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Superiority Of Symplectic Methods For Stochastic Hamiltonian System Via Asymptotic Error Distribution

Jialin Hong, Ge Liang, Derui Sheng

TL;DR

The paper investigates why symplectic methods outperform in stochastic Hamiltonian systems with additive noise by analyzing the asymptotic distribution of the normalized error. It proves that the stochastic $\theta$-method is symplectic only at $\theta=\tfrac{1}{2}$ (the midpoint method) and that this choice minimizes the long-time asymptotic error constant, with a limit SDE characterizing the error dynamics. For the linear stochastic oscillator, explicit asymptotic error constants show symplectic schemes have error variance growing linearly with time $T$, while non-symplectic schemes grow as $T^3$, implying exponentially faster decay of error deviations for symplectic methods in long-time computations. The results are reinforced by numerical experiments that confirm the linear vs cubic scaling in variance for symplectic and non-symplectic methods, respectively, highlighting the practical advantage of symplectic integrators in long-time stochastic simulations.

Abstract

The superiority of symplectic methods for stochastic Hamiltonian systems has been widely recognized, yet the probabilistic mechanism behind this superiority remains incompletely understood. This paper studies the superiority of symplectic methods from the perspective of the asymptotic error distribution, i.e., the limit distribution of normalized error. Focusing on stochastic Hamiltonian systems driven by additive noise, we obtain the asymptotic limit of the normalized error distribution of the $θ$ method $(θ\in[0,1])$ that is symplectic if and only if $θ=\frac12$. By establishing upper bounds for the second-order moment of the asymptotic error distribution, we show that the midpoint method minimizes the error constant of the $θ$ method for a large time horizon $T$. Furthermore, we take the linear stochastic oscillator as a test equation and investigate exact asymptotic error constants of several symplectic and non-symplectic methods. Our result suggests that in the long-time computation, the probability that the error deviates from zero decays exponentially faster for the symplectic methods than that for the non-symplectic ones.

Superiority Of Symplectic Methods For Stochastic Hamiltonian System Via Asymptotic Error Distribution

TL;DR

The paper investigates why symplectic methods outperform in stochastic Hamiltonian systems with additive noise by analyzing the asymptotic distribution of the normalized error. It proves that the stochastic -method is symplectic only at (the midpoint method) and that this choice minimizes the long-time asymptotic error constant, with a limit SDE characterizing the error dynamics. For the linear stochastic oscillator, explicit asymptotic error constants show symplectic schemes have error variance growing linearly with time , while non-symplectic schemes grow as , implying exponentially faster decay of error deviations for symplectic methods in long-time computations. The results are reinforced by numerical experiments that confirm the linear vs cubic scaling in variance for symplectic and non-symplectic methods, respectively, highlighting the practical advantage of symplectic integrators in long-time stochastic simulations.

Abstract

The superiority of symplectic methods for stochastic Hamiltonian systems has been widely recognized, yet the probabilistic mechanism behind this superiority remains incompletely understood. This paper studies the superiority of symplectic methods from the perspective of the asymptotic error distribution, i.e., the limit distribution of normalized error. Focusing on stochastic Hamiltonian systems driven by additive noise, we obtain the asymptotic limit of the normalized error distribution of the method that is symplectic if and only if . By establishing upper bounds for the second-order moment of the asymptotic error distribution, we show that the midpoint method minimizes the error constant of the method for a large time horizon . Furthermore, we take the linear stochastic oscillator as a test equation and investigate exact asymptotic error constants of several symplectic and non-symplectic methods. Our result suggests that in the long-time computation, the probability that the error deviates from zero decays exponentially faster for the symplectic methods than that for the non-symplectic ones.
Paper Structure (12 sections, 8 theorems, 99 equations, 2 figures, 1 table)

This paper contains 12 sections, 8 theorems, 99 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Asymptotic error distribution of $\theta$ method
  3. Error distribution of numerical methods for test equation
  4. Comparison of symplectic and non-symplectic methods for test equation
  5. Error evolution for symplectic methods
  6. Symplectic $\beta$ methods ($\beta\in[0,1]$ )
  7. General symplectic methods with $\xi=h$
  8. Error evolution for non-symplectic methods
  9. Stochastic $\theta$-methods ($\theta\in[0,\frac{1}{2})\cup(\frac{1}{2},1]$ )
  10. PC(EM-BEM) method
  11. Superiority of symplectic methods
  12. Numerical experiments

Key Result

Lemma 2.3

Let Assumption ass1 hold. Then for any $\theta\in[0,1]$, we have

Figures (2)

  • Figure 1: The relation between Var$(e_N)/h^2$ and $T$ of symplectic methods for the test equation \ref{['model_zz']}.
  • Figure 2: The relation between Var$(e_N)/h^2$ and $T$ of non-symplectic methods for the test equation \ref{['model_zz']}.

Theorems & Definitions (19)

  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • ...and 9 more