A. Iserles, Numerical methods on (and off) manifolds, in "Foundations of Computational Mathematics " (F. Cucker & M. Shub, eds), Springer-Verlag, New York (1997), 180--189.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....these invariants. Typical examples are the symplectic structure in a Hamiltonian system, the energy in a conservative mechanical system and the angular momentum of a rotating rigid body in space. Classical numerical methods normally fail to preserve such as the above mentioned quantities (see e.g. [7]) We view the constraints or invariants as a manifold embedded in some Euclidean space, on which the numerical solution should evolve. The numerical methodswe use in this work are constructed so that the computed solution stays on the manifold, and the numerical error will only result in error ....

A. Iserles. Numerical methods on (and off) manifolds. In F. Cucker and M. Shub, editors, Foundation of Computational Mathematics, pages 180--189. Springer-Verlag, 1997.

....equations [7] This can lead to practical difficulties as the solution can drift out of feasibility, unless steps are taken to bring q(t) back to the feasible set C = f q j f (j) q) 0; j = 1; p g. This has become a particularly important issue for integration on manifolds [23] and rigid body dynamics without unilateral constraints (see, for example, 6, pp. 150 157] and [7] In this case, the constraints are not manifolds, but more general sets defined by smooth inequalities. Projection methods, for example, can be implemented in conjunction with the time stepping ....

A. Iserles. Numerical methods on (and off) manifolds. In F. Cucker and M. Shub, editors, Foundations of Computational Mathematics, pages 180--189, Berlin, Heidelberg, New York, 1997. Springer. Selected papers of a conference held at IMPA in Rio de Janeiro, January 1997.

....how much of this qualitative structure can be preserved by a numerical method. Of course some of these questions have been considered for some time, for example the energy preserving schemes introduced by Arakawa [1] and schemes for ordinary di erential equations that conserve Lie point symmetries [25]. Furthermore, for some speci c problems such as Hamiltonian ODEs or time reversible systems, some excellent geometry preserving integrators have been developed which can be analysed using backward error analysis [36] However, a general theory, applicable to a wide class of problems is still ....

A. Iserles, \Numerical methods on (and o) manifolds". In Foundations of Computational Mathematics, (ed. F. Cucker and M. Shub), (1997), pp. 180-189, New York: Springer

....these invariants. Typical examples are the symplectic structure in a Hamiltonian system, the energy in a conservative mechanical system and the angular momentum of a rotating rigid body in space. Classical numerical methods normally fail to preserve such as the above mentioned quantities (see e.g. [7]) We view the constraints or invariants as a manifold embedded in some Euclidean space, on which the numerical solution should evolve. The numerical methodswe use in this work are constructed so that the computed solution stays on the manifold, and the numerical error will only result in error on ....

A. Iserles. Numerical methods on (and off) manifolds. In F. Cucker and M. Shub, editors, Foundation of Computational Mathematics, pages 180--189. Springer-Verlag, 1997.

....theoretical) interest, a given problem is often reduced to the solution of a differential equation or a family of differential equations evolving on some prescribed manifold. In the framework of devising structural discretization methods for Ordinary Differential Equations (ODEs) on manifolds (cf. [6, 7] for a discussion on such matters) we describe the method of iterated commutators for those manifolds that posses an underlying Lie group structure. Such manifolds are interesting per s e, as for instance the orthogonal group, the unitary group etc. but also because, once we have Lie group ....

Iserles, A.: Numerical methods on (and off) manifolds. This volume.

....intrinsic methods. In the first of these one embeds the manifold in R n and employs a classical integration scheme here. The problem of this approach is that, except in very special cases, it is generally impossible to find classical integration schemes which will stay on the correct manifold [5, 9, 11]. The alternative, intrinsic methods, are based on expressing the algorithm via a set of basic flows which in each point span all the directions of the manifold. These methods have the advantage of Received August 1996. Revised August 1997. y This work is sponsored by NFR under contract no. ....

A. Iserles. Numerical methods on (and off) manifolds. In F. Cucker, editor, Foundation of Computational Mathematics, pages 180--189. Springer-Verlag, 1997.

