Pub Date : 2023-11-17DOI: 10.1007/s10092-023-00547-z
Thomas Mejstrik, Valdimir Yu. Protasov
Elliptic polytopes are convex hulls of several concentric plane ellipses in ({{mathbb {R}}}^d). They arise in applications as natural generalizations of usual polytopes. In particular, they define invariant convex bodies of linear operators, optimal Lyapunov norms for linear dynamical systems, etc. To construct elliptic polytopes one needs to decide whether a given ellipse is contained in the convex hull of other ellipses. We analyse the computational complexity of this problem and show that for (d=2, 3), it admits an explicit solution. For larger d, two geometric methods for approximate solution are presented. Both use the convex optimization tools. The efficiency of the methods is demonstrated in two applications: the construction of extremal norms of linear operators and the computation of the joint spectral radius/Lyapunov exponent of a family of matrices.
{"title":"Elliptic polytopes and invariant norms of linear operators","authors":"Thomas Mejstrik, Valdimir Yu. Protasov","doi":"10.1007/s10092-023-00547-z","DOIUrl":"https://doi.org/10.1007/s10092-023-00547-z","url":null,"abstract":"<p>Elliptic polytopes are convex hulls of several concentric plane ellipses in <span>({{mathbb {R}}}^d)</span>. They arise in applications as natural generalizations of usual polytopes. In particular, they define invariant convex bodies of linear operators, optimal Lyapunov norms for linear dynamical systems, etc. To construct elliptic polytopes one needs to decide whether a given ellipse is contained in the convex hull of other ellipses. We analyse the computational complexity of this problem and show that for <span>(d=2, 3)</span>, it admits an explicit solution. For larger <i>d</i>, two geometric methods for approximate solution are presented. Both use the convex optimization tools. The efficiency of the methods is demonstrated in two applications: the construction of extremal norms of linear operators and the computation of the joint spectral radius/Lyapunov exponent of a family of matrices.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525782","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 : 2023-11-15DOI: 10.1007/s10092-023-00546-0
Stefano Cipolla, Jacek Gondzio
Embedding randomization procedures in the Alternating Direction Method of Multipliers (ADMM) has recently attracted an increasing amount of interest as a remedy to the fact that the direct multi-block generalization of ADMM is not necessarily convergent. Even if, in practice, the introduction of such techniques could mitigate the diverging behaviour of the multi-block extension of ADMM, from the theoretical point of view, it can ensure just the convergence in expectation, which may not be a good indicator of its robustness and efficiency. In this work, analysing the strongly convex quadratic programming case from a linear algebra perspective, we interpret the block Gauss–Seidel sweep performed by the multi-block ADMM in the context of the inexact Augmented Lagrangian Method. Using the proposed analysis, we are able to outline an alternative technique to those present in the literature which, supported from stronger theoretical guarantees, is able to ensure the convergence of the multi-block generalization of the ADMM method.
{"title":"A linear algebra perspective on the random multi-block ADMM: the QP case","authors":"Stefano Cipolla, Jacek Gondzio","doi":"10.1007/s10092-023-00546-0","DOIUrl":"https://doi.org/10.1007/s10092-023-00546-0","url":null,"abstract":"<p>Embedding randomization procedures in the Alternating Direction Method of Multipliers (ADMM) has recently attracted an increasing amount of interest as a remedy to the fact that the direct multi-block generalization of ADMM is not necessarily convergent. Even if, in practice, the introduction of such techniques could <i>mitigate</i> the diverging behaviour of the multi-block extension of ADMM, from the theoretical point of view, it can ensure just the <i>convergence in expectation</i>, which may not be a good indicator of its robustness and efficiency. In this work, analysing the strongly convex quadratic programming case from a linear algebra perspective, we interpret the block Gauss–Seidel sweep performed by the multi-block ADMM in the context of the inexact Augmented Lagrangian Method. Using the proposed analysis, we are able to outline an alternative technique to those present in the literature which, supported from stronger theoretical guarantees, is able to ensure the convergence of the multi-block generalization of the ADMM method.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513132","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 : 2023-11-15DOI: 10.1007/s10092-023-00550-4
Shunfeng Yang, Shuhuang Xiang
Based on Cauchy’s integral formula and conformal maps, this paper presents a new method for constructing barycentric rational interpolation formulae for complex functions, which may contain singularities such as poles, branch cuts, or essential singularities. The resulting interpolations are pole-free, exponentially convergent, and numerically stable, requiring only ({mathcal {O}}(N)) operations. Inspired by the logarithm equilibrium potential, we introduce a Möbius transform to concentrate nodes to the vicinity of singularity to get a spectacular improvement on approximation quality. A thorough convergence analysis is provided, alongside numerous numerical examples that illustrate the theoretical results and demonstrate the accuracy and efficiency of the methodology. Meanwhile, the paper also discusses some applications of the method including the numerical solutions of boundary value problems and the zero locations of holomorphic functions.
