An exponential map free implicit midpoint method for stochastic Lie-Poisson systems

TL;DR

The paper introduces an exponential-map-free isospectral stochastic implicit midpoint method for Lie--Poisson systems with Stratonovich noise. By performing discrete Lie--Poisson reduction of a stochastic canonical Hamiltonian integrator, it yields a Lie--Poisson integrator that almost surely preserves Casimirs and coadjoint orbits while remaining scalable to high dimensions. The authors prove strong and weak convergence (RMSE order $1/2$ and order $1$, respectively) and demonstrate the method on several LP models, including the rigid body, Manakov system, point vortices on the sphere, and Zeitlin--Euler discretizations. The approach avoids costly algebra-to-group maps and provides a framework for structure-preserving simulations under uncertainty with broad applicability to stochastic fluid and spin systems.

Abstract

An integrator for a class of stochastic Lie-Poisson systems driven by Stratonovich noise is developed. The integrator is suited for Lie-Poisson systems that also admit an isospectral formulation, which enables scalability to high-dimensional systems. Its derivation follows from discrete Lie-Poisson reduction of the symplectic midpoint scheme for stochastic Hamiltonian systems. We prove almost sure preservation of Casimir functions and coadjoint orbits under the numerical flow and provide strong and weak convergence rates of the proposed method. The scalability, structure-conservation, and convergence rates are illustrated numerically for the (generalized) rigid body, point vortex dynamics, and the two-dimensional Euler equations on the sphere.

Figures (11)

• Figure 1: Left: Trajectories of components of the angular velocity in the rigid body equations. The lines show numerical solutions to the deterministic equations (black) and the stochastic equations (blue). Note that both trajectories remain on the unit sphere, depicted in gray. Right: Trajectories of four point vortices on the sphere, following the deterministic dynamics (black) and stochastic dynamics (colored).
• Figure 2: Instantaneous vorticity snapshots of the stochastic two-dimensional Euler equations on the sphere, presented on a latitude-longitude grid. The stochastic LP system evolves on $\mathfrak{su}(512)^*$. Shown are the initial condition consisting only of large-scale components (left); turbulent mixing during a transitory phase (middle left, middle right); formation of large-scale vorticity condensates (right).
• Figure 3: Schematic representation of the construction of the Lie--Poisson integrator, as in modin2023spatio. We consider an equivariant symplectic method $\Phi_h\colon T^*G\to T^*G$, i.e., satisfying $\Phi_h(g\cdot a) = g\cdot\Phi_h(a)$ for $g\in G$ and $a\in T^*G$. This method descends to a Lie--Poisson method $\Psi_h\colon\mathfrak{g}^*\to\mathfrak{g}^*$ on the coadjoint orbit $\mathcal{O}$ after applying the momentum map $\mu$.
• Figure 4: A single realization of the stochastic rigid body. Left: absolute departure of the eigenvalues of the solution matrix $X$, measured from the initial eigenvalues. Right: relative departure of the Hamiltonian, normalized by the initial value.
• Figure 5: Left: Strong errors for the rigid body system using 500 independent realizations. Right: Weak error for the rigid body system based on $10^7$ independent realizations.

Theorems & Definitions (19)

• Remark 2.1
• Remark 3.1
• Remark 4.1
• Lemma 4.2
• proof
• Remark 4.3
• Lemma 4.4
• proof
• Theorem 4.5
• proof