Pub Date : 2025-01-15DOI: 10.1016/j.apnum.2025.01.005
Meng Li , Dan Wang , Junjun Wang , Xiaolong Zhao
In this paper, the variable-time-step weighted implicit-explicit (IMEX) finite element methods (FEMs) are developed for some types of nonlinear real- or complex-valued evolution equations. Extensive research is conducted on the discrete orthogonal convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels of the variable-time-step weighted IMEX scheme, elucidating their crucial properties in both real- and complex-valued scenarios. We prove that the scheme exhibits optimal convergence without any restrictions on the time-space step ratio. At last, several numerical examples are provided to demonstrate our theoretical results. With the weighted parameter , the scheme in this work can degenerate into a special case: variable-time-step two-step backward differentiation formula (BDF2) scheme, and the convergence analysis in this special case was introduced in Liao et al. (2020) [15] and Liao et al. (2021) [29].
{"title":"Variable-time-step weighted IMEX FEMs for nonlinear evolution equations","authors":"Meng Li , Dan Wang , Junjun Wang , Xiaolong Zhao","doi":"10.1016/j.apnum.2025.01.005","DOIUrl":"10.1016/j.apnum.2025.01.005","url":null,"abstract":"<div><div>In this paper, the variable-time-step weighted implicit-explicit (IMEX) finite element methods (FEMs) are developed for some types of nonlinear real- or complex-valued evolution equations. Extensive research is conducted on the discrete orthogonal convolution (DOC) kernels and the discrete complementary convolution (DCC) kernels of the variable-time-step weighted IMEX scheme, elucidating their crucial properties in both real- and complex-valued scenarios. We prove that the scheme exhibits optimal convergence without any restrictions on the time-space step ratio. At last, several numerical examples are provided to demonstrate our theoretical results. With the weighted parameter <span><math><mi>θ</mi><mo>=</mo><mn>1</mn></math></span>, the scheme in this work can degenerate into a special case: variable-time-step two-step backward differentiation formula (BDF2) scheme, and the convergence analysis in this special case was introduced in Liao et al. (2020) <span><span>[15]</span></span> and Liao et al. (2021) <span><span>[29]</span></span>.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 123-143"},"PeriodicalIF":2.2,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141889","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 : 2025-01-15DOI: 10.1016/j.apnum.2025.01.006
Fengna Yan , Yinhua Xia
We propose a high-order average bound-preserving limiter for implicit backward differentiation formula (BDF) and local discontinuous Galerkin (LDG) discretizations applied to convection-diffusion-reaction equations. Our approach first imposes cell average bounds of the numerical solution using the Karush-Kuhn-Tucker (KKT) limiter and then enforces pointwise bounds with an explicit bound-preserving limiter. This method reduces the number of constraints compared to using only the KKT system to directly ensure pointwise bounds, resulting in a relatively small system of nonlinear equations to solve at each time step. We prove the unique solvability of the proposed average bound-preserving BDF-LDG discretizations. Furthermore, we establish the stability and optimal error estimates for the second-order average bound-preserving BDF2-LDG discretization. The unique solvability and stability are derived by transforming the KKT-limited cell average bounds-preserving LDG discretizations into a variational inequality. The error estimates are derived using the cell average bounds-preserving inequality constraints. Numerical results are presented to validate the accuracy and effectiveness of the proposed method in preserving the bounds.
{"title":"Analysis of average bound preserving time-implicit discretizations for convection-diffusion-reaction equation","authors":"Fengna Yan , Yinhua Xia","doi":"10.1016/j.apnum.2025.01.006","DOIUrl":"10.1016/j.apnum.2025.01.006","url":null,"abstract":"<div><div>We propose a high-order average bound-preserving limiter for implicit backward differentiation formula (BDF) and local discontinuous Galerkin (LDG) discretizations applied to convection-diffusion-reaction equations. Our approach first imposes cell average bounds of the numerical solution using the Karush-Kuhn-Tucker (KKT) limiter and then enforces pointwise bounds with an explicit bound-preserving limiter. This method reduces the number of constraints compared to using only the KKT system to directly ensure pointwise bounds, resulting in a relatively small system of nonlinear equations to solve at each time step. We prove the unique solvability of the proposed average bound-preserving BDF-LDG discretizations. Furthermore, we establish the stability and optimal error estimates for the second-order average bound-preserving BDF2-LDG discretization. The unique solvability and stability are derived by transforming the KKT-limited cell average bounds-preserving LDG discretizations into a variational inequality. The error estimates are derived using the cell average bounds-preserving inequality constraints. Numerical results are presented to validate the accuracy and effectiveness of the proposed method in preserving the bounds.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 103-122"},"PeriodicalIF":2.2,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141891","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 : 2025-01-14DOI: 10.1016/j.apnum.2025.01.004
Kaido Lätt, Arvet Pedas, Hanna Britt Soots
We investigate a class of singular fractional integro-differential equations with non-constant coefficients. After reformulating the original problem as a cordial Volterra integral equation we study the unique solvability of the underlying problem. We construct a numerical method based on collocation techniques to find an approximation to the solution of the original problem and analyse the convergence and the convergence order of the proposed method. Additionally, we present the results of some numerical experiments.
