GEOMETRY PRESERVING WEAK ITO-TAYLOR SCHEMES FOR STOCHASTIC DYNAMICS ON MANIFOLDS

The dynamics of physically occurring systems under the influence of random noise are often associated with constraints of geometric nature. For example, the rigid body rotations are constrained to a manifold defined by the space of special orthogonal groups, or in control theory, for instance, the movement of a robot arm is constrained to have constant length. Therefore, to simulate/predict the movement of such a system, it is sensible to choose a numerical technique that preserves the underlying geometry while considering the stochasticity in the system. The current state-of-the-art approaches either preserve the geometry of the manifold without considering the stochasticity or fail to preserve the geometry of the manifold while accounting for the randomness. In this work, we study matrix stochastic differential equations (SDEs) on a manifold that has a Lie group structure and is driven by k-dimensional Brownian motion. Weak geometric integration schemes to derive geometrically accurate responses with coarser integration step size are explored. Mainly two geometric integration schemes based on Ito-Taylor expansion are proposed: Weak 1.0 and Weak 2.0. New formulations for the Kolmogorov operators associated with the higher-order scheme considering the geometric aspects of the stochastic differential equations are developed. The proposed numerical techniques premises on an approximation to the SDE within the tangent space, taking advantage of the algebraic structure, before mapping back to the manifold using the exponential map. Numerical simulations on systems evolving on SO(3) and S^2 manifolds are carried out to validate the accuracy of the proposed stochastic geometric schemes.