Numerical integration of ordinary differential equations on manifolds

Crouch, P. E.; Grossman, Robert L.

doi:10.1007/bf02429858

Cited by 295 publications

(188 citation statements)

References 23 publications

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“…Numerical integration schemes of ordinary differential equations on manifolds are presented by Crouch and Grossman (1991), Crouch, Grossman and Yan (1992a; and Ge-Zhong and Marsden (1988). Discrete-time versions of some classical integrable systems are analyzed by Moser and Veselov (1991).…”

Section: Notes For Chaptermentioning

confidence: 99%

Optimization and Dynamical Systems

Helmke¹,

Moore²

1994

Communications and Control Engineering

547

419

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PrefaceThis work is aimed at mathematics and engineering graduate students and researchers in the areas of optimization, dynamical systems, control systems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emergence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems theory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control.The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical systems implementation, either in continuous time or discrete time , which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been the recent discovery of rather deep existence and uniqueness results for the solution of certain matrix least squares optimization problems in geometric invariant theory. These problems, as well as many other optimization problems arising in linear algebra and systems theory, do not always admit solutions which can be found by algebraic methods. Even for such problems that do admit solutions via algebraic methods, as for example the classical task of singular value decomposition, there is merit in viewing the task as a certain matrix optimization problem, so as to shift the focus from algebraic methods to geometric methods. It is in this context that gradient flows on manifolds appear as a natural approach to achieve construction methods that complement the existence and uniqueness results of geometric invari- There has been an attempt to bridge the disciplines of engineering and mathematics in such a way that the work is lucid for engineers and yet suitably rigorous for mathematicians. There is also an attempt to teach, to provide insight, and to make connections, rather than to present the results as a fait accompli.The research for this work has been carried out by the authors while visiting each other's institutions. Some of the work has been written in conjunction with the writing of research papers in collaboration with PhD students Jane Perkins and Robert Mahony, and post doctoral fellow Weiyong Yan. Indeed, the papers benefited from the book and vice versa, and consequently many of the paragraphs are common to both. Uwe Helmke has a background in global analysis with a strong interest in systems theory, and John Moore has a background in control systems and signal processing. AcknowledgementsThis work was partially supported by grants from Boeing Commercial Airplane Company, the Cooperative Research Centre for Robust and Adaptive Systems, and the German-Israel...

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Section: Notes For Chaptermentioning

confidence: 99%

Optimization and Dynamical Systems

Helmke¹,

Moore²

1994

Communications and Control Engineering

547

419

View full text Add to dashboard Cite

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“…The other development is motivated by the philosophy of geometric integration and its purpose is to recover under discretization important qualitative and geometric features of the underlying dynamical system. Examples of such methods can be found inter alia in (Casas 1996, Crouch & Grossman 1993). An important technique in geometric integration is the use of Lie-group actions, which lend themselves to the design of very effective time-stepping methods for ODEs evolving on homogeneous manifolds.…”

Section: Introductionmentioning

confidence: 99%

Methods for the approximation of the matrix exponential in a Lie-algebraic setting

Celledoni¹,

Iserles²

2001

IMA Journal of Numerical Analysis

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Discretization methods for ordinary differential equations based on the use of matrix exponentials have been known for decades. This set of ideas has come off age and acquired greater urgency recently, within the context of geometric integration and discretization methods on manifolds based on the use of Lie-group actions.In the present paper we study the approximation of the matrix exponential in a particular context: given a Lie group G and its Lie algebra g, we seek approximants F (tB) of exp(tB) such that F (tB) ∈ G if B ∈ g. Having fixed a basis V1, . . . , V d of g, we write F (tB) as a composition of exponentials of the type exp(αi(t)Vi), where αi for i = 1, 2, . . . , d are scalar functions. In this manner it becomes possible to increase the order of the approximation without increasing the number of exponentials to evaluate and multiply together. We study order conditions and implementation details and conclude the paper with some numerical experiments.

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“…. , σ q are scalar functions (Crouch & Grossman 1993, Owren & Marthinsen 2001). All the above are based on the exponential trivialisation.…”

Section: Alternative Lie-algebraic Expansionsmentioning

confidence: 99%

Magnus expansions and beyond

Iserles¹

2011

Contemporary Mathematics

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In this brief review we describe the coming of age of Magnus expansions as an asymptotic and numerical tool in the investigation of linear differential equations in a Lie-group and homogeneous-space setting. Special attention is afforded to the many connections between modern theory of geometric numerical integration and other parts of mathematics: from abstract algebra to differential geometry and combinatorics, all the way to classical numerical analysis. Lie-group equationsNumerical solution of evolutionary differential equations is as old as the theory of differential equations itself: although proper numerical analysis of differential equations commenced with Leonhard Euler, earlier ad hoc numerical ideas abound in the works of Sir Isaac Newton and of Gottfried von Leibnitz. (A brief, yet outstanding historical synopsis can be found in (Hairer, Nørsett & Wanner 1986).) In the last fifty years numerical analysis of differential equations has developed in leaps and bounds, in parallel with the evolution in computing power and speed.On the face of it, all is well in the numerical kingdom. However, a closer look reveals a worrying gap between the efforts of numerical analysts and of pure mathematicians. Thus, pure mathematicians expand a very great deal of effort to analyse qualitative properties of differential equations but they usually fall short of fleshing out numbers. At the same time, numerical analysts are extraordinarily successful in producing numbers and figures with appropriately small errors but these numbers and figures typically fail to respect qualitative properties of differential equations. This disparity between analysis and computation motivated in the last decade the emergence of a new paradigm of geometric numerical integration (GNI): to seek computational methods that render exactly important qualitative features of differential equations. Examples of qualitative features whose preservation under discretization is important include the symplectic structure of Hamiltonian and Lie-Poisson systems

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Numerical integration of ordinary differential equations on manifolds

Cited by 295 publications

References 23 publications

Optimization and Dynamical Systems

Optimization and Dynamical Systems

Methods for the approximation of the matrix exponential in a Lie-algebraic setting

Magnus expansions and beyond

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