{"title":"Singular fractional integro-differential equations with non-constant coefficients","authors":"Kaido Lätt, Arvet Pedas, Hanna Britt Soots","doi":"10.1016/j.apnum.2025.01.004","DOIUrl":"10.1016/j.apnum.2025.01.004","url":null,"abstract":"<div><div>We investigate a class of singular fractional integro-differential equations with non-constant coefficients. After reformulating the original problem as a cordial Volterra integral equation we study the unique solvability of the underlying problem. We construct a numerical method based on collocation techniques to find an approximation to the solution of the original problem and analyse the convergence and the convergence order of the proposed method. Additionally, we present the results of some numerical experiments.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 179-192"},"PeriodicalIF":2.2,"publicationDate":"2025-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143142752","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}
A novel approach is presented to tackle the Allen-Cahn equation arising from phase separation in alloys, by utilizing the hybrid high-order (HHO) method on polygonal meshes. The primary challenge in this equation lies in employing a straightforward Gronwall inequality-type argument for error estimation with exponential growth factor as ε approaches zero. The application of the discrete Lyapunov functional and the discrete HHO spectrum estimate of the linearized Allen-Cahn operator are used to overcome this exponential growth factor and achieve polynomial growth of order for error bounds in error estimations. Rigorous convergence analyses are established for the fully implicit schemes, which are energy stable. However, due to the implicit processing of the nonlinear term, the computational cost significantly increases. To enhance computational efficiency, a static condensation process is hired by using the HHO method, resulting in optimal convergence rates in norm. Finally, various numerical experiments on diverse meshes are conducted to validate our theoretical findings.
{"title":"Error estimates with polynomial growth O(ε−1) for the HHO method on polygonal meshes of the Allen-Cahn model","authors":"Naresh Kumar , Ajeet Singh , Ram Jiwari , J.Y. Yuan","doi":"10.1016/j.apnum.2025.01.001","DOIUrl":"10.1016/j.apnum.2025.01.001","url":null,"abstract":"<div><div>A novel approach is presented to tackle the Allen-Cahn equation arising from phase separation in alloys, by utilizing the hybrid high-order (HHO) method on polygonal meshes. The primary challenge in this equation lies in employing a straightforward Gronwall inequality-type argument for error estimation with exponential growth factor <span><math><msup><mrow><mi>e</mi></mrow><mrow><mo>(</mo><mi>C</mi><mi>T</mi><mo>/</mo><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></msup></math></span> as <em>ε</em> approaches zero. The application of the discrete Lyapunov functional and the discrete HHO spectrum estimate of the linearized Allen-Cahn operator <span><math><msubsup><mrow><mi>λ</mi></mrow><mrow><mi>A</mi><mi>C</mi></mrow><mrow><mi>H</mi><mi>H</mi><mi>O</mi></mrow></msubsup></math></span> are used to overcome this exponential growth factor and achieve polynomial growth of order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>ε</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> for error bounds in error estimations. Rigorous convergence analyses are established for the fully implicit schemes, which are energy stable. However, due to the implicit processing of the nonlinear term, the computational cost significantly increases. To enhance computational efficiency, a static condensation process is hired by using the HHO method, resulting in optimal convergence rates in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm. Finally, various numerical experiments on diverse meshes are conducted to validate our theoretical findings.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 78-102"},"PeriodicalIF":2.2,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141886","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 : 2025-01-09DOI: 10.1016/j.apnum.2025.01.003
Liya Liu , Xiaolong Qin , Jen-Chih Yao
In this paper, we consider a Cartesian stochastic variational inequality with a high dimensional solution space. This mathematical formulation captures a wide range of optimization problems including stochastic Nash games and stochastic minimization problems. By combining the advantages of the forward-backward-forward method and the stochastic approximated method, a novel distributed algorithm is developed for addressing this large-scale problem without any kind of monotonicity. A salient feature of the proposed algorithm is to compute two independent queries of a stochastic oracle at each iteration. The main contributions include: (i) The necessary condition imposed on the involved operator is related merely to the Lipschitz continuity, which are quite general. (ii) At each iteration, the suggested algorithm only requires one computation of the projection onto each feasible set, which can be easily evaluated. (iii) The distributed implementation of the stochastic approximation based Armijo-type line search strategy is adopted to weaken the line search condition and define variable adaptive non-monotonic stepsizes, when the Lipschitz constant is unknown. Some theoretical results of the almost sure convergence, the optimal rate statement, and the oracle complexity bound are established with conditions weaker than the conditions of other methods studied in the literature. Finally, preliminary numerical results are presented to show the efficiency and the competitiveness of our algorithm.
