用于逼近Robin边值问题与反射扩散过程遍历极限的最简随机游走方法
Simplest random walk for approximating Robin boundary value problems and ergodic limits of reflected diffusions
AI总结：
本文提出一种一阶弱收敛的反射随机微分方程弱逼近数值方法，可结合蒙特卡洛求解带Robin边界条件的PDE，还能计算RSDEs的遍历极限以实现紧支撑分布高效采样，并探讨了多项扩展，理论结果经数值实验验证。
摘要：
本文提出并分析了一种实现简单的弱意义数值方法，用于逼近反射随机微分方程（reflected stochastic differential equations, RSDEs）。研究证明该方法具有一阶弱收敛阶。结合蒙特卡洛技术，该方法可用于数值求解带Robin边界条件的线性抛物型和椭圆型偏微分方程（PDEs）。本文的核心成果之一是将所提方法用于计算遍历极限，即关于RSDEs不变测度的期望，涵盖$\boldsymbol{\rm R}^d$中区域内部及其边界上的情形，这使得从紧支撑分布中高效采样成为可能。文中同时考虑并分析了时间平均估计量与集合平均估计量。此外还探讨了多项扩展内容，包括二阶弱逼近、任意斜反射方向的情形，以及一种用于求解带Neumann边界条件的泊松PDE的新型自适应弱格式。若干数值实验验证了本文给出的理论结果。
英文摘要：
A simple-to-implement weak-sense numerical method to approximate reflected stochastic differential equations (RSDEs) is proposed and analysed. It is proved that the method has the first order of weak convergence. Together with the Monte Carlo technique, it can be used to numerically solve linear parabolic and elliptic PDEs with Robin boundary condition. One of the key results of this paper is the use of the proposed method for computing ergodic limits, i.e. expectations with respect to the invariant law of RSDEs, both inside a domain in $\mathbb{R}^{d}$ and on its boundary. This allows to efficiently sample from distributions with compact support. Both time-averaging and ensemble-averaging estimators are considered and analysed. A number of extensions are considered including a second-order weak approximation, the case of arbitrary oblique direction of reflection, and a new adaptive weak scheme to solve a Poisson PDE with Neumann boundary condition. The presented theoretical results are supported by several numerical experiments.
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