....these invariants. Typical examples are the symplectic structure in a Hamiltonian system, the energy in a conservative mechanical system and the angular momentum of a rotating rigid body in space. Classical numerical methods normally fail to preserve such as the above mentioned quantities (see e.g. [7]) We view the constraints or invariants as a manifold embedded in some Euclidean space, on which the numerical solution should evolve. The numerical methodswe use in this work are constructed so that the computed solution stays on the manifold, and the numerical error will only result in error on ....

A. Iserles. Numerical methods on (and off) manifolds. In F. Cucker and M. Shub, editors, Foundation of Computational Mathematics, pages 180--189. Springer-Verlag, 1997.

....these invariants. Typical examples are the symplectic structure in a Hamiltonian system, the energy in a conservative mechanical system and the angular momentum of a rotating rigid body in space. Classical numerical methods normally fail to preserve such as the above mentioned quantities (see e.g. [7]) We view the constraints or invariants as a manifold embedded in some Euclidean space, on which the numerical solution should evolve. The numerical methods we use in this work are constructed so that the computed solution stays on the manifold, and the numerical error will only result in error ....

A. Iserles. Numerical methods on (and off) manifolds. In F. Cucker and M. Shub, editors, Foundation of Computational Mathematics, pages 180--189. Springer-Verlag, 1997.

....techniques which preserve important qualitative properties of differential equations. This includes the recent work on preserving the symplectic structure of Hamiltonian systems [15] orthogonality [5] isospectrality [2] and on designing numerical methods which stay on a prescribed manifold [3, 4, 6, 7, 11, 12, 13]. In this paper we are concerned with the latter problem, in particular numerical integration of differential equations evolving on homogeneous spaces. The structure of a homogeneous space is both specific enough to allow the definition and the analysis of numerical integration methods, and, at ....

....M. There are two main approaches for integrating differential equations on manifolds, associated with embedded and intrinsic methods. The first consists of methods where the manifold is embedded in a vector space and a classical integration scheme is employed. The work of Calvo, Iserles and Zanna [3, 7] shows This work is sponsored by NFR under contract no. 111038 410, through the SYNODE project. WWW: http: www.imf.unit.no num synode that except for a few special, yet important cases, it is impossible to devise classical integration schemes such that the numerical solution will stay on ....

Iserles, A.: Numerical methods on (and off) manifolds. This volume.

No context found.

A. Iserles, Numerical methods on (and off) manifolds, in "Foundations of Computational Mathematics " (F. Cucker & M. Shub, eds), Springer-Verlag, New York (1997), 180--189.

....finite dimensional Lie group is isomorphic to a subgroup of a matricial Lie group, we henceforth assume that G is a matricial Lie group. Obviously we can solve (1) by a classical method, e.g. a linear multistep method or a Runge Kutta scheme. However, it has been pointed out by Iserles [6] that the approximations obtained by these methods do not stay on general nonlinear manifolds (except for some Runge Kutta schemes that retain quadratic conservation laws) Hence there is a need for new methods which ensure that the numerical discretization stays on the correct manifold, thereby ....

A. Iserles, Numerical methods on (and off) manifolds, in "Foundations of Computational Mathematics " (F. Cucker & M. Shub, eds), Springer-Verlag, New York (1997), 180--189.

....finite dimensional Lie group is isomorphic to a subgroup of a matricial Lie group, we henceforth assume that G is a matricial Lie group. Obviously we can solve (1) by a classical method, e.g. a linear multistep method or a Runge Kutta scheme. However, it has been pointed out by Iserles [6] that the approximations obtained by these methods do not stay on general nonlinear manifolds (except for some Runge Kutta schemes that retain quadratic conservation laws) Hence there is a need for new methods which ensure that the numerical discretization stays on the correct manifold, thereby ....

A. Iserles, Numerical methods on (and off) manifolds, in "Foundations of Computational Mathematics " (F. Cucker & M. Shub, eds), Springer-Verlag, New York (1997), 180--189.

No context found.

A. Iserles, Numerical methods on (and off) manifolds, in Foundations of Computational Mathematics (F. Cucker & M. Shub, eds), Springer-Verlag, New York (1997), 180--189.

No context found.

A. Iserles. Numerical methods on (and off) manifolds. In F. Cucker, editor, Foundation of Computational Mathematics, pages 180--189. SpringerVerlag, 1997.

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