{"title":"A distributed stochastic forward-backward-forward self-adaptive algorithm for Cartesian stochastic variational inequalities","authors":"Liya Liu , Xiaolong Qin , Jen-Chih Yao","doi":"10.1016/j.apnum.2025.01.003","DOIUrl":"10.1016/j.apnum.2025.01.003","url":null,"abstract":"<div><div>In this paper, we consider a Cartesian stochastic variational inequality with a high dimensional solution space. This mathematical formulation captures a wide range of optimization problems including stochastic Nash games and stochastic minimization problems. By combining the advantages of the forward-backward-forward method and the stochastic approximated method, a novel distributed algorithm is developed for addressing this large-scale problem without any kind of monotonicity. A salient feature of the proposed algorithm is to compute two independent queries of a stochastic oracle at each iteration. The main contributions include: (i) The necessary condition imposed on the involved operator is related merely to the Lipschitz continuity, which are quite general. (ii) At each iteration, the suggested algorithm only requires one computation of the projection onto each feasible set, which can be easily evaluated. (iii) The distributed implementation of the stochastic approximation based Armijo-type line search strategy is adopted to weaken the line search condition and define variable adaptive non-monotonic stepsizes, when the Lipschitz constant is unknown. Some theoretical results of the almost sure convergence, the optimal rate statement, and the oracle complexity bound are established with conditions weaker than the conditions of other methods studied in the literature. Finally, preliminary numerical results are presented to show the efficiency and the competitiveness of our algorithm.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 17-41"},"PeriodicalIF":2.2,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141893","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 : 2025-01-09DOI: 10.1016/j.apnum.2025.01.002
Quang Huy Nguyen , Van Chien Le , Phuong Cuc Hoang , Thi Thanh Mai Ta
This paper presents a space-time interface-fitted finite element method for solving a parabolic advection-diffusion problem with a nonstationary interface. The jumping diffusion coefficient gives rise to the discontinuity of the solution gradient across the interface. We use the Banach-Nečas-Babuška theorem to show the well-posedness of the continuous variational problem. A fully discrete finite-element based scheme is analyzed using the Galerkin method and unstructured interface-fitted meshes. An optimal error estimate is established in a discrete energy norm under a globally low but locally high regularity condition. Some numerical results corroborate our theoretical results.
{"title":"A fitted space-time finite element method for an advection-diffusion problem with moving interfaces","authors":"Quang Huy Nguyen , Van Chien Le , Phuong Cuc Hoang , Thi Thanh Mai Ta","doi":"10.1016/j.apnum.2025.01.002","DOIUrl":"10.1016/j.apnum.2025.01.002","url":null,"abstract":"<div><div>This paper presents a space-time interface-fitted finite element method for solving a parabolic advection-diffusion problem with a nonstationary interface. The jumping diffusion coefficient gives rise to the discontinuity of the solution gradient across the interface. We use the Banach-Nečas-Babuška theorem to show the well-posedness of the continuous variational problem. A fully discrete finite-element based scheme is analyzed using the Galerkin method and unstructured interface-fitted meshes. An optimal error estimate is established in a discrete energy norm under a globally low but locally high regularity condition. Some numerical results corroborate our theoretical results.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 61-77"},"PeriodicalIF":2.2,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141890","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 : 2025-01-07DOI: 10.1016/j.apnum.2024.12.014
Jiaxi Gu , Daniel Olmos-Liceaga , Jae-Hun Jung
Many reaction-diffusion systems exhibit traveling wave solutions that evolve on multiple spatiotemporal scales, where obtaining fast and accurate numerical solutions is challenging. In this work, we employ sixth-order weighted essentially non-oscillatory (WENO) methods within the finite difference framework to solve the reaction-diffusion system for the traveling wave solution with the sharp fronts. It is shown that those WENO methods achieve the expected sixth-order accuracy in the Fisher's, Zeldovich, bistable equations, and the Lotka-Volterra competition-diffusion system. However, we find that the WENO methods converge very slowly in the Newell-Whitehead-Segel equation because of the speed issue, in which one possible way to match the exact speed is to coarsen the spatial grid and decrease the time step simultaneously. It is also seen that the central WENO method could carry the larger time step while preserving the essentially non-oscillatory behavior for the approximations.
