Vol. 08  No. 02 ( 2019 ), Article ID: 28760 , 7 pages
10.12677/AAM.2019.82020

The Method of Monte Carlo Markov Chain for Solving the Poisson Equation on Irregular Domain

Chengfeng Zheng1,2, Zhizhong Yan1,2*

1School of Mathematics and Statistics, Beijing Institute of Technology, Beijing

2Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing

Received: Jan. 10th, 2019; accepted: Jan. 25th, 2019; published: Feb. 1st, 2019

ABSTRACT

In this paper, the Monte Carlo Markov chain method for solving Poisson equation in irregular domain is studied. In irregular domain, the numerical calculation of differential equations is usually difficult. Based on the idea of finite difference, an absorbable Markov chain is constructed to solve differential equations in irregular domain. The numerical experiments in irregular domain show that the method is feasible and effective. This method provides a new idea for solving Poisson’s equation in irregular region and keeps the second order convergence order.

Keywords:Monte Carlo Chain Method, Poisson Equation, Irregular Domain

1北京理工大学数学与统计学院，北京

2北京理工大学复杂信息数学表征分析与应用北京市重点实验室，北京

1. 引言

2. 蒙特卡洛随机模型

$\left\{\begin{array}{l}-\Delta u=f\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{Ω}\\ u=g\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{\hspace{0.17em}}\text{on}\text{\hspace{0.17em}}\text{Γ}\end{array}$ (1)

Figure 1. The irregular solution domain of the Poisson equation

Figure 2. Grid points obtained by mesh generation for irregular domain

$2\frac{\Delta {x}_{i-1}\left({U}_{i+1,j}-{U}_{i,j}\right)+\Delta {x}_{i}\left({U}_{i-1,j}-{U}_{i,j}\right)}{\Delta {x}_{i-1}\Delta {x}_{i}\left(\Delta {x}_{i}+\Delta {x}_{i-1}\right)}+2\frac{\Delta {y}_{j-1}\left({U}_{i,j+1}-{U}_{i,j}\right)+\Delta {y}_{j}\left({U}_{i,j-1}-{U}_{i,j}\right)}{\Delta {y}_{j-1}\Delta {y}_{j}\left(\Delta {y}_{j}+\Delta {y}_{j-1}\right)}=-{f}_{i,j}$ (2)

$\begin{array}{l}\frac{1}{\Delta {x}_{i}\left(\Delta {x}_{i}+\Delta {x}_{i-1}\right)}{U}_{i+1,j}-\left(\frac{1}{\Delta {x}_{i}\left(\Delta {x}_{i}+\Delta {x}_{i-1}\right)}+\frac{1}{\Delta {x}_{i-1}\left(\Delta {x}_{i}+\Delta {x}_{i-1}\right)}\right){U}_{i,j}\\ +\frac{1}{\Delta {x}_{i-1}\left(\Delta {x}_{i}+\Delta {x}_{i-1}\right)}{U}_{i-1,j}+\frac{1}{\Delta {y}_{j}\left(\Delta {y}_{j}+\Delta {y}_{j-1}\right)}{U}_{i,j+1}\\ -\left(\frac{1}{\Delta {y}_{j}\left(\Delta {y}_{j}+\Delta {y}_{j-1}\right)}+\frac{1}{\Delta {y}_{j-1}\left(\Delta {y}_{j}+\Delta {y}_{j-1}\right)}\right){U}_{i,j}\\ +\frac{1}{\Delta {y}_{j-1}\left(\Delta {y}_{j}+\Delta {y}_{j-1}\right)}{U}_{i,j-1}=-\frac{1}{2}{f}_{i,j}\end{array}$ (3)

