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Order Conditions for Sampling the Invariant Measure of Ergodic Stochastic Differential Equations on Manifolds
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2021-06-07 , DOI: 10.1007/s10208-021-09495-y
Adrien Laurent , Gilles Vilmart

We derive a new methodology for the construction of high-order integrators for sampling the invariant measure of ergodic stochastic differential equations with dynamics constrained on a manifold. We obtain the order conditions for sampling the invariant measure for a class of Runge–Kutta methods applied to the constrained overdamped Langevin equation. The analysis is valid for arbitrarily high order and relies on an extension of the exotic aromatic Butcher-series formalism. To illustrate the methodology, a method of order two is introduced, and numerical experiments on the sphere, the torus and the special linear group confirm the theoretical findings.



中文翻译:

流形上遍历随机微分方程不变测度的抽样阶条件

我们推导出一种用于构造高阶积分器的新方法,用于对动力学约束在流形上的遍历随机微分方程的不变测度进行采样。我们获得了应用于约束过阻尼 Langevin 方程的一类 Runge-Kutta 方法的不变测度采样的阶条件。该分析适用于任意高阶,并依赖于异国芳香 Butcher 系列形式主义的扩展。为了说明该方法,引入二阶方法,并在球体、环面和特殊线性群上的数值实验证实了理论发现。

更新日期:2021-06-08
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