{"title":"A numerical study of WENO approximations to sharp propagating fronts for reaction-diffusion systems","authors":"Jiaxi Gu , Daniel Olmos-Liceaga , Jae-Hun Jung","doi":"10.1016/j.apnum.2024.12.014","DOIUrl":"10.1016/j.apnum.2024.12.014","url":null,"abstract":"<div><div>Many reaction-diffusion systems exhibit traveling wave solutions that evolve on multiple spatiotemporal scales, where obtaining fast and accurate numerical solutions is challenging. In this work, we employ sixth-order weighted essentially non-oscillatory (WENO) methods within the finite difference framework to solve the reaction-diffusion system for the traveling wave solution with the sharp fronts. It is shown that those WENO methods achieve the expected sixth-order accuracy in the Fisher's, Zeldovich, bistable equations, and the Lotka-Volterra competition-diffusion system. However, we find that the WENO methods converge very slowly in the Newell-Whitehead-Segel equation because of the speed issue, in which one possible way to match the exact speed is to coarsen the spatial grid and decrease the time step simultaneously. It is also seen that the central WENO method could carry the larger time step while preserving the essentially non-oscillatory behavior for the approximations.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 1-16"},"PeriodicalIF":2.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141894","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 : 2025-01-07DOI: 10.1016/j.apnum.2024.12.013
Philipp Öffner , Louis Petri , Davide Torlo
In this manuscript, we present the development of implicit and implicit-explicit ADER and DeC methodologies within the DeC framework using the two-operators formulation, with a focus on their stability analysis both as solvers for ordinary differential equations (ODEs) and within the context of linear partial differential equations (PDEs). To analyze their stability, we reinterpret these methods as Runge-Kutta schemes and uncover significant variations in stability behavior, ranging from A-stable to bounded stability regions, depending on the chosen order, method, and quadrature nodes. This differentiation contrasts with their explicit counterparts. When applied to advection-diffusion and advection-dispersion equations employing finite difference spatial discretization, the von Neumann stability analysis demonstrates stability under CFL-like conditions. Particularly noteworthy is the stability maintenance observed for the advection-diffusion equation, even under spatial-independent constraints. Furthermore, we establish precise boundaries for relevant coefficients and provide suggestions regarding the suitability of specific schemes for different problem.
{"title":"Analysis for implicit and implicit-explicit ADER and DeC methods for ordinary differential equations, advection-diffusion and advection-dispersion equations","authors":"Philipp Öffner , Louis Petri , Davide Torlo","doi":"10.1016/j.apnum.2024.12.013","DOIUrl":"10.1016/j.apnum.2024.12.013","url":null,"abstract":"<div><div>In this manuscript, we present the development of implicit and implicit-explicit ADER and DeC methodologies within the DeC framework using the two-operators formulation, with a focus on their stability analysis both as solvers for ordinary differential equations (ODEs) and within the context of linear partial differential equations (PDEs). To analyze their stability, we reinterpret these methods as Runge-Kutta schemes and uncover significant variations in stability behavior, ranging from A-stable to bounded stability regions, depending on the chosen order, method, and quadrature nodes. This differentiation contrasts with their explicit counterparts. When applied to advection-diffusion and advection-dispersion equations employing finite difference spatial discretization, the von Neumann stability analysis demonstrates stability under CFL-like conditions. Particularly noteworthy is the stability maintenance observed for the advection-diffusion equation, even under spatial-independent constraints. Furthermore, we establish precise boundaries for relevant coefficients and provide suggestions regarding the suitability of specific schemes for different problem.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"212 ","pages":"Pages 110-134"},"PeriodicalIF":2.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143155013","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 : 2025-01-07DOI: 10.1016/j.apnum.2024.12.016
Si-Da Lin , Ya-Jing Zhang , Ming Huang , Jin-Long Yuan , Hong-Han Bei
In this paper, the convex constrained optimization problems are studied via the alternating linearization approach. The objective function f is assumed to be complex, and its exact oracle information (function values and subgradients) is not easy to obtain, while the constraint function h is expected to be “simple” relatively. With the help of the exact penalty function, we present an alternating linearization method with inexact information. In this method, the penalty problem is replaced by two relatively simple linear subproblems with regularized form which are needed to be solved successively in each iteration. An approximate solution is utilized instead of an exact form to solve each of the two subproblems. Moreover, it is proved that the generated sequence converges to some solution of the original problem. The dual form of this approach is discussed and described. Finally, some preliminary numerical test results are reported. Numerical experiences provided show that the inexact scheme has good performance, certificate and reliability.
