论文标题
用于高维分数泊松方程的修改后的车手步骤方法
A Modified Walk-on-sphere Method for High Dimensional Fractional Poisson Equation
论文作者
论文摘要
我们开发了在高维度下具有Dirichilet边界条件的分数泊松方程的步骤。速段方法基于概率代表分数泊松方程的代表。我们提出了效率正交规则，以评估球中的积分表示，并将拒绝采样方法应用于从一般域中计算的概率绘制的绘制。此外，当域是球时，我们提供了该方法平均值中的步行数量的估计。我们表明，步行的数量在分数顺序和起点与原点的距离之间增加。我们还提供了分数拉普拉斯方程的绿色函数与经典拉普拉斯方程的关系之间的关系。 2-10维中问题的数值结果验证了我们的理论以及修改后的步伐方法的效率。
We develop walk-on-sphere for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk-on-sphere method is based on probabilistic represen tation of the fractional Poisson equation. We propose effcient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the distance of the starting point to the origin. We also give the relationship between the Green function of fractional Laplace equation and that of the classical Laplace equation. Numerical results for problems in 2-10 dimensions verify our theory and the effciency of the modified walk-on-sphere method.