${\lambda }_{1}=\frac{1}{\Delta {x}_{i}\left(\Delta {x}_{i}+\Delta {x}_{i-1}\right)}$${\lambda }_{2}=\frac{1}{\Delta {x}_{i-1}\left(\Delta {x}_{i}+\Delta {x}_{i-1}\right)}$${\lambda }_{3}=\frac{1}{\Delta {y}_{j}\left(\Delta {y}_{j}+\Delta {y}_{j-1}\right)}$${\lambda }_{4}=\frac{1}{\Delta {y}_{j-1}\left(\Delta {y}_{j}+\Delta {y}_{j-1}\right)}$$\lambda ={\lambda }_{1}+{\lambda }_{2}+{\lambda }_{3}+{\lambda }_{4}$ ，可以得到：

${U}_{i,j}=\frac{{\lambda }_{1}}{\lambda }{U}_{i+1,j}+\frac{{\lambda }_{2}}{\lambda }{U}_{i-1,j}+\frac{{\lambda }_{3}}{\lambda }{U}_{i,j+1}+\frac{{\lambda }_{4}}{\lambda }{U}_{i,j-1}+\frac{1}{2\lambda }{f}_{i,j}$ (4)

$U\left({\Omega }_{k}\right)=\underset{b}{\sum }g\left({\Gamma }_{b}\right)\frac{{P}_{b}}{{N}_{p}}+\frac{1}{{N}_{p}}\underset{i=1}{\overset{{N}_{p}}{\sum }}\underset{m=1}{\overset{{n}_{i}-1}{\sum }}\frac{1}{2{\lambda }^{m}}f\left({\Omega }_{m}\right)$ (5)

3. Poisson方程的蒙特卡洛马尔科夫链方法

Figure 3. Serial numbers of interior points and boundary points in irregular domain

${P}_{k}={\left[0,\cdots ,\frac{{\lambda }_{1}^{k}}{{\lambda }^{k}},0,\cdots ,\frac{{\lambda }_{2}^{k}}{{\lambda }^{k}},0,\cdots \frac{{\lambda }_{3}^{k}}{{\lambda }^{k}},0,\cdots \frac{{\lambda }_{4}^{k}}{{\lambda }^{k}},0,\cdots ,0\right]}_{1×\left({N}_{f}+{N}_{b}\right)}$ (6)

$\mathcal{P}={\left[\begin{array}{cc}I& 0\\ R& Q\end{array}\right]}_{\left({N}_{f}+{N}_{b}\right)×\left({N}_{f}+{N}_{b}\right)}$ (7)

$N={\left(E-Q\right)}^{-1}$

${N}_{i,j}$ 具有一定的实际意义，表示从状态i到状态j的传输路径数。那么对应的可吸收概率矩阵为：

$B=N\cdot R$

${U}_{\Omega }=B\cdot {g}_{\Gamma }+N\cdot {\mathcal{F}}_{\Omega }$ (8)

4. 数值实验

${E}^{N}\left(h\right)={|U\left({x}_{c},{y}_{j}\right)-u\left({x}_{c},{y}_{j}\right)|}_{\infty },\text{\hspace{0.17em}}\text{\hspace{0.17em}}rat{e}_{h}={\mathrm{log}}_{2}\left(\frac{{E}^{N}\left(h\right)}{{E}^{N}\left(2h\right)}\right)$

$\left\{\begin{array}{l}\frac{{\partial }^{2}u}{\partial {x}^{2}}+\frac{{\partial }^{2}u}{\partial {y}^{2}}=2{\text{e}}^{x+y},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,y\right)\in \text{Ω}\\ u={\text{e}}^{x+y},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(x,y\right)\in \text{Γ}\end{array}$

Figure 4. The comparison between the exact solution (right panel) and the numerical solution (left panel) with $M=N=80$

Figure 5. The error obtained by Monte Carlo Markov chain method with $M=N=80$

Table 1. The numerical convergence order by Monte Carlo Markov chain method

5. 结论

The Method of Monte Carlo Markov Chain for Solving the Poisson Equation on Irregular Domain[J]. 应用数学进展, 2019, 08(02): 181-187. https://doi.org/10.12677/AAM.2019.82020

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NOTES

*通讯作者