{"title":"Inexact proximal penalty alternating linearization decomposition scheme of nonsmooth convex constrained optimization problems","authors":"Si-Da Lin , Ya-Jing Zhang , Ming Huang , Jin-Long Yuan , Hong-Han Bei","doi":"10.1016/j.apnum.2024.12.016","DOIUrl":"10.1016/j.apnum.2024.12.016","url":null,"abstract":"<div><div>In this paper, the convex constrained optimization problems are studied via the alternating linearization approach. The objective function <em>f</em> is assumed to be complex, and its exact oracle information (function values and subgradients) is not easy to obtain, while the constraint function <em>h</em> is expected to be “simple” relatively. With the help of the exact penalty function, we present an alternating linearization method with inexact information. In this method, the penalty problem is replaced by two relatively simple linear subproblems with regularized form which are needed to be solved successively in each iteration. An approximate solution is utilized instead of an exact form to solve each of the two subproblems. Moreover, it is proved that the generated sequence converges to some solution of the original problem. The dual form of this approach is discussed and described. Finally, some preliminary numerical test results are reported. Numerical experiences provided show that the inexact scheme has good performance, certificate and reliability.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"211 ","pages":"Pages 42-60"},"PeriodicalIF":2.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141892","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}
Renowned for offering a more precise approximation of the objective function, a third-order tensor expansion is deemed superior to the traditional second-order Taylor expansion, a viewpoint supported by various academics. Despite its acknowledged benefits, the adoption of this advanced expansion within the widely utilized quasi-Newton method remains notably rare. This research endeavors to construct update equations for the quasi-Newton method, based on a third-order tensor expansion, and to introduce an innovative quasi-Newton equation. The main contributions of this study include: (i) the development of a unique quasi-Newton equation based on a third-order tensor expansion; (ii) a detailed comparative analysis of the new BFGS quasi-Newton update method versus the traditional BFGS methodologies; (iii) the demonstration of convergence outcomes for the newly developed BFGS quasi-Newton technique; and (iv) the introduction of novel methodologies for conjugate gradients inspired by this distinctive quasi-Newton formula. Through exhaustive numerical experimentation, the algorithms derived from this pioneering quasi-Newton equation have shown superior performance.
{"title":"Analysis of a new BFGS algorithm and conjugate gradient algorithms and their applications in image restoration and machine learning","authors":"Yijia Wang , Chen Ouyang , Liangfu Lv , Gonglin Yuan","doi":"10.1016/j.apnum.2024.12.015","DOIUrl":"10.1016/j.apnum.2024.12.015","url":null,"abstract":"<div><div>Renowned for offering a more precise approximation of the objective function, a third-order tensor expansion is deemed superior to the traditional second-order Taylor expansion, a viewpoint supported by various academics. Despite its acknowledged benefits, the adoption of this advanced expansion within the widely utilized quasi-Newton method remains notably rare. This research endeavors to construct update equations for the quasi-Newton method, based on a third-order tensor expansion, and to introduce an innovative quasi-Newton equation. The main contributions of this study include: (i) the development of a unique quasi-Newton equation based on a third-order tensor expansion; (ii) a detailed comparative analysis of the new BFGS quasi-Newton update method versus the traditional BFGS methodologies; (iii) the demonstration of convergence outcomes for the newly developed BFGS quasi-Newton technique; and (iv) the introduction of novel methodologies for conjugate gradients inspired by this distinctive quasi-Newton formula. Through exhaustive numerical experimentation, the algorithms derived from this pioneering quasi-Newton equation have shown superior performance.</div></div>","PeriodicalId":8199,"journal":{"name":"Applied Numerical Mathematics","volume":"210 ","pages":"Pages 199-221"},"PeriodicalIF":2.2,"publicationDate":"2025-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143175063","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}
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