Journal of Computational Dynamics最新文献_第6页

Using Lie group integrators to solve two and higher dimensional variational problems with symmetry 利用李群积分器求解二维及高维对称变分问题

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Journal of Computational Dynamics Pub Date : 2019-12-01 DOI: 10.3934/jcd.2019025

Michele Zadra, E. Mansfield

{"title":"Using Lie group integrators to solve two and higher dimensional variational problems with symmetry","authors":"Michele Zadra, E. Mansfield","doi":"10.3934/jcd.2019025","DOIUrl":"https://doi.org/10.3934/jcd.2019025","url":null,"abstract":"The theory of moving frames has been used successfully to solve one dimensional (1D) variational problems invariant under a Lie group symmetry. In the one dimensional case, Noether's laws give first integrals of the Euler–Lagrange equations. In higher dimensional problems, the conservation laws do not enable the exact integration of the Euler–Lagrange system. In this paper we use the theory of moving frames to help solve, numerically, some higher dimensional variational problems, which are invariant under a Lie group action. In order to find a solution to the variational problem, we need first to solve the Euler Lagrange equations for the relevant differential invariants, and then solve a system of linear, first order, compatible, coupled partial differential equations for a moving frame, evolving on the Lie group. We demonstrate that Lie group integrators may be used in this context. We show first that the Magnus expansions on which one dimensional Lie group integrators are based, may be taken sequentially in a well defined way, at least to order 5; that is, the exact result is independent of the order of integration. We then show that efficient implementations of these integrators give a numerical solution of the equations for the frame, which is independent of the order of integration, to high order, in a range of examples. Our running example is a variational problem invariant under a linear action of begin{document}$ SU(2) $end{document} . We then consider variational problems for evolving curves which are invariant under the projective action of begin{document}$ SL(2) $end{document} and finally the standard affine action of begin{document}$ SE(2) $end{document} .","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"5 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78500700","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}

引用次数: 1

On the influence of cross-diffusion in pattern formation 交叉扩散对模式形成的影响

IF 1

Journal of Computational Dynamics Pub Date : 2019-10-08 DOI: 10.3934/JCD.2021010

M. Breden, C. Kuehn, C. Soresina

引用次数: 13

The geometry of convergence in numerical analysis 数值分析中的收敛几何

IF 1

Journal of Computational Dynamics Pub Date : 2019-09-20 DOI: 10.3934/jcd.2021003

G. Patrick

引用次数: 0

A self-consistent dynamical system with multiple absolutely continuous invariant measures 具有多个绝对连续不变测度的自洽动力系统

IF 1

Journal of Computational Dynamics Pub Date : 2019-09-10 DOI: 10.3934/jcd.2021002

Fanni M. S'elley

引用次数: 5

Principal symmetric space analysis 主对称空间分析

IF 1

Journal of Computational Dynamics Pub Date : 2019-08-13 DOI: 10.3934/jcd.2019013

S. Marsland, R. McLachlan, Charles Curry

引用次数: 4

An incremental approach to online dynamic mode decomposition for time-varying systems with applications to EEG data modeling 时变系统在线动态模态分解的增量方法及其在脑电数据建模中的应用

IF 1

Journal of Computational Dynamics Pub Date : 2019-08-02 DOI: 10.3934/jcd.2020009

M. Alfatlawi, Vaibhav Srivastava

{"title":"An incremental approach to online dynamic mode decomposition for time-varying systems with applications to EEG data modeling","authors":"M. Alfatlawi, Vaibhav Srivastava","doi":"10.3934/jcd.2020009","DOIUrl":"https://doi.org/10.3934/jcd.2020009","url":null,"abstract":"Dynamic Mode Decomposition (DMD) is a data-driven technique to identify a low dimensional linear time invariant dynamics underlying high-dimensional data. For systems in which such underlying low-dimensional dynamics is time-varying, a time-invariant approximation of such dynamics computed through standard DMD techniques may not be appropriate. We focus on DMD techniques for such time-varying systems and develop incremental algorithms for systems without and with exogenous control inputs. We build upon the work in [35] to scenarios in which high dimensional data are governed by low dimensional time-varying dynamics. We consider two classes of algorithms that rely on (i) a discount factor on previous observations, and (ii) a sliding window of observations. Our algorithms leverage existing techniques for incremental singular value decomposition and allow us to determine an appropriately reduced model at each time and are applicable even if data matrix is singular. We apply the developed algorithms for autonomous systems to Electroencephalographic (EEG) data and demonstrate their effectiveness in terms of reconstruction and prediction. Our algorithms for non-autonomous systems are illustrated using randomly generated linear time-varying systems.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"1 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80797145","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}

引用次数: 14

Numerical efficacy study of data assimilation for the 2D magnetohydrodynamic equations 二维磁流体动力学方程数据同化的数值有效性研究

IF 1

Journal of Computational Dynamics Pub Date : 2019-08-01 DOI: 10.3934/JCD.2019006

Joshua Hudson, M. Jolly

引用次数: 14

Computing Covariant Lyapunov Vectors in Hilbert spaces 希尔伯特空间中协变Lyapunov向量的计算

IF 1

Journal of Computational Dynamics Pub Date : 2019-07-29 DOI: 10.3934/jcd.2021014

Florian Noethen

引用次数: 4

Re-factorising a QRT map 重构QRT映射

IF 1

Journal of Computational Dynamics Pub Date : 2019-06-02 DOI: 10.3934/jcd.2019016

N. Joshi, P. Kassotakis

引用次数: 4

The Lie algebra of classical mechanics 经典力学中的李代数

IF 1

Journal of Computational Dynamics Pub Date : 2019-05-18 DOI: 10.3934/jcd.2019017

R. McLachlan, A. Murua

{"title":"The Lie algebra of classical mechanics","authors":"R. McLachlan, A. Murua","doi":"10.3934/jcd.2019017","DOIUrl":"https://doi.org/10.3934/jcd.2019017","url":null,"abstract":"Classical mechanical systems are defined by their kinetic and potential energies. They generate a Lie algebra under the canonical Poisson bracket. This Lie algebra, which is usually infinite dimensional, is useful in analyzing the system, as well as in geometric numerical integration. But because the kinetic energy is quadratic in the momenta, the Lie algebra obeys identities beyond those implied by skew symmetry and the Jacobi identity. Some Poisson brackets, or combinations of brackets, are zero for all choices of kinetic and potential energy, regardless of the dimension of the system. Therefore, we study the universal object in this setting, the `Lie algebra of classical mechanics' modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. We show that it is the direct sum of an abelian algebra $mathcal X$, spanned by `modified' potential energies isomorphic to the free commutative nonassociative algebra with one generator, and an algebra freely generated by the kinetic energy and its Poisson bracket with $mathcal X$. We calculate the dimensions $c_n$ of its homogeneous subspaces and determine the value of its entropy $lim_{ntoinfty} c_n^{1/n}$. It is $1.8249dots$, a fundamental constant associated to classical mechanics. We conjecture that the class of systems with Euclidean kinetic energy metrics is already free, i.e., the only linear identities satisfied by the Lie brackets of all such systems are those satisfied by the Lie algebra of classical mechanics.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"35 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2019-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81072488","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}

引用次数: 3