Geometry-preserving Numerical Scheme for Riemannian Stochastic Differential Equations

TL;DR

This work addresses the need for geometry-preserving numerical schemes for SDEs on general Riemannian manifolds. It introduces the Exponential Euler–Maruyama (Exp-EM) scheme, which updates along the tangent space and re-enters the manifold via the exponential map, avoiding costly manifold projections. Under mild Lipschitz and curvature-related assumptions, the authors prove a strong convergence rate of $O\left(\delta^{\frac{1-ε}{2}}\right)$ and demonstrate the method's effectiveness through Brownian motion on spheres in both moderate and high dimensions, showing stable geometry preservation and practical efficiency. The results generalize geometry-preserving discretizations beyond specific manifolds and offer a scalable tool for simulating constrained stochastic dynamics with high fidelity.

Abstract

Stochastic differential equations (SDEs) on Riemannian manifolds have numerous applications in system identification and control. However, geometry-preserving numerical methods for simulating Riemannian SDEs remain relatively underdeveloped. In this paper, we propose the Exponential Euler-Maruyama (Exp-EM) scheme for approximating solutions of SDEs on Riemannian manifolds. The Exp-EM scheme is both geometry-preserving and computationally tractable. We establish a strong convergence rate of $\mathcal{O}(δ^{\frac{1 - ε}{2}})$ for the Exp-EM scheme, which extends previous results obtained for specific manifolds to a more general setting. Numerical simulations are provided to illustrate our theoretical findings.

Figures (3)

• Figure 1: Performance of Schemes for $n=20$
• Figure 2: Performance of Schemes for $n=2000$ and $\delta = 0.001$
• Figure 3: Performance of Schemes for $n=2000$ and $\delta = 0.01$

Theorems & Definitions (6)

• Lemma 1
• Theorem 1
• Lemma 2
• Lemma 3
• Theorem 2
• Lemma 4