利用漂移Brown族的概率算法

A probabilistic computing method by Brownian family with drift

在一般区域上 ,所得三维Dirichlet问题的解不精确 ,因此 ,需要考虑它的数值解 .作者在概率论的基础上提出了一种新的有效的求数值解的概率算法 .其步骤是 :首先 ,从随机分析的理论出发 ,将Dirichlet问题的解表示为漂移Brown族在区域边界上的首中时和首中位置的函数的期望 ,这样 ,就将三维问题转化为边界上的二维问题 ;然后 ,通过对区域边界作有限元剖分 ,使问题离散化 ;最后 ,构作辅助球 ,利用已知的漂移Brown族在球面上的首中时和首中位置的分布 ,以及漂移Brown族对停时的强马氏性 ,得到一般区域上的一类Dirichlet问题的数值解 .应用结果表明 ,此概率算法具有计算量较小、精度较高的特点 ,适宜于更一般区域的三维Dirichlet问题求解 .

In general, the solutions to 3 dimensional Dirichlet problems over ordinary domains can not be actually computed, so the numerical solutions are considered. In this paper, a new numerical method named probabilistic computing method is proposed. The calculating procedure of using the stochastic calculus is as follows. First, the solutions to the 3 dimensional Dirichlet problems are represented as expectations of stochastic functions, so the 3 dimensional problems are turned into 2 dimensional problems. Second, the boundary is subdivided into a finite number of subsets and finite element spaces are constructed. To obtain numerical solutions, an auxiliary ball is needed and its boundary is also subdivided into the same number of subsets. Finally, from the known joint distributions of the time and place of hitting spheres and the strong Markov property for Brownian family with drift, the numerical solutions to the Dirichlet problems over ordinary domains are obtained. The results show that the probabilistic computing method has less computational amount and higher accuracy than some other methods.