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Stochastic conformal integrators for linearly damped stochastic Poisson systems

Charles-Edouard Bréhier, David Cohen, Yoshio Komori

TL;DR

This work tackles the numerical integration of linearly damped stochastic Poisson systems by introducing stochastic conformal exponential integrators that preserve conformal invariants. The method leverages Strang splitting and discrete gradients to respect the Poisson structure, and employs truncated Wiener increments to ensure well-posed implicit updates. Key contributions include proofs of conformal Casimir preservation for quadratic Casimirs, energy balance for homogeneous Hamiltonians, and strong/weak convergence rates (strong: $1/2$ generally, $1$ when $M=1$; weak: $1$) under appropriate moment bounds. Theoretical results are complemented by extensive numerical experiments on pendulum, rigid body with random inertia, Lotka–Volterra, Maxwell–Bloch, and other damped Poisson systems, demonstrating structure preservation, stability, and accuracy with practical significance for stochastic geometric computation.

Abstract

We propose and study conformal integrators for linearly damped stochastic Poisson systems. We analyse the qualitative and quantitative properties of these numerical integrators: preservation of dynamics of certain Casimir and Hamiltonian functions, almost sure bounds of the numerical solutions, and strong and weak rates of convergence under appropriate conditions. These theoretical results are illustrated with several numerical experiments on, for example, the linearly damped free rigid body with random inertia tensor or the linearly damped stochastic Lotka--Volterra system.

Stochastic conformal integrators for linearly damped stochastic Poisson systems

TL;DR

This work tackles the numerical integration of linearly damped stochastic Poisson systems by introducing stochastic conformal exponential integrators that preserve conformal invariants. The method leverages Strang splitting and discrete gradients to respect the Poisson structure, and employs truncated Wiener increments to ensure well-posed implicit updates. Key contributions include proofs of conformal Casimir preservation for quadratic Casimirs, energy balance for homogeneous Hamiltonians, and strong/weak convergence rates (strong: generally, when ; weak: ) under appropriate moment bounds. Theoretical results are complemented by extensive numerical experiments on pendulum, rigid body with random inertia, Lotka–Volterra, Maxwell–Bloch, and other damped Poisson systems, demonstrating structure preservation, stability, and accuracy with practical significance for stochastic geometric computation.

Abstract

We propose and study conformal integrators for linearly damped stochastic Poisson systems. We analyse the qualitative and quantitative properties of these numerical integrators: preservation of dynamics of certain Casimir and Hamiltonian functions, almost sure bounds of the numerical solutions, and strong and weak rates of convergence under appropriate conditions. These theoretical results are illustrated with several numerical experiments on, for example, the linearly damped free rigid body with random inertia tensor or the linearly damped stochastic Lotka--Volterra system.

Paper Structure

This paper contains 18 sections, 12 theorems, 161 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Notation
  4. Examples
  5. Qualitative properties
  6. Stochastic conformal exponential integrator
  7. Description of the integrator
  8. Qualitative properties
  9. Strong and weak convergence results
  10. Preliminary results
  11. Proofs of the convergence results
  12. Numerical experiments
  13. A linearly damped stochastic mathematical pendulum
  14. Linearly damped free rigid body with random inertia tensor
  15. Linearly damped stochastic Lotka--Volterra system
  16. ...and 3 more sections

Key Result

Proposition 3

Consider the linearly damped stochastic Poisson system prob. Assume that $C$ is a Casimir function of class $\mathcal{C}^2$, which is an homogeneous function of degree $p\in(0,\infty)$. Then, the mapping $C$ is a conformal Casimir function for the linearly damped stochastic Poisson system prob: if $ Consider the linearly damped stochastic Poisson system with one noise prob1. Assume that the Hamilt

Figures (10)

  • Figure 1: Linearly damped stochastic mathematical pendulum \ref{['pendul']}: Strong convergence.
  • Figure 2: Linearly damped stochastic rigid body system \ref{['eq:srb']}: Evolution of the quadratic Casimir $C(y)=\frac{1}{2}\left( y_1^2+y_2^2+y_3^2 \right)$.
  • Figure 3: Linearly damped stochastic rigid body system \ref{['eq:srb']}: Strong convergence of the numerical schemes.
  • Figure 4: Linearly damped stochastic rigid body system \ref{['eq:srb']} (with $3$ noises): Weak convergence of the stochastic conformal exponential integrator.
  • Figure 5: Linearly damped stochastic rigid body system \ref{['eq:srbH']}: Evolution of the quadratic Hamiltonian $H_0(y)=\frac{1}{2}\left( \frac{y_1^2}{I_1}+\frac{y_2^2}{I_2}+\frac{y_3^2}{I_3} \right)$.
  • ...and 5 more figures

Theorems & Definitions (33)

  • Example 1: Linearly damped stochastic Hamiltonian systems
  • Example 2: Linearly damped stochastic rigid body system
  • Example 3: Linearly damped stochastic Lotka--Volterra system
  • Example 4: Linearly damped stochastic Maxwell--Bloch system
  • Example 5: Linearly damped stochastic three-dimensional Poisson system
  • Remark 1
  • Remark 2
  • Proposition 3
  • proof
  • Corollary 4
  • ...and 23 more