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Liegroup methods
 ACTA NUMERICA
, 2000
"... Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having ..."
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Cited by 93 (18 self)
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Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Liegroup structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Liegroup structure, highlighting theory, algorithmic issues and a number of applications.
Newton’s method on Riemannian manifolds: convariant alpha theory
 IMA J. Numer. Anal
, 2003
"... In this paper, Smale’s α theory is generalized to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds. Results are valid for analytic mappings from a manifold to a linear space of the same dimension, or for analytic vector fields on the mani ..."
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Cited by 18 (2 self)
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In this paper, Smale’s α theory is generalized to the context of intrinsic Newton iteration on geodesically complete analytic Riemannian and Hermitian manifolds. Results are valid for analytic mappings from a manifold to a linear space of the same dimension, or for analytic vector fields on the manifold. The invariant γ is defined by means of high order covariant derivatives. Bounds on the size of the basin of quadratic convergence are given. If the ambient manifold has negative sectional curvature, those bounds depend on the curvature. A criterion of quadratic convergence for Newton iteration from the information available at a point is also given. 1 Introduction and main results. Numerical problems posed in manifolds arise in many natural contexts. Classical examples are given by the eigenvalue problem, the symmetric eigenvalue problem, invariant subspace computations, minimization problems with orthogonality constraints, optimization problems with equality constraints... etc. In the first
Generalized Polar Decompositions for the Approximation of the Matrix Exponential
, 2000
"... In this paper we describe the use of the theory of generalized polar decompositions (MuntheKaas, Quispel & Zanna 2000b) to approximate a matrix exponential. The algorithms presented in this paper have the property that, if Z 2 g, a Lie algebra of matrices, then the approximation for exp(Z) resides i ..."
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Cited by 14 (5 self)
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In this paper we describe the use of the theory of generalized polar decompositions (MuntheKaas, Quispel & Zanna 2000b) to approximate a matrix exponential. The algorithms presented in this paper have the property that, if Z 2 g, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of g. This property is very relevant when solving Liegroup ODEs and is not usually fulfilled by standard approximations to the matrix exponential. We propose algorithms based on a splitting of Z into matrices having a very simple structure, usually one row and one column (or a few rows and a few columns), whose exponential is computed very cheaply to machine accuracy. The proposed methods have a complexity of O \Gamma n 3 \Delta , the constant is small, depending on the order and the Lie algebra g. The algorithms are recommended in cases where it is of fundamental importance that the approximation for the exponential resides in G, and when the order of approx...
Lie Group Methods for Rigid Body Dynamics and Time Integration on Manifolds
 Computer Methods in Applied Mechanics and Engineering
, 1999
"... Recently there has been an increasing interest in time integrators for ordinary dierential equations which use Lie group actions as a primitive in the design of the methods. These methods are usually phrased in an abstract sense for arbitrary Lie groups and actions. We show here how the methods l ..."
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Cited by 10 (2 self)
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Recently there has been an increasing interest in time integrators for ordinary dierential equations which use Lie group actions as a primitive in the design of the methods. These methods are usually phrased in an abstract sense for arbitrary Lie groups and actions. We show here how the methods look when applied to the rigid body equations in particular and indicate how the methods work in general. An important part of the Lie group methods involves the computation of a coordinate map and its derivative. Various options are available, and they vary in cost, accuracy and ability to approximately conserve invariants. We discuss how the computation of these maps can be optimized for the rigid body case, and we provide numerical experiments which give an idea of the performance of Lie group methods compared to other known integration schemes. AMS Subject Classication: 65L05 Key Words: time integration, geometric integration, numerical integration of ordinary dierential equati...
Complexity theory for Liegroup solvers
, 1999
"... Commencing with a brief survey of Liegroup theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Liegroup structure: RungeKuttaMuntheKaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and ..."
