Abstract - This paper presents the multiharmonic analysis of a distributed parabolic optimal control problem in a time-periodic setting. We prove the existence and uniqueness of the solution of some weak space-time variational formulation for the parabolic time-periodic boundary value problem appearing in the constraints for the optimal control problem. Since the cost functional is quadratic, the optimal control problem is uniquely solvable as well. In order to solve the optimal control problem numerically, we state its optimality system and discretize it by the multiharmonic finite element method leading to a system of linear algebraic equations which decouples into smaller systems. We construct preconditioners for these systems which yield robust convergence rates and optimal complexity for the preconditioned minimal residual method. All systems can be solved totally in parallel. Furthermore, we present a complete analysis for the error introduced by the multiharmonic finite element discretization as well as some numerical results confirming our theoretical findings.
{"title":"Multiharmonic finite element analysis of a time-periodic parabolic optimal control problem","authors":"Ulrich Langer, M. Wolfmayr","doi":"10.1515/jnum-2013-0011","DOIUrl":"https://doi.org/10.1515/jnum-2013-0011","url":null,"abstract":"Abstract - This paper presents the multiharmonic analysis of a distributed parabolic optimal control problem in a time-periodic setting. We prove the existence and uniqueness of the solution of some weak space-time variational formulation for the parabolic time-periodic boundary value problem appearing in the constraints for the optimal control problem. Since the cost functional is quadratic, the optimal control problem is uniquely solvable as well. In order to solve the optimal control problem numerically, we state its optimality system and discretize it by the multiharmonic finite element method leading to a system of linear algebraic equations which decouples into smaller systems. We construct preconditioners for these systems which yield robust convergence rates and optimal complexity for the preconditioned minimal residual method. All systems can be solved totally in parallel. Furthermore, we present a complete analysis for the error introduced by the multiharmonic finite element discretization as well as some numerical results confirming our theoretical findings.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"237 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121049136","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we consider a family of hybrid linear multistep methods (LMM) with reasonable error order for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The methods are A(ɑ)-stable for k = 1, ...,9.
{"title":"A class of hybrid linear multistep methods with A(ɑ)-stability properties for stiff IVPs in ODEs","authors":"R. Okuonghae, M. Ikhile","doi":"10.1515/jnum-2013-0006","DOIUrl":"https://doi.org/10.1515/jnum-2013-0006","url":null,"abstract":"Abstract In this paper, we consider a family of hybrid linear multistep methods (LMM) with reasonable error order for the numerical solution of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The methods are A(ɑ)-stable for k = 1, ...,9.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127288708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Jolivet, V. Dolean, F. Hecht, F. Nataf, C. Prud'homme, N. Spillane
Abstract - In this document, we present a parallel implementation in freefem++ of scalable two-level domain decomposition methods. Numerical studies with highly heterogeneous problems are then performed on large clusters in order to assert the performance of our code.
{"title":"High performance domain decomposition methods on massively parallel architectures with freefem++","authors":"P. Jolivet, V. Dolean, F. Hecht, F. Nataf, C. Prud'homme, N. Spillane","doi":"10.1515/jnum-2012-0015","DOIUrl":"https://doi.org/10.1515/jnum-2012-0015","url":null,"abstract":"Abstract - In this document, we present a parallel implementation in freefem++ of scalable two-level domain decomposition methods. Numerical studies with highly heterogeneous problems are then performed on large clusters in order to assert the performance of our code.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115077602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract -This is a short presentation of the freefem++ software. In Section 1, we recall most of the characteristics of the software, In Section 2, we recall how to to build the weak form of a partial differential equation (PDE) from the strong form. In the 3 last sections, we present different examples and tools to illustrated the power of the software. First we deal with mesh adaptation for problems in two and three dimension, second, we solve numerically a problem with phase change and natural convection, and the finally to show the possibilities for HPC we solve a Laplace equation by a Schwarz domain decomposition problem on parallel computer.
