Numerical methods for stochastic differential equations in the Stiefel manifold made simple

Abstract

Several engineering applications require numerical schemes for the simulation of stochastic differential equations (SDEs) on the Stiefel manifold of n × p real matrices with orthonormal columns (n ≥ p). The geometry of the Stiefel manifold causes classical schemes to fail, as they leave the manifold after one time step. Motivated by deterministic schemes on the Stiefel manifold, we develop a scheme for SDEs which always remains on the manifold. Our scheme can be easily applied to any SDE on the Stiefel manifold. The derivation is direct and made accessible to an engineering audience. Simulations with n = 1000, p = 10 are provided.