Numerical Solution of Stochastic Differential Equations in Stiefel Manifolds via Tangent Space Parametrization

Abstract

Stochastic differential equations (SDEs) evolving in Stiefel manifold have numerous applications in Science and Engineering. While numerical schemes for ordinary differential equations (ODEs) in Stiefel manifolds are reasonably well established, much less has been done for numerical SDEs schemes in Stiefel manifolds. A crucial challenge is to ensure that the trajectory remains on the manifold. But many existing SDE numerical schemes fail to do this. Here we achieve this by extending the so-called 'tangent space parameterization’ (TaSP) for ODEs to SDEs. In so doing we discover a previously missed constraint. We give simulations to illustrate the new numerical scheme.