约束随机力学系统半显式Euler格式的Strong convergence
Strong convergence of a half-explicit Euler scheme for constrained stochastic mechanical systems
摘要：
本文研究具有非线性完整约束的随机力学系统的数值逼近问题。这类系统由二阶随机微分-代数方程描述，其中包含一个隐式给定的拉格朗日乘子过程。拉格朗日乘子的显式表示导致潜在的随机常微分方程，其漂移系数通常不是全局单侧Lipschitz连续的。我们研究了一个半显式的漂移截断欧拉格式，它完全满足约束。建立了路径一致$L_p$ -收敛性.证明是基于一个合适的分解的离散拉格朗日乘子和范数估计的单个组件，使验证的一致性，半稳定性和时刻增长性能的计划。据我们所知，所提出的结果是第一个强收敛结果的约束保持计划在考虑设置。
英文摘要：
This paper is concerned with the numerical approximation of stochastic mechanical systems with nonlinear holonomic constraints. Such systems are described by second order stochastic differential-algebraic equations involving an implicitly given Lagrange multiplier process. The explicit representation of the Lagrange multiplier leads to an underlying stochastic ordinary differential equation, the drift coefficient of which is typically not globally one-sided Lipschitz continuous. We investigate a half-explicit drift-truncated Euler scheme which fulfills the constraint exactly. Pathwise uniform $L_p$-convergence is established. The proof is based on a suitable decomposition of the discrete Lagrange multipliers and on norm estimates for the single components, enabling the verification of consistency, semi-stability and moment growth properties of the scheme. To the best of our knowledge, the presented result is the first strong convergence result for a constraint-preserving scheme in the considered setting.
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