{"title":"A Sharp $alpha$-Robust $L1$ Scheme on Graded Meshes for Two-Dimensional Time Tempered Fractional Fokker-Planck Equation","authors":"Can Wang, Weihua Deng null, Xiangong Tang","doi":"10.4208/ijnam2023-1033","DOIUrl":"https://doi.org/10.4208/ijnam2023-1033","url":null,"abstract":"","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"283 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195302","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}
. In this paper, we design and analyze new residual-type a posteriori error estimators for the local discontinuous Galerkin (LDG) method applied to semilinear second-order elliptic problems in two dimensions of the type (cid:0) ∆ u = f ( x ;u ). We use our recent superconvergence results derived in Commun. Appl. Math. Comput. (2021) to prove that the LDG solution is superconvergent with an order p +2 towards the p -degree right Radau interpolating polynomial of the exact solution, when tensor product polynomials of degree at most p are considered as basis for the LDG method. Moreover, we show that the global discretization error can be decomposed into the sum of two errors. The first error can be expressed as a linear combination of two ( p +1)- degree Radau polynomials in the x - and y (cid:0) directions. The second error converges to zero with order p + 2 in the L 2 -norm. This new result allows us to construct a posteriori error estimators of residual type. We prove that the proposed a posteriori error estimators converge to the true errors in the L 2 -norm under mesh refinement at the optimal rate. The order of convergence is proved to be p + 2. We further prove that our a posteriori error estimates yield upper and lower bounds for the actual error. Finally, a series of numerical examples are presented to validate the theoretical results and numerically demonstrate the convergence of the proposed a posteriori error estimators.
{"title":"A Posteriori Error Estimates for a Local Discontinuous Galerkin Approximation of Semilinear Second-Order Elliptic Problems on Cartesian Grids","authors":"Mahboub Baccouch","doi":"10.4208/ijnam2023-1034","DOIUrl":"https://doi.org/10.4208/ijnam2023-1034","url":null,"abstract":". In this paper, we design and analyze new residual-type a posteriori error estimators for the local discontinuous Galerkin (LDG) method applied to semilinear second-order elliptic problems in two dimensions of the type (cid:0) ∆ u = f ( x ;u ). We use our recent superconvergence results derived in Commun. Appl. Math. Comput. (2021) to prove that the LDG solution is superconvergent with an order p +2 towards the p -degree right Radau interpolating polynomial of the exact solution, when tensor product polynomials of degree at most p are considered as basis for the LDG method. Moreover, we show that the global discretization error can be decomposed into the sum of two errors. The first error can be expressed as a linear combination of two ( p +1)- degree Radau polynomials in the x - and y (cid:0) directions. The second error converges to zero with order p + 2 in the L 2 -norm. This new result allows us to construct a posteriori error estimators of residual type. We prove that the proposed a posteriori error estimators converge to the true errors in the L 2 -norm under mesh refinement at the optimal rate. The order of convergence is proved to be p + 2. We further prove that our a posteriori error estimates yield upper and lower bounds for the actual error. Finally, a series of numerical examples are presented to validate the theoretical results and numerically demonstrate the convergence of the proposed a posteriori error estimators.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195303","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}
. A special (cid:12)nite volume element method based on postprocessing technique is proposed to solve the anisotropic di(cid:11)usion problem on arbitrary convex polygonal meshes. The shape function of polygonal (cid:12)nite element method is constructed by Wachspress generalized barycentric coordinate, and by adding some element-wise bubble functions to the (cid:12)nite element solution, we get a new (cid:12)nite volume element solution that satis(cid:12)es the local conservation law on a certain dual mesh. The postprocessing algorithm only needs to solve a local linear algebraic system on each primary cell, so that it is easy to implement. More interesting is that, a general construction of the bubble functions is introduced on each polygonal cell, which enables us to prove the existence and uniqueness of the post-processed solution on arbitrary convex polygonal meshes with full anisotropic di(cid:11)usion tensor. The optimal H 1 and L 2 error estimates of the post-processed solution are also obtained. Finally, the local conservation property and convergence of the new polygonal (cid:12)nite volume element solution are veri(cid:12)ed by numerical experiments.
