Thursday,
February 5, 1998, 3:05 p.m. in GWC Room 604
Arne Marthinsen and Brynjulf Owren
Department of Mathematical Sciences, NTNU, Norway
Numerical Methods on Manifolds
Abstract
In many applications, for instance in mechanics, the state space
on which the solution evolves is a manifold and the dynamics is
naturally described as a vector field on this manifold.
The numerical analyst needs to devise techniques for approximating
the flow of the vector field. Using local coordinates as an intrinsic
part of the integration method can lead to a very complicated representation
of the vector field. Global embedding in a higher dimensional Euclidean
space can be done, but we then want to make sure that the approximation
respects the manifold.
A completely different technique was proposed by Crouch and Grossman
in 1993, and similar ideas have been pursued in the SYNODE project over
that past two years. The key point is to formulate both the vector field
and the approximation by means of an action on the manifold by a Lie
algebra. In this way the choice of representation of the manifold, as
well as the action primitives can be tailored to each application.
In particular, an object oriented software package called DIFFMAN has
been designed and implemented using Matlab 5.0.
