDE)的数值解法。首先,我们在均匀网格上采用 Crank-Nicolson 方法和积积分(PI)规则来逼近时域导数和多期回火型积分项,从而建立了一个二阶时域离散方案。然后,采用二阶有限差分法进行空间离散化,并结合 ADI 技术提高计算效率。基于正则条件,通过能量论证给出了 ADI 方案的稳定性和收敛性分析。最后,数值实例证实了理论分析的结果,并表明该方法是有效的。
{"title":"An accurate second-order ADI scheme for three-dimensional tempered evolution problems arising in heat conduction with memory","authors":"Mengmeng Liu , Tao Guo , Mahmoud A. Zaky , Ahmed S. Hendy","doi":"10.1016/j.apnum.2024.06.006","DOIUrl":"10.1016/j.apnum.2024.06.006","url":null,"abstract":"<div><p>An alternating direction implicit (ADI) scheme is proposed to study the numerical solution of a three-dimensional integrodifferential equation (IDE) with multi-term tempered singular kernels. Firstly, we employ the Crank-Nicolson method and the product integral (PI) rule on a uniform grid to approximate the temporal derivative and the multi-term tempered-type integral terms, thus establishing a second-order temporal discrete scheme. Then, a second-order finite difference method is used for spatial discretization and combined with the ADI technique to improve computational efficiency. Based on regularity conditions, the stability and convergence analysis of the ADI scheme is given by the energy argument. Finally, numerical examples confirm the results of the theoretical analysis and show that the method is effective.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141391137","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}
Pub Date : 2024-06-07DOI: 10.1016/j.apnum.2024.05.025
Pep Mulet , Juan Ruiz-Álvarez , Chi-Wang Shu , Dionisio F. Yáñez
The weighted essentially non-oscillatory technique using a stencil of 2r points (WENO-2r) is an interpolatory method that consists in obtaining a higher approximation order from the non-linear combination of interpolants of nodes. The result is an interpolant of order 2r at the smooth parts and order when an isolated discontinuity falls at any grid interval of the large stencil except at the central one. Recently, a new WENO method based on Aitken-Neville's algorithm has been designed for interpolation of equally spaced data at the mid-points and presents progressive order of accuracy close to discontinuities. This paper is devoted to constructing a general progressive WENO method for non-necessarily uniformly spaced data and several variables interpolating in any point of the central interval. Also, we provide explicit formulas for linear and non-linear weights and prove the order obtained. Finally, some numerical experiments are presented to check the theoretical results.
{"title":"A non-separable progressive multivariate WENO-2r point value","authors":"Pep Mulet , Juan Ruiz-Álvarez , Chi-Wang Shu , Dionisio F. Yáñez","doi":"10.1016/j.apnum.2024.05.025","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.05.025","url":null,"abstract":"<div><p>The weighted essentially non-oscillatory technique using a stencil of 2<em>r</em> points (WENO-2<em>r</em>) is an interpolatory method that consists in obtaining a higher approximation order from the non-linear combination of interpolants of <span><math><mi>r</mi><mo>+</mo><mn>1</mn></math></span> nodes. The result is an interpolant of order 2<em>r</em> at the smooth parts and order <span><math><mi>r</mi><mo>+</mo><mn>1</mn></math></span> when an isolated discontinuity falls at any grid interval of the large stencil except at the central one. Recently, a new WENO method based on Aitken-Neville's algorithm has been designed for interpolation of equally spaced data at the mid-points and presents progressive order of accuracy close to discontinuities. This paper is devoted to constructing a general progressive WENO method for non-necessarily uniformly spaced data and several variables interpolating in any point of the central interval. Also, we provide explicit formulas for linear and non-linear weights and prove the order obtained. Finally, some numerical experiments are presented to check the theoretical results.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001399/pdfft?md5=9095bb1320e9de7aaba4607ca987adba&pid=1-s2.0-S0168927424001399-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141322522","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}
Pub Date : 2024-06-07DOI: 10.1016/j.apnum.2024.06.007
Yuyu He , Hongtao Chen , Bolin Chen
In this paper, we develop efficient compact difference schemes for the coupled nonlinear Schrödinger-KdV (CNLS-KdV) equations to conserve all the physical invariants, namely, the energy of oscillations, the number of plasmon, the number of particle and the momentum. By combining the exponential scalar auxiliary variable (E-SAV) approach, we reconstruct the original CNLS-KdV equations and adopt the compact difference method and Crank-Nicolson method to develop energy stable scheme. The E-SAV compact difference scheme preserves the total energy and the number of particle. We further introduce two Lagrange multipliers for the E-SAV reformulation system to develop compact difference scheme, which preserves exactly the number of plasmon and the momentum. At each time step for the second scheme, we only need to solve linear systems with constant coefficients and nonlinear quadratic algebraic equations which can be efficiently solved by Newton's iteration. Numerical experiments are given to show the effectiveness, accuracy and performance of the proposed schemes.
