Exponentially accurate spectral Monte Carlo method for linear PDEs and their error estimates

TL;DR

This paper presents a spectral Monte Carlo method for solving linear PDEs driven by stable Lévy processes, achieving exponential error convergence and demonstrating high accuracy and efficiency through extensive numerical validation.

Contribution

The paper introduces a unified space-time spectral Monte Carlo method that achieves exponential convergence for fractional Laplacian PDEs with comprehensive error estimates.

Findings

Spectral accuracy demonstrated through numerical experiments.

Exponential convergence of error bounds rigorously established.

Method effectively handles both fractional and classical cases.

Abstract

This paper introduces a spectral Monte Carlo iterative method (SMC) for solving linear Poisson and parabolic equations driven by $\alpha$-stable L\'evy process with $\alpha\in (0,2)$, which was initially proposed and developed by Gobet and Maire in their pioneering works (Monte Carlo Methods Appl 10(3-4), 275--285, 2004, and SIAM J Numer Anal 43(3), 1256--1275, 2005) for the case $\alpha=2$. The novel method effectively integrates multiple computational techniques, including the interpolation based on generalized Jacobi functions (GJFs), space-time spectral methods, control variates techniques, and a novel walk-on-sphere method (WOS). The exponential convergence of the error bounds is rigorously established through finite iterations for both Poisson and parabolic equations involving the integral fractional Laplacian operator. Remarkably, the proposed space-time spectral Monte Carlo method (ST-SMC) for the parabolic equation is unified for both $\alpha\in(0,2)$ and $\alpha=2$. Extensive numerical results are provided to demonstrate the spectral accuracy and efficiency of the proposed method, thereby validating the theoretical findings.