Results 1  10
of
13
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 ..."
Abstract

Cited by 93 (18 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
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
Hamilton–Pontryagin integrators on Lie groups
, 2007
"... Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
Abstract In this paper, structurepreserving timeintegrators for rigid bodytype mechanical
Geometric integration algorithms on homogeneous manifolds
 Foundations of Computational Mathematics
, 2002
"... Given an ordinary differential equation on a homogeneous manifold, one can construct a “geometric integrator ” by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curve ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
Given an ordinary differential equation on a homogeneous manifold, one can construct a “geometric integrator ” by determining a compatible ordinary differential equation on the associated Lie group, using a Lie group integration scheme to construct a discrete time approximation of the solution curves in the group, and then mapping the discrete trajectories onto the homogeneous manifold using the group action. If the points of the manifold have continuous isotropy, a vector field on the manifold determines a continuous family of vector fields on the group, typically with distinct discretizations. If sufficient isotropy is present, an appropriate choice of vector field can yield improved capture of key features of the original system. In particular, if the algebra of the group is “full”, then the order of accuracy of orbit capture (i.e. approximation of trajectories modulo time reparametrization) within a specified family of integration schemes can be increased by an appropriate choice of isotropy element. We illustrate the approach developed here with comparisons of several integration schemes for the reduced rigid body equations on the sphere. 1 Introduction. Geometric integration techniques have become increasingly popular in the modern approach to numerical
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 ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
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...
High order optimized geometric integrators for linear differential equations
, 2000
"... In this paper new integration algorithms for linear differential equations up to eighth order are obtained. Starting from Magnus expansion, methods based on Cayley transformation and Fer expansion are also built. The structure of the exact solution is retained while the computational cost is reduced ..."
Abstract

Cited by 7 (1 self)
 Add to MetaCart
In this paper new integration algorithms for linear differential equations up to eighth order are obtained. Starting from Magnus expansion, methods based on Cayley transformation and Fer expansion are also built. The structure of the exact solution is retained while the computational cost is reduced compared to similar methods. Their relative performance is tested on some illustrative examples.
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 ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
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...
Numerical integration of LiePoisson systems while preserving coadjoint orbits and energy
, 1999
"... In this paper we apply geometric integrators of the RKMK type to the problem of integrating LiePoisson systems numerically. By using the coadjoint action of the Lie group G on the dual Lie algebra g* to advance the numerical flow, we devise methods of arbitrary order that automatically stay on the ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
In this paper we apply geometric integrators of the RKMK type to the problem of integrating LiePoisson systems numerically. By using the coadjoint action of the Lie group G on the dual Lie algebra g* to advance the numerical flow, we devise methods of arbitrary order that automatically stay on the coadjoint orbits. First integrals known as Casimirs are retained to machine accuracy by the numerical algorithm. Within the proposed class of methods we find integrators that also conserve the energy. These schemes are implicit and of second order. Nonlinear iteration in the Lie algebra and linear error growth of the global error are discussed. Numerical experiments with the rigid body, the heavy top and a finitedimensional truncation of the Euler equations for a 2D incompressible fluid are used to illustrate the properties of the algorithm.
The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as TimeDependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial resummation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related nonperturbative
Partitioned RungeKutta methods in Liegroup setting
, 2000
"... We introduce partitioned RungeKutta (PRK) methods as geometric integrators in the Runge KuttaMuntheKaas (RKMK) method hierarchy. This is done by first noticing that tangent and cotangent bundles are the natural domains for the differential equations to be solved. Next, we equip the (co)tangen ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
We introduce partitioned RungeKutta (PRK) methods as geometric integrators in the Runge KuttaMuntheKaas (RKMK) method hierarchy. This is done by first noticing that tangent and cotangent bundles are the natural domains for the differential equations to be solved. Next, we equip the (co)tangent bundle of a Lie group with a group structure and treat it as a Lie group. The structure of the differential equations on the (co)tangentbundle Lie group is such that partitioned versions of the RKMK methods are naturally introduced. Numerical examples are included to illustrate the new methods. AMS Subject Classification: 65L06, 34C40 Key Words: Partitioned RungeKutta method, RKMK method, Magnus method, Cayley method, CrouchGrossman method, geometric integration, tangent bundle of Lie group, semidirect product, differential equations on manifolds 1 Introduction The RungeKuttaMuntheKaas (RKMK) method was introduced in [17] to solve differential equations on homogeneous spaces, t...