Numerical integration of ODEs on the Stiefel manifold

The problem of computing the numerical solution of odes on Stiefel and Grassmann manifolds arises in a variety of applications. The Stiefel manifold for example is a homogeneous space whose elements can be represented by nxp matrices with orthogonal columns. The naive application of a Lie group integrator to odes on the Stiefel manifold give rise to computations of order n^3 complexity, that by careful implementation might be reduced to n^2p. Even lower complexity can be however achieved using projection methods, in which the orthogonality if not preserved is in any case recovered. In this talk we will address the problem of constructing intrinsic numerical integrators preserving orthogonality with complexity O(np^2). The case of more general homogeneous spaces is also considered.