{"title":"A Finite Volume Element Solution Based on Postprocessing Technique Over Arbitrary Convex Polygonal Meshes","authors":"Yanlong Zhang null, Yanhui Zhou","doi":"10.4208/ijnam2023-1026","DOIUrl":"https://doi.org/10.4208/ijnam2023-1026","url":null,"abstract":". A special (cid:12)nite volume element method based on postprocessing technique is proposed to solve the anisotropic di(cid:11)usion problem on arbitrary convex polygonal meshes. The shape function of polygonal (cid:12)nite element method is constructed by Wachspress generalized barycentric coordinate, and by adding some element-wise bubble functions to the (cid:12)nite element solution, we get a new (cid:12)nite volume element solution that satis(cid:12)es the local conservation law on a certain dual mesh. The postprocessing algorithm only needs to solve a local linear algebraic system on each primary cell, so that it is easy to implement. More interesting is that, a general construction of the bubble functions is introduced on each polygonal cell, which enables us to prove the existence and uniqueness of the post-processed solution on arbitrary convex polygonal meshes with full anisotropic di(cid:11)usion tensor. The optimal H 1 and L 2 error estimates of the post-processed solution are also obtained. Finally, the local conservation property and convergence of the new polygonal (cid:12)nite volume element solution are veri(cid:12)ed by numerical experiments.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135143757","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}
. This paper focuses on neural network-based learning methods for identifying nonlinear dynamic systems. The Takagi-Sugeno (T-S) fuzzy model is introduced to represent nonlinear systems in a linear way. Fractional calculus is integrated to minimize the cost function, yielding a fractional-order learning algorithm that can derive optimal parameters in the T-S fuzzy model. The proposed algorithm is evaluated by comparing it with an integer-order method for identifying numerical nonlinear systems and a water quality system. Both evaluations demonstrate that the proposed algorithm can e(cid:11)ectively reduce errors and improve model accuracy.
{"title":"Fractional Order Learning Methods for Nonlinear System Identification Based on Fuzzy Neural Network","authors":"Jie Ding, Sen Xu null, Zhijie Li","doi":"10.4208/ijnam2023-1031","DOIUrl":"https://doi.org/10.4208/ijnam2023-1031","url":null,"abstract":". This paper focuses on neural network-based learning methods for identifying nonlinear dynamic systems. The Takagi-Sugeno (T-S) fuzzy model is introduced to represent nonlinear systems in a linear way. Fractional calculus is integrated to minimize the cost function, yielding a fractional-order learning algorithm that can derive optimal parameters in the T-S fuzzy model. The proposed algorithm is evaluated by comparing it with an integer-order method for identifying numerical nonlinear systems and a water quality system. Both evaluations demonstrate that the proposed algorithm can e(cid:11)ectively reduce errors and improve model accuracy.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135143754","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}
In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton's method for nonlinear systems in which the Jacobian is singular at a solution. For these problems, the standard Newton algorithm converges linearly in a region about the solution; and, it has been previously observed that Anderson acceleration can substantially improve convergence without additional a priori knowledge, and with little additional computation cost. We present an analysis of the Newton-Anderson algorithm in this context, and introduce a novel and theoretically supported safeguarding strategy. The convergence results are demonstrated with the Chandrasekhar H-equation and some standard benchmark examples.
{"title":"Newton-Anderson at Singular Points","authors":"Matt Dallas null, Sara Pollock","doi":"10.4208/ijnam2023-1029","DOIUrl":"https://doi.org/10.4208/ijnam2023-1029","url":null,"abstract":"In this paper we develop convergence and acceleration theory for Anderson acceleration applied to Newton's method for nonlinear systems in which the Jacobian is singular at a solution. For these problems, the standard Newton algorithm converges linearly in a region about the solution; and, it has been previously observed that Anderson acceleration can substantially improve convergence without additional a priori knowledge, and with little additional computation cost. We present an analysis of the Newton-Anderson algorithm in this context, and introduce a novel and theoretically supported safeguarding strategy. The convergence results are demonstrated with the Chandrasekhar H-equation and some standard benchmark examples.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"86 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135143755","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}
. We apply orthogonal spline collocation with splines of degree r (cid:21) 3 to solve, on the unit square, Poisson’s equation with Neumann boundary conditions. We show that the H 1 norm error is of order r and explain how to compute e(cid:14)ciently the approximate solution using a matrix decomposition algorithm involving the solution of a symmetric generalized eigenvalue problem.