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Cited by 5 (0 self)
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Commencing with a brief survey of Liegroup theory and differential equations evolving on Lie groups, we describe a number of numerical algorithms designed to respect Liegroup structure: RungeKuttaMuntheKaas schemes, Fer and Magnus expansions. This is followed by complexity analysis of Fer and Magnus expansions, whose conclusion is that for order four, six and eight an appropriately discretized Magnus method is always cheaper than a Fer method of the same order. Each Liegroup method of the kind surveyed in this paper requires the computation of a matrix exponential. Classical methods, e.g. Krylovsubspace and rational approximants, may fail to map elements in a Lie algebra to a Lie group. Therefore we survey a number of approximants based on the splitting approach and demonstrate that their cost is compatible (and often superior) to classical methods. 1 Introduction A central goal of classical numerical analysis is to design, implement and analyse computational algorithms that ...
On The Dimension Of Certain Graded Lie Algebras Arising In Geometric Integration Of Differential Equations
, 1999
"... Many discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropri ..."
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Cited by 4 (3 self)
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Many discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and exploitation of redundancies inherent in a Liealgebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of mnary multilinear operations which display alternating symmetry and a `hierarchy condition'. We present explicit formulae for the dimension of graded subspaces of free hirarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Liealgebraic setting. 1. Graded algebras and Liegroup methods 1.1. Geometric integration and general Liegroup solvers An increasing attention has been paid in recent years to discretization of differential equations that evolve on smooth manifolds. The main reason is that invariants and conservation laws of a differential system can be phrased by restricting the configuration space to a manifold. Discretization methods that respect manifold structure are an important example of geometric integrators, computational methods that preserve the underlying geometry and qualitative attributes of the differential system [1]. Perhaps the most ubiquitous (and arguably most important) type of a m...
A Class of Low Complexity Intrinsic Schemes for Orthogonal Integration
, 2001
"... Numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is considered. Points on this manifold are represented as n k matrices with orthonormal columns, of particular interest is the case when n >> k. Mainly two requirements are imposed on the integration schemes. ..."
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Cited by 4 (1 self)
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Numerical integration of ordinary differential equations on the orthogonal Stiefel manifold is considered. Points on this manifold are represented as n k matrices with orthonormal columns, of particular interest is the case when n >> k. Mainly two requirements are imposed on the integration schemes. First, they should have arithmetic complexity of order nk 2 . Second, they should be intrinsic in the sense that they only require the ODE vector field to be defined on the Stiefel manifold, as opposed to for instance projection methods. The design of the methods makes use of retractions maps. Two algorithms are proposed, one where the retraction map is based on the QR decomposition of a matrix, and one where it is based on the polar decomposition. Numerical experiments show that the new methods are superior to standard Lie group methods with respect to arithmetic complexity, and may be more reliable than projection methods, owing to their intrinsic nature.
On the construction of geometric integrators in the RKMK class
, 1998
"... We consider the construction of geometric integrators in the class of RKMK methods. Any differential equation in the form of an infinitesimal generator on a homogeneous space is shown to be locally equivalent to a differential equation on the Lie algebra corresponding to the Lie group acting on the ..."
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Cited by 3 (1 self)
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We consider the construction of geometric integrators in the class of RKMK methods. Any differential equation in the form of an infinitesimal generator on a homogeneous space is shown to be locally equivalent to a differential equation on the Lie algebra corresponding to the Lie group acting on the homogenous space. This way we obtain a distinction between the coordinatefree phrasing of the differential equation and the local coordinates used. In this paper we study methods based on arbitrary local coordinates on the Lie group manifold. By choosing the coordinates to be canonical coordinates of the first kind we obtain the original method of MuntheKaas [14]. Methods similar to the RKMK method are developed based on the different coordinatizations of the Lie group manifold, given by the Cayley transform, diagonal Pad'e approximants of the exponential map, canonical coordinates of the second kind, etc. Some numerical experiments are also given. 1 Introduction In the past few years the...
A note on the numerical integration of the KdV equation via isospectral deformations
, 2000
"... The main purpose of this note is to test the performance of Lie group integrators as applied to a semi discrete version of the KdV equation. ..."
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The main purpose of this note is to test the performance of Lie group integrators as applied to a semi discrete version of the KdV equation.