{"title":"Physical invariants-preserving compact difference schemes for the coupled nonlinear Schrödinger-KdV equations","authors":"Yuyu He , Hongtao Chen , Bolin Chen","doi":"10.1016/j.apnum.2024.06.007","DOIUrl":"10.1016/j.apnum.2024.06.007","url":null,"abstract":"<div><p>In this paper, we develop efficient compact difference schemes for the coupled nonlinear Schrödinger-KdV (CNLS-KdV) equations to conserve all the physical invariants, namely, the energy of oscillations, the number of plasmon, the number of particle and the momentum. By combining the exponential scalar auxiliary variable (E-SAV) approach, we reconstruct the original CNLS-KdV equations and adopt the compact difference method and Crank-Nicolson method to develop energy stable scheme. The E-SAV compact difference scheme preserves the total energy and the number of particle. We further introduce two Lagrange multipliers for the E-SAV reformulation system to develop compact difference scheme, which preserves exactly the number of plasmon and the momentum. At each time step for the second scheme, we only need to solve linear systems with constant coefficients and nonlinear quadratic algebraic equations which can be efficiently solved by Newton's iteration. Numerical experiments are given to show the effectiveness, accuracy and performance of the proposed schemes.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141399769","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}
Pub Date : 2024-06-07DOI: 10.1016/j.apnum.2024.06.002
S. Blanes , F. Casas , A. Escorihuela-Tomàs
New families of composition methods with processing of order 4 and 6 are presented and analyzed. They are specifically designed to be used for the numerical integration of differential equations whose vector field is separated into three or more parts which are explicitly solvable. The new schemes are shown to be more efficient than previous state-of-the-art splitting methods.
{"title":"Families of efficient low order processed composition methods","authors":"S. Blanes , F. Casas , A. Escorihuela-Tomàs","doi":"10.1016/j.apnum.2024.06.002","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.002","url":null,"abstract":"<div><p>New families of composition methods with processing of order 4 and 6 are presented and analyzed. They are specifically designed to be used for the numerical integration of differential equations whose vector field is separated into three or more parts which are explicitly solvable. The new schemes are shown to be more efficient than previous state-of-the-art splitting methods.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141322525","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}
Pub Date : 2024-06-07DOI: 10.1016/j.apnum.2024.06.005
Mariusz Frackiewicz, Henryk Palus
Objective image quality assessment involves the use of mathematical models to quantitatively describe image quality. FR-IQA (Full-Reference Image Quality Assessment) methods using reference images are also often used to evaluate image processing and computer vision algorithms. Quality indices often use gradient operators to express relevant visual information, such as edges. Fractional calculus has been applied in the last two decades in various fields such as signal processing, image processing, and pattern recognition. Fractional derivatives are generalizations of integer-order derivatives and can be computed using various operators such as the Riemann-Liouville, Caputo-Fabrizio, and Grünwald-Letnikov operators. In this paper, we propose a modification of the FSIMc image quality index by including fractional derivatives to extract and enhance edges. A study of the usefulness of fractional derivative in the FSIMc model was conducted by assessing Pearson, Spearman and Kendall correlations with MOS scores for images from the TID2013 and KADID-10k databases. Comparison of FD_FSIMc with the classic FSIMc shows an increase of several percent in the correlation coefficients for the modified index. The results obtained are superior to those other known approaches to FR-IQA that use fractional derivatives. The results encourage the use of fractional calculus.
