A note on spectral Monte-Carlo method for fractional Poisson equation on high-dimensional ball

Abstract

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.