{"title":"New development in freefem++","authors":"F. Hecht","doi":"10.1515/jnum-2012-0013","DOIUrl":"https://doi.org/10.1515/jnum-2012-0013","url":null,"abstract":"Abstract -This is a short presentation of the freefem++ software. In Section 1, we recall most of the characteristics of the software, In Section 2, we recall how to to build the weak form of a partial differential equation (PDE) from the strong form. In the 3 last sections, we present different examples and tools to illustrated the power of the software. First we deal with mesh adaptation for problems in two and three dimension, second, we solve numerically a problem with phase change and natural convection, and the finally to show the possibilities for HPC we solve a Laplace equation by a Schwarz domain decomposition problem on parallel computer.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133832194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract - Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such that their singular values are equal to the tangents of PABS, using several approaches: orthonormal and non-orthonormal bases for subspaces, as well as projectors. Such a construction has applications, e.g., in analysis of convergence of subspace iterations for eigenvalue problems.
{"title":"Angles between subspaces and their tangents","authors":"Peizhen Zhu, A. Knyazev","doi":"10.1515/jnum-2013-0013","DOIUrl":"https://doi.org/10.1515/jnum-2013-0013","url":null,"abstract":"Abstract - Principal angles between subspaces (PABS) (also called canonical angles) serve as a classical tool in mathematics, statistics, and applications, e.g., data mining. Traditionally, PABS are introduced via their cosines. The cosines and sines of PABS are commonly defined using the singular value decomposition. We utilize the same idea for the tangents, i.e., explicitly construct matrices, such that their singular values are equal to the tangents of PABS, using several approaches: orthonormal and non-orthonormal bases for subspaces, as well as projectors. Such a construction has applications, e.g., in analysis of convergence of subspace iterations for eigenvalue problems.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122406115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract -We consider here different family of Boussinesq systems in two space dimensions. These systems approximate the three-dimensional Euler equations and consist of three coupled nonlinear dispersive wave equations that describe propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom and which was studied in [7,9,10].We present here a freefem++ code aimed at solving numerically these systems where a discretization using P1 finite element for these systems was taken in space and a second order Runge-Kutta scheme in time.We give the detail of our code where we use a mesh adaptation technique. An optimization of the used algorithm is done and a comparison of the solution for different Boussinesq family is done too. The results we obtained agree with those of the literature.
{"title":"Solution of 2D Boussinesq systems with freefem++: the flat bottom case","authors":"G. Sadaka","doi":"10.1515/jnum-2012-0016","DOIUrl":"https://doi.org/10.1515/jnum-2012-0016","url":null,"abstract":"Abstract -We consider here different family of Boussinesq systems in two space dimensions. These systems approximate the three-dimensional Euler equations and consist of three coupled nonlinear dispersive wave equations that describe propagation of long surface waves of small amplitude in ideal fluids over a horizontal bottom and which was studied in [7,9,10].We present here a freefem++ code aimed at solving numerically these systems where a discretization using P1 finite element for these systems was taken in space and a second order Runge-Kutta scheme in time.We give the detail of our code where we use a mesh adaptation technique. An optimization of the used algorithm is done and a comparison of the solution for different Boussinesq family is done too. The results we obtained agree with those of the literature.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"169 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121622254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract -The purpose of this paper is to explain the phenomenon of symmetry breaking for optimal functions in functional inequalities by the numerical computations of some well chosen solutions of the corresponding Euler-Lagrange equations. For many of those inequalities it was believed that the only source of symmetry breaking would be the instability of the symmetric optimizer in the class of all admissible functions. But recently, it was shown by an indirect argument that for some Caffarelli-Kohn-Nirenberg inequalities this conjecture was not true. In order to understand this new symmetry breaking mechanism we have computed the branch of minimal solutions for a simple problem. A reparametrization of this branch allows us to build a scenario for the new phenomenon of symmetry breaking. The computations have been performed using freefem++.
{"title":"A scenario for symmetry breaking in Caffarelli–Kohn–Nirenberg inequalities","authors":"J. Dolbeault, M. Esteban","doi":"10.1515/jnum-2012-0012","DOIUrl":"https://doi.org/10.1515/jnum-2012-0012","url":null,"abstract":"Abstract -The purpose of this paper is to explain the phenomenon of symmetry breaking for optimal functions in functional inequalities by the numerical computations of some well chosen solutions of the corresponding Euler-Lagrange equations. For many of those inequalities it was believed that the only source of symmetry breaking would be the instability of the symmetric optimizer in the class of all admissible functions. But recently, it was shown by an indirect argument that for some Caffarelli-Kohn-Nirenberg inequalities this conjecture was not true. In order to understand this new symmetry breaking mechanism we have computed the branch of minimal solutions for a simple problem. A reparametrization of this branch allows us to build a scenario for the new phenomenon of symmetry breaking. The computations have been performed using freefem++.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132518824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Demand-forecasting problems frequently arise in logistics and supply chain management. The Newsboy problem is one such problem. In this paper, we present an improved solution method by application of the Black–Scholes model incorporating a stochastic process used in financial engineering for option pricing. The proposed model is shown to be effective through numerical experiments using real-world data.
