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

TL;DR

This work addresses solving the fractional Poisson equation $(-\Delta)^{\frac{s}{2}}u=f$ on high-dimensional balls by extending a spectral Monte-Carlo framework to radial solutions. It introduces a radial eigenfunction formulation with a weight $\rho(x)=(1-|x|^2)^{s/2}$ and a new interpolation operator derived from the change of variables $t=2|x|^2-1$, enabling spectral accuracy in dimensions $n\ge 2$. The key contributions include the development of radial eigenfunctions, a corresponding interpolation scheme, and comprehensive numerical experiments that demonstrate exponential (spectral) convergence even in high dimensions (e.g., $n=10$). This provides a dimension-scalable, boundary-singularity-aware method for nonlocal fractional PDEs on balls with practical computational efficiency.

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.

Figures (6)

• Figure 1: $n=1$, the discrete maximum error $E_{N_t}^{\infty}$ about iterative step number $K$.
• Figure 2: $n=2$, the discrete maximum error $E_{N_t}^{\infty}$ for Example \ref{['exm:2']} about iterative steps $K$.
• Figure 3: $n=2$, the numerical solution $u_{\star}(x)$ for Example \ref{['exm:2']} with various fractional power $s$.
• Figure 4: $n=2$, the discrete maximum error $E_{N_t}^{\infty}$ for Example \ref{['exm:3']} about iterative steps $K$.
• Figure 5: $n=2$, the numerical solution $u_{\star}(x)$ for Example \ref{['exm:3']} with various fractional power $s$.

Theorems & Definitions (4)

• Example 4.1
• Example 4.2
• Example 4.3
• Example 4.4