{"title":"Fast barycentric rational interpolations for complex functions with some singularities","authors":"Shunfeng Yang, Shuhuang Xiang","doi":"10.1007/s10092-023-00550-4","DOIUrl":"https://doi.org/10.1007/s10092-023-00550-4","url":null,"abstract":"<p>Based on Cauchy’s integral formula and conformal maps, this paper presents a new method for constructing barycentric rational interpolation formulae for complex functions, which may contain singularities such as poles, branch cuts, or essential singularities. The resulting interpolations are pole-free, exponentially convergent, and numerically stable, requiring only <span>({mathcal {O}}(N))</span> operations. Inspired by the logarithm equilibrium potential, we introduce a Möbius transform to concentrate nodes to the vicinity of singularity to get a spectacular improvement on approximation quality. A thorough convergence analysis is provided, alongside numerous numerical examples that illustrate the theoretical results and demonstrate the accuracy and efficiency of the methodology. Meanwhile, the paper also discusses some applications of the method including the numerical solutions of boundary value problems and the zero locations of holomorphic functions.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138525839","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 : 2023-11-01DOI: 10.1007/s10092-023-00548-y
Felipe Lepe, Jesus Vellojin
{"title":"A posteriori analysis for a mixed formulation of the Stokes spectral problem","authors":"Felipe Lepe, Jesus Vellojin","doi":"10.1007/s10092-023-00548-y","DOIUrl":"https://doi.org/10.1007/s10092-023-00548-y","url":null,"abstract":"","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135714495","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 : 2023-10-20DOI: 10.1007/s10092-023-00544-2
Sergio Caucao, Gabriel N. Gatica, Luis F. Gatica
{"title":"A Banach spaces-based mixed finite element method for the stationary convective Brinkman–Forchheimer problem","authors":"Sergio Caucao, Gabriel N. Gatica, Luis F. Gatica","doi":"10.1007/s10092-023-00544-2","DOIUrl":"https://doi.org/10.1007/s10092-023-00544-2","url":null,"abstract":"","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513760","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 : 2023-10-18DOI: 10.1007/s10092-023-00545-1
A. K. Mittal
{"title":"Two-dimensional Jacobi pseudospectral quadrature solutions of two-dimensional fractional Volterra integral equations","authors":"A. K. Mittal","doi":"10.1007/s10092-023-00545-1","DOIUrl":"https://doi.org/10.1007/s10092-023-00545-1","url":null,"abstract":"","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135823962","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 : 2023-10-09DOI: 10.1007/s10092-023-00543-3
Pin Lyu, Seakweng Vong
{"title":"A weighted ADI scheme with variable time steps for diffusion-wave equations","authors":"Pin Lyu, Seakweng Vong","doi":"10.1007/s10092-023-00543-3","DOIUrl":"https://doi.org/10.1007/s10092-023-00543-3","url":null,"abstract":"","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135095623","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 : 2023-10-08DOI: 10.1007/s10092-023-00542-4
Ivo Dravins, Maya Neytcheva
Abstract We consider the iterative solution of algebraic systems, arising in optimal control problems constrained by a partial differential equation with additional box constraints on the state and the control variables, and sparsity imposed on the control. A nonsymmetric two-by-two block preconditioner is analysed and tested for a wide range of problem, regularization and discretization parameters. The constraint equation characterizes convection-diffusion processes.
{"title":"Preconditioning of discrete state- and control-constrained optimal control convection-diffusion problems","authors":"Ivo Dravins, Maya Neytcheva","doi":"10.1007/s10092-023-00542-4","DOIUrl":"https://doi.org/10.1007/s10092-023-00542-4","url":null,"abstract":"Abstract We consider the iterative solution of algebraic systems, arising in optimal control problems constrained by a partial differential equation with additional box constraints on the state and the control variables, and sparsity imposed on the control. A nonsymmetric two-by-two block preconditioner is analysed and tested for a wide range of problem, regularization and discretization parameters. The constraint equation characterizes convection-diffusion processes.","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135198276","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 : 2023-10-06DOI: 10.1007/s10092-023-00541-5
Naresh Kumar, Bhupen Deka
{"title":"A numerical method for analysis and simulation of diffusive viscous wave equations with variable coefficients on polygonal meshes","authors":"Naresh Kumar, Bhupen Deka","doi":"10.1007/s10092-023-00541-5","DOIUrl":"https://doi.org/10.1007/s10092-023-00541-5","url":null,"abstract":"","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135346594","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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