{"title":"Validation of a demand forecasting method based on a stochastic process using real-world data","authors":"Y. Zheng, H. Suito, H. Kawarada","doi":"10.1515/jnum.2010.007","DOIUrl":"https://doi.org/10.1515/jnum.2010.007","url":null,"abstract":"Abstract Demand-forecasting problems frequently arise in logistics and supply chain management. The Newsboy problem is one such problem. In this paper, we present an improved solution method by application of the Black–Scholes model incorporating a stochastic process used in financial engineering for option pricing. The proposed model is shown to be effective through numerical experiments using real-world data.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"99 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126534622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper develops a combined a posteriori analysis for the discretization and iteration errors in the computation of finite element approximations to elliptic boundary value problems. The emphasis is on the multigrid method, but for comparison also simple iterative schemes such as the Gauß–Seidel and the conjugate gradient method are considered. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of these a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The efficiency of the proposed method is demonstrated for several model situations including the simple Poisson equation, the Stokes equations in fluid mechanics and the KKT system of linear-quadratic elliptic optimal control problems.
{"title":"Goal-oriented error control of the iterative solution of finite element equations","authors":"Dominik Meidner, R. Rannacher, Jevgeni Vihharev","doi":"10.1515/JNUM.2009.009","DOIUrl":"https://doi.org/10.1515/JNUM.2009.009","url":null,"abstract":"Abstract This paper develops a combined a posteriori analysis for the discretization and iteration errors in the computation of finite element approximations to elliptic boundary value problems. The emphasis is on the multigrid method, but for comparison also simple iterative schemes such as the Gauß–Seidel and the conjugate gradient method are considered. The underlying theoretical framework is that of the Dual Weighted Residual (DWR) method for goal-oriented error estimation. On the basis of these a posteriori error estimates the algebraic iteration can be adjusted to the discretization within a successive mesh adaptation process. The efficiency of the proposed method is demonstrated for several model situations including the simple Poisson equation, the Stokes equations in fluid mechanics and the KKT system of linear-quadratic elliptic optimal control problems.","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128250336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The main goal of this article is two fold: (i) To discuss a methodology for the numerical solution of the Dirichlet problem for a Pucci equation in dimension two. (ii) Use the ensuing algorithms to investigate the homogenization properties of the solutions when a coefficient in the Pucci equation oscillates periodically or randomly in space. The solution methodology relies on the combination of a least-squares formulation of the Pucci equation in an appropriate Hilbert space with operator-splitting techniques and mixed finite element approximations. The results of numerical experiments suggest second order accuracy when globally continuous piecewise affine space approximations are used; they also show that the solution of the problem under consideration can be reduced to a sequence of discrete Poisson–Dirichlet problems coupled with one-dimensional optimization problems (one per grid point).
{"title":"Numerical solution of the Dirichlet problem for a Pucci equation in dimension two. Application to homogenization","authors":"L. Caffarelli, R. Glowinski","doi":"10.1515/JNUM.2008.009","DOIUrl":"https://doi.org/10.1515/JNUM.2008.009","url":null,"abstract":"Abstract The main goal of this article is two fold: (i) To discuss a methodology for the numerical solution of the Dirichlet problem for a Pucci equation in dimension two. (ii) Use the ensuing algorithms to investigate the homogenization properties of the solutions when a coefficient in the Pucci equation oscillates periodically or randomly in space. The solution methodology relies on the combination of a least-squares formulation of the Pucci equation in an appropriate Hilbert space with operator-splitting techniques and mixed finite element approximations. The results of numerical experiments suggest second order accuracy when globally continuous piecewise affine space approximations are used; they also show that the solution of the problem under consideration can be reduced to a sequence of discrete Poisson–Dirichlet problems coupled with one-dimensional optimization problems (one per grid point).","PeriodicalId":342521,"journal":{"name":"J. Num. Math.","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2008-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114468741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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