{"title":"Application of fractional derivatives in image quality assessment indices","authors":"Mariusz Frackiewicz, Henryk Palus","doi":"10.1016/j.apnum.2024.06.005","DOIUrl":"10.1016/j.apnum.2024.06.005","url":null,"abstract":"<div><p>Objective image quality assessment involves the use of mathematical models to quantitatively describe image quality. FR-IQA (Full-Reference Image Quality Assessment) methods using reference images are also often used to evaluate image processing and computer vision algorithms. Quality indices often use gradient operators to express relevant visual information, such as edges. Fractional calculus has been applied in the last two decades in various fields such as signal processing, image processing, and pattern recognition. Fractional derivatives are generalizations of integer-order derivatives and can be computed using various operators such as the Riemann-Liouville, Caputo-Fabrizio, and Grünwald-Letnikov operators. In this paper, we propose a modification of the FSIMc image quality index by including fractional derivatives to extract and enhance edges. A study of the usefulness of fractional derivative in the FSIMc model was conducted by assessing Pearson, Spearman and Kendall correlations with MOS scores for images from the TID2013 and KADID-10k databases. Comparison of FD_FSIMc with the classic FSIMc shows an increase of several percent in the correlation coefficients for the modified index. The results obtained are superior to those other known approaches to FR-IQA that use fractional derivatives. The results encourage the use of fractional calculus.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001442/pdfft?md5=c5b941ec0099c65d784aced80ddf399e&pid=1-s2.0-S0168927424001442-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141403595","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}
Pub Date : 2024-06-06DOI: 10.1016/j.apnum.2024.06.001
Xuewei Liu , Zhanwen Yang , Qiang Ma , Xiaohua Ding
This paper proposes a local discontinuous Galerkin (LDG) method for the stochastic Korteweg-de Vries (KdV) equation with multi-dimensional multiplicative noise. In the mean square sense, we show that the numerical method is stable and it preserves energy conservation and energy dissipation. If the degree of the polynomial is n, the optimal error estimate in the mean square sense can reach as . Finally, structure-preserving and convergence are verified by numerical experiments.
{"title":"A structure-preserving local discontinuous Galerkin method for the stochastic KdV equation","authors":"Xuewei Liu , Zhanwen Yang , Qiang Ma , Xiaohua Ding","doi":"10.1016/j.apnum.2024.06.001","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.06.001","url":null,"abstract":"<div><p>This paper proposes a local discontinuous Galerkin (LDG) method for the stochastic Korteweg-de Vries (KdV) equation with multi-dimensional multiplicative noise. In the mean square sense, we show that the numerical method is <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> stable and it preserves energy conservation and energy dissipation. If the degree of the polynomial is <em>n</em>, the optimal error estimate in the mean square sense can reach as <span><math><mi>n</mi><mo>+</mo><mn>1</mn></math></span>. Finally, structure-preserving and convergence are verified by numerical experiments.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141322521","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}
Pub Date : 2024-06-05DOI: 10.1016/j.apnum.2024.05.024
Chengyu Chen , Xue-Lei Lin
It is well known that the implicit central difference discretization for unsteady advection diffusion equation (ADE) suffers from being time-consuming to solve when the advection term dominates. In this paper, we propose an operator-splitting scheme for the unsteady ADE, in which the ADE is firstly discretized by Crank-Nicolson (CN) scheme in time and central difference scheme in space; and then the discrete advection-diffusion problem is split as advection sub-problem and diffusion sub-problem at each time-level. The significance of the new scheme is that these sub-problems can be fast and directly solved within a linearithmic complexity (a linear-times-logarithm complexity) by means of fast sine transforms (FSTs). In particular, the complexity is independent of the dominance of the advection term. Theoretically, we show that proposed scheme is unconditionally stable and of second-order convergence in time and space. Numerical results are reported to show the efficiency of the proposed scheme.
{"title":"A fast solvable operator-splitting scheme for time-dependent advection diffusion equation","authors":"Chengyu Chen , Xue-Lei Lin","doi":"10.1016/j.apnum.2024.05.024","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.05.024","url":null,"abstract":"<div><p>It is well known that the implicit central difference discretization for unsteady advection diffusion equation (ADE) suffers from being time-consuming to solve when the advection term dominates. In this paper, we propose an operator-splitting scheme for the unsteady ADE, in which the ADE is firstly discretized by Crank-Nicolson (CN) scheme in time and central difference scheme in space; and then the discrete advection-diffusion problem is split as advection sub-problem and diffusion sub-problem at each time-level. The significance of the new scheme is that these sub-problems can be fast and directly solved within a linearithmic complexity (a linear-times-logarithm complexity) by means of fast sine transforms (FSTs). In particular, the complexity is independent of the dominance of the advection term. Theoretically, we show that proposed scheme is unconditionally stable and of second-order convergence in time and space. Numerical results are reported to show the efficiency of the proposed scheme.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141322523","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}
Pub Date : 2024-06-04DOI: 10.1016/j.apnum.2024.05.023
Federico Nudo
In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix–Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method.
