论文标题
有效的蒙特卡洛方法用于多个维
Efficient Monte Carlo Method for Integral Fractional Laplacian in Multiple Dimensions
论文作者
论文摘要
在本文中，我们开发了一种蒙特卡洛方法，用于求解涉及多个维度的积分分数拉普拉斯（IFL）的PDE。我们首先根据任意维度的单位球上的分数laplacian操作员的绿色函数构建新的Feynman-kac表示。受到[24]中提出的“步骤”算法的启发，我们扩展了用于求解复杂域中的分数PDE的算法。然后，我们可以计算具有已知密度函数的多维随机变量的期望，以有效地获得数值解决方案。所提出的算法发现它在求解分数PDE方面非常有效：它只需要评估具有已知绿色功能的一系列内部球形切线边界上的期望积分。此外，我们对$ n $维单元球的拟议方法进行了错误估计。最后，提出了足够的数值结果，以证明这种方法对单位磁盘和复杂域中的分数PDE的鲁棒性和有效性，甚至在十维单位球中。
In this paper, we develop a Monte Carlo method for solving PDEs involving an integral fractional Laplacian (IFL) in multiple dimensions. We first construct a new Feynman-Kac representation based on the Green function for the fractional Laplacian operator on the unit ball in arbitrary dimensions. Inspired by the "walk-on-spheres" algorithm proposed in [24], we extend our algorithm for solving fractional PDEs in the complex domain. Then, we can compute the expectation of a multi-dimensional random variable with a known density function to obtain the numerical solution efficiently. The proposed algorithm finds it remarkably efficient in solving fractional PDEs: it only needs to evaluate the integrals of expectation form over a series of inside ball tangent boundaries with the known Green function. Moreover, we carry out the error estimates of the proposed method for the $n$-dimensional unit ball. Finally, ample numerical results are presented to demonstrate the robustness and effectiveness of this approach for fractional PDEs in unit disk and complex domains, and even in ten-dimensional unit balls.
