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.
翻译:在本文中,我们开发了一个蒙特卡洛方法来解决多维的分数拉普拉西亚(IFL) 的 PDE 。 我们首先为单球任意尺寸的分数拉普拉西亚运算员根据绿色函数构建一个新的 Feynman- Kac 代表。 在[ 24] 中提议的“行走在球上”算法的启发下, 我们扩展了在复杂域中解决分数 PDE 的算法。 然后, 我们可以计算出一个多维随机变量的预期值, 该变量具有已知密度函数, 以有效获得数字解决方案 。 提议的算法发现它非常高效地解决分数 PDE : 它只需要对已知绿色函数的球切线内一系列球切线的预期形式进行评估即可。 此外, 我们还对$- 维单位球的拟议方法进行了错误估计。 最后, 我们提出了大量的数字结果, 以证明在单位磁盘和复杂域, 甚至十维单位球中, 的分数式PDE 方法的稳健性和有效性 。