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Tangent Space Parametrization for Stochastic Differential Equations on SO(n)

Xi Wang, Victor Solo

TL;DR

This work develops the stochastic tangent space parametrization (S-TaSP) for simulating multiplicative SDEs on the special orthogonal group SO(n) while preserving the manifold constraint without explicit exponential-map computations. It derives a simple, explicit normal correction C = sqrt(I − Z^T Z) − I that ensures updates stay on SO(n) and reduces the dynamics to a Euclidean SDE for the tangent variable Z, enabling a one-step Euler update. The authors establish a strong convergence rate of ${\frac{1-\epsilon}{2}}$ for any $\epsilon>0$, via two key lemmas bounding the discretization error and drift–diffusion contributions. Numerical experiments on Brownian motion and state-dependent noise demonstrate geometry preservation, validate the convergence theory, and reveal substantial computational savings (≈60% of the cost of competing Magnus-based SL-EM schemes) for large-scale problems. The approach provides a practical and scalable GPNS for SDEs on SO(n) and sets the stage for extensions to general Riemannian manifolds.

Abstract

In this paper, we study the numerical simulation of stochastic differential equations (SDEs) on the special orthogonal Lie group $\text{SO}(n)$. We propose a geometry-preserving numerical scheme based on the stochastic tangent space parametrization (S-TaSP) method for state-dependent multiplicative SDEs on $\text{SO}(n)$. The convergence analysis of the S-TaSP scheme establishes a strong convergence order of $\mathcal{O}(δ^{\frac{1-ε}{2}})$, which matches the convergence order of the previous stochastic Lie Euler-Maruyama scheme while avoiding the computational cost of the exponential map. Numerical simulation illustrates the theoretical results.

Tangent Space Parametrization for Stochastic Differential Equations on SO(n)

TL;DR

This work develops the stochastic tangent space parametrization (S-TaSP) for simulating multiplicative SDEs on the special orthogonal group SO(n) while preserving the manifold constraint without explicit exponential-map computations. It derives a simple, explicit normal correction C = sqrt(I − Z^T Z) − I that ensures updates stay on SO(n) and reduces the dynamics to a Euclidean SDE for the tangent variable Z, enabling a one-step Euler update. The authors establish a strong convergence rate of for any , via two key lemmas bounding the discretization error and drift–diffusion contributions. Numerical experiments on Brownian motion and state-dependent noise demonstrate geometry preservation, validate the convergence theory, and reveal substantial computational savings (≈60% of the cost of competing Magnus-based SL-EM schemes) for large-scale problems. The approach provides a practical and scalable GPNS for SDEs on SO(n) and sets the stage for extensions to general Riemannian manifolds.

Abstract

In this paper, we study the numerical simulation of stochastic differential equations (SDEs) on the special orthogonal Lie group . We propose a geometry-preserving numerical scheme based on the stochastic tangent space parametrization (S-TaSP) method for state-dependent multiplicative SDEs on . The convergence analysis of the S-TaSP scheme establishes a strong convergence order of , which matches the convergence order of the previous stochastic Lie Euler-Maruyama scheme while avoiding the computational cost of the exponential map. Numerical simulation illustrates the theoretical results.

Paper Structure

This paper contains 14 sections, 6 theorems, 47 equations, 5 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM STATEMENT
  3. Geometry of $\text{SO}(n)$
  4. Multiplicative SDE on $\text{SO}(n)$
  5. Problem Definition
  6. STOCHASTIC TaSP NUMERICAL SCHEME
  7. CONVERGENCE ANALYSIS
  8. Numerical simulation
  9. Brownian Motion on $\text{SO}(n)$
  10. State-Dependent Noise
  11. Conclusion
  12. Appendix
  13. Proof of Theorem \ref{['thm:correction']}
  14. Proof of Lemma \ref{['lem:1']}

Key Result

Theorem 1

Let $R_0 \in \text{SO}(n)$ and $\mathcal{Z} \in \mathfrak{so}(n)$. If $I - \mathcal{Z}^T \mathcal{Z}$ is positive definite, then the normal correction term $\mathbf{C}_0 \in N_{R_0} \text{SO}(n)$ that ensures $R = R_0 + R_0 \mathcal{Z} + R_0 \mathbf{C} \in \text{SO}(n)$ is given by where $\sqrt{A}$ denotes the unique positive definite square root of the positive definite matrix $A$.

Figures (5)

  • Figure 1: Normality test of $\log \hat{R}(t)$
  • Figure 2: Geometry Preservation of S-TaSP for Brownian Motion
  • Figure 3: Trajectory Comparison between S-TaSP and SL-EM
  • Figure 4: Geometry Preservation of S-TaSP and SL-EM
  • Figure 5: Running Time Comparison between S-TaSP and SL-EM

Theorems & Definitions (6)

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