{"title":"Orthogonal Spline Collocation for Poisson’S Equation with Neumann Boundary Conditions","authors":"Bernard Bialecki null, Nick Fisher","doi":"10.4208/ijnam2023-1036","DOIUrl":"https://doi.org/10.4208/ijnam2023-1036","url":null,"abstract":". We apply orthogonal spline collocation with splines of degree r (cid:21) 3 to solve, on the unit square, Poisson’s equation with Neumann boundary conditions. We show that the H 1 norm error is of order r and explain how to compute e(cid:14)ciently the approximate solution using a matrix decomposition algorithm involving the solution of a symmetric generalized eigenvalue problem.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135195305","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}
. A new weak Galerkin method with weakly enforced Dirichlet boundary condition is proposed and analyzed for the second order elliptic problems. Two penalty terms are incorporated into the weak Galerkin method to enforce the boundary condition in the weak sense. The new numerical scheme is designed by using the locally constructed weak gradient. Optimal order error estimates are established for the numerical approximation in the energy norm and usual L 2 norm. Moreover, by using the Schur complement technique, the unknowns of the numerical scheme are only de(cid:12)ned on the boundary of each piecewise element and an e(cid:11)ective implementation of the reduced global system is presented. Some numerical experiments are reported to demonstrate the accuracy and e(cid:14)ciency of the proposed method.
{"title":"A New Weak Galerkin Method with Weakly Enforced Dirichlet Boundary Condition","authors":"Dan Li, Yiqiang Li null, Zhanbin Yuan","doi":"10.4208/ijnam2023-1028","DOIUrl":"https://doi.org/10.4208/ijnam2023-1028","url":null,"abstract":". A new weak Galerkin method with weakly enforced Dirichlet boundary condition is proposed and analyzed for the second order elliptic problems. Two penalty terms are incorporated into the weak Galerkin method to enforce the boundary condition in the weak sense. The new numerical scheme is designed by using the locally constructed weak gradient. Optimal order error estimates are established for the numerical approximation in the energy norm and usual L 2 norm. Moreover, by using the Schur complement technique, the unknowns of the numerical scheme are only de(cid:12)ned on the boundary of each piecewise element and an e(cid:11)ective implementation of the reduced global system is presented. Some numerical experiments are reported to demonstrate the accuracy and e(cid:14)ciency of the proposed method.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135143357","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}
Pelin G. Geredeli, Carson Givens null, Ahmed Zytoon
{"title":"Numerical Approximations for the Null Controllers of Structurally Damped Plate Dynamics","authors":"Pelin G. Geredeli, Carson Givens null, Ahmed Zytoon","doi":"10.4208/ijnam2023-1013","DOIUrl":"https://doi.org/10.4208/ijnam2023-1013","url":null,"abstract":"","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135421667","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}
. In this paper, we present and analyze a posteriori error estimates in the L 2 -norm of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order boundary-value problems for ordinary di(cid:11)erential equations of the form u ′′ = f ( x;u ). We (cid:12)rst use the superconvergence results proved in the (cid:12)rst part of this paper ( J. Appl. Math. Comput. 69, 1507-1539, 2023) to prove that the UWDG solution converges, in the L 2 -norm, towards a special p -degree interpolating polynomial, when piecewise polynomials of degree at most p (cid:21) 2 are used. The order of convergence is proved to be p + 2. We then show that the UWDG error on each element can be divided into two parts. The dominant part is proportional to a special ( p +1)-degree Baccouch polynomial, which can be written as a linear combination of Legendre polynomials of degrees p (cid:0) 1, p , and p + 1. The second part converges to zero with order p + 2 in the L 2 - norm. These results allow us to construct a posteriori UWDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the L 2 -norm under mesh re(cid:12)nement. The order of convergence is proved to be p + 2. Finally, we prove that the global e(cid:11)ectivity index converges to unity at O ( h ) rate. Numerical results are presented exhibiting the reliability and the e(cid:14)ciency of the proposed error estimator.
