关于高维球上分数阶泊松方程的谱蒙特卡洛方法的注记
A note on spectral Monte-Carlo method for fractional Poisson equation on high-dimensional ball
AI总结：
针对现有分数阶泊松方程谱蒙特卡洛方法本质为一维的局限，研究通过采用新的分数阶拉普拉斯本征函数与插值公式，将其推广至高维球径向解场景，经数值验证有效。
摘要：
近期，文献[Feng2025ExponentiallyAS]中提出了一类用于求解分数阶泊松方程的高效谱蒙特卡洛方法。这类方法充分考虑了解在边界附近的低正则性，同时利用了球上行走（walk-on-spheres）算法的高效性，能够达到谱精度。不过，其基础公式本质上是一维的。在本研究中，我们将该方法推广到一般高维球中的径向解情形。推广过程采用了一组不同的分数阶拉普拉斯算子本征函数，并推导了新的插值公式。我们对所提方法进行了全面阐述，并与现有技术做了详细对比。数值实验证实了该推广方法的有效性。
英文摘要：
Recently, a class of efficient spectral Monte-Carlo methods was developed in \cite{Feng2025ExponentiallyAS} for solving fractional Poisson equations. These methods fully consider the low regularity of the solution near boundaries and leverage the efficiency of walk-on-spheres algorithms, achieving spectral accuracy. However, the underlying formulation is essentially one-dimensional. In this work, we extend this approach to radial solutions in general high-dimensional balls. This is accomplished by employing a different set of eigenfunctions for the fractional Laplacian and deriving new interpolation formulas. We provide a comprehensive description of our methodology and a detailed comparison with existing techniques. Numerical experiments confirm the efficacy of the proposed extension.
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