{"title":"Two one-parameter families of nonconforming enrichments of the Crouzeix–Raviart finite element","authors":"Federico Nudo","doi":"10.1016/j.apnum.2024.05.023","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.05.023","url":null,"abstract":"<div><p>In this paper, we introduce two one-parameter families of quadratic polynomial enrichments designed to enhance the accuracy of the classical Crouzeix–Raviart finite element. These enrichments are realized by using weighted line integrals as enriched linear functionals and quadratic polynomial functions as enrichment functions. To validate the effectiveness of our approach, we conduct numerical experiments that confirm the improvement achieved by the proposed method.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0168927424001375/pdfft?md5=94fcf67e97df0d893b8352a366187794&pid=1-s2.0-S0168927424001375-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141292365","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}
Pub Date : 2024-06-04DOI: 10.1016/j.apnum.2024.05.022
Xin Li , Xiuling Hu
In this paper, based on the framework of the supplementary variable method, we present two classes of high-order, linearized, structure-preserving algorithms for simulating the regularized long-wave equation. The suggested schemes are as accurate and efficient as the recently proposed schemes in Jiang et al. (2022) [20], but share the nice features in two folds: (i) the first type of schemes conserves the original energy conservation, as opposed to a modified quadratic energy in [20]; (ii) the second type of schemes fills the gap of [20] by constructing high-order linear algorithms that preserve both two invariants of mass and momentum. We discretize the SVM systems by employing the algebraically stable Runge-Kutta method together with the prediction-correction technique in time and the Fourier pseudo-spectral method in space. The implementation benefits from solving the optimization problems subject to PDE constraints. Numerical examples and some comparisons are provided to show the effectiveness, accuracy and performance of the proposed schemes.
{"title":"Algebraically stable high-order multi-physical property-preserving methods for the regularized long-wave equation","authors":"Xin Li , Xiuling Hu","doi":"10.1016/j.apnum.2024.05.022","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.05.022","url":null,"abstract":"<div><p>In this paper, based on the framework of the supplementary variable method, we present two classes of high-order, linearized, structure-preserving algorithms for simulating the regularized long-wave equation. The suggested schemes are as accurate and efficient as the recently proposed schemes in Jiang et al. (2022) <span>[20]</span>, but share the nice features in two folds: (i) the first type of schemes conserves the original energy conservation, as opposed to a modified quadratic energy in <span>[20]</span>; (ii) the second type of schemes fills the gap of <span>[20]</span> by constructing high-order linear algorithms that preserve both two invariants of mass and momentum. We discretize the SVM systems by employing the algebraically stable Runge-Kutta method together with the prediction-correction technique in time and the Fourier pseudo-spectral method in space. The implementation benefits from solving the optimization problems subject to PDE constraints. Numerical examples and some comparisons are provided to show the effectiveness, accuracy and performance of the proposed schemes.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141290982","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}
Pub Date : 2024-05-29DOI: 10.1016/j.apnum.2024.05.021
Xin He
In this paper, we introduce two accelerated primal-dual methods tailored to address linearly constrained composite convex optimization problems, where the objective function is expressed as the sum of a possibly nondifferentiable function and a differentiable function with Lipschitz continuous gradient. The first method is the accelerated linearized augmented Lagrangian method (ALALM), which permits linearization to the differentiable function; the second method is the accelerated linearized proximal point algorithm (ALPPA), which enables linearization of both the differentiable function and the augmented term. By incorporating adaptive parameters, we demonstrate that ALALM achieves the convergence rate and the linear convergence rate under the assumption of convexity and strong convexity, respectively. Additionally, we establish that ALPPA enjoys the convergence rate in convex case and the convergence rate in strongly convex case. We provide numerical results to validate the effectiveness of the proposed methods.
{"title":"Accelerated primal-dual methods with adaptive parameters for composite convex optimization with linear constraints","authors":"Xin He","doi":"10.1016/j.apnum.2024.05.021","DOIUrl":"https://doi.org/10.1016/j.apnum.2024.05.021","url":null,"abstract":"<div><p>In this paper, we introduce two accelerated primal-dual methods tailored to address linearly constrained composite convex optimization problems, where the objective function is expressed as the sum of a possibly nondifferentiable function and a differentiable function with Lipschitz continuous gradient. The first method is the accelerated linearized augmented Lagrangian method (ALALM), which permits linearization to the differentiable function; the second method is the accelerated linearized proximal point algorithm (ALPPA), which enables linearization of both the differentiable function and the augmented term. By incorporating adaptive parameters, we demonstrate that ALALM achieves the <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> convergence rate and the linear convergence rate under the assumption of convexity and strong convexity, respectively. Additionally, we establish that ALPPA enjoys the <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><mi>k</mi><mo>)</mo></math></span> convergence rate in convex case and the <span><math><mi>O</mi><mo>(</mo><mn>1</mn><mo>/</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> convergence rate in strongly convex case. We provide numerical results to validate the effectiveness of the proposed methods.</p></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":null,"pages":null},"PeriodicalIF":2.8,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141249733","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}