{"title":"A Posteriori Error Analysis for an Ultra-Weak Discontinuous Galerkin Approximations of Nonlinear Second-Order Two-Point Boundary-Value Problems","authors":"Mahboub Baccouch","doi":"10.4208/ijnam2023-1027","DOIUrl":"https://doi.org/10.4208/ijnam2023-1027","url":null,"abstract":". In this paper, we present and analyze a posteriori error estimates in the L 2 -norm of an ultra-weak discontinuous Galerkin (UWDG) method for nonlinear second-order boundary-value problems for ordinary di(cid:11)erential equations of the form u ′′ = f ( x;u ). We (cid:12)rst use the superconvergence results proved in the (cid:12)rst part of this paper ( J. Appl. Math. Comput. 69, 1507-1539, 2023) to prove that the UWDG solution converges, in the L 2 -norm, towards a special p -degree interpolating polynomial, when piecewise polynomials of degree at most p (cid:21) 2 are used. The order of convergence is proved to be p + 2. We then show that the UWDG error on each element can be divided into two parts. The dominant part is proportional to a special ( p +1)-degree Baccouch polynomial, which can be written as a linear combination of Legendre polynomials of degrees p (cid:0) 1, p , and p + 1. The second part converges to zero with order p + 2 in the L 2 - norm. These results allow us to construct a posteriori UWDG error estimates. The proposed error estimates are computationally simple and are obtained by solving a local problem with no boundary conditions on each element. Furthermore, we prove that, for smooth solutions, these a posteriori error estimates converge to the exact errors in the L 2 -norm under mesh re(cid:12)nement. The order of convergence is proved to be p + 2. Finally, we prove that the global e(cid:11)ectivity index converges to unity at O ( h ) rate. Numerical results are presented exhibiting the reliability and the e(cid:14)ciency of the proposed error estimator.","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135143756","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}
Robust development of biological organisms in the presence of genetic and epi-genetic perturbations is important for time spans short relative to evolutionary time. Gradients of receptor bound signaling morphogens are responsible for patterning formation and development. A variety of inhibitors for reducing ectopic signaling activities are known to exist and their specific role in down-regulating the undesirable ectopic activities reasonably well understood. However, how a developing organism manages to adjust inhibition/stimulation in response to genetic and/or environmental changes remains to be uncovered. The need to adjust for ectopic signaling activities requires the presence of one or more feedback mechanisms to stimulate the needed adjustment. As the ultimate effect of many inhibitors (including those of the nonreceptor type) is to reduce the availability of signaling morphogens for binding with signaling receptors, a negative feedback on signaling morphogen synthesis rate based on a root-mean-square measure of the spatial distribution of signaling concentration offers a simple approach to robusness and has been demonstrated to be effective in a proof-of-concept implementation. In this paper, we complement the previous investigation of feedback in steady state by examining the effect of one or more feedback adjustments during the transient phase of the biological development.
{"title":"TRANSIENT FEEDBACK AND ROBUST SIGNALING GRADIENTS.","authors":"Aghavni Simonyan, Frederic Y M Wan","doi":"","DOIUrl":"","url":null,"abstract":"<p><p>Robust development of biological organisms in the presence of genetic and epi-genetic perturbations is important for time spans short relative to evolutionary time. Gradients of receptor bound signaling morphogens are responsible for patterning formation and development. A variety of inhibitors for reducing ectopic signaling activities are known to exist and their specific role in down-regulating the undesirable ectopic activities reasonably well understood. However, how a developing organism manages to adjust inhibition/stimulation in response to genetic and/or environmental changes remains to be uncovered. The need to adjust for ectopic signaling activities requires the presence of one or more feedback mechanisms to stimulate the needed adjustment. As the ultimate effect of many inhibitors (including those of the nonreceptor type) is to reduce the availability of signaling morphogens for binding with signaling receptors, a negative feedback on signaling morphogen synthesis rate based on a root-mean-square measure of the spatial distribution of signaling concentration offers a simple approach to robusness and has been demonstrated to be effective in a proof-of-concept implementation. In this paper, we complement the previous investigation of feedback in steady state by examining the effect of one or more feedback adjustments during the transient phase of the biological development.</p>","PeriodicalId":50301,"journal":{"name":"International Journal of Numerical Analysis and Modeling","volume":"13 2","pages":"179-204"},"PeriodicalIF":1.1,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5102427/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140289488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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