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

TL;DR

The paper tackles the problem of solving Poisson-type and parabolic PDEs driven by $\alpha$-stable Lévy processes, including fractional Laplacians, on bounded domains. It develops a spectral Monte Carlo (SMC) framework that combines generalized Jacobi functions, walk-on-spheres, control variates, and space-time interpolation to achieve high-accuracy solutions without solving linear systems. The authors prove exponential (geometric) convergence of the iterative schemes and provide explicit error bounds, with numerical experiments showing spectral accuracy across $\alpha\in(0,2]$ and the classical case $\alpha=2$, along with favorable parallelizability. This work yields a unified, versatile method (ST-SMC) that extends to space-time problems and offers significant computational advantages for nonlocal PDEs where stochastic representations are advantageous.

Abstract

This paper introduces a spectral Monte Carlo iterative method (SMC) for solving linear Poisson and parabolic equations driven by $α$-stable Lévy process with $α\in (0,2)$, 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 $α=2$. 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 $α\in(0,2)$ and $α=2$. Extensive numerical results are provided to demonstrate the spectral accuracy and efficiency of the proposed method, thereby validating the theoretical findings.

Figures (8)

• Figure 1: A comparison of the convergence rates between the Monte Carlo method and the spectral Monte Carlo method for solving the fractional Poisson equation $(-\Delta)^{\frac{\alpha}{2}}u(x)=f(x) \;{\rm in}\,\Omega$; $u(x)=0\;{\rm on}\,\Omega^c.$ Here, we take the exact solution $u(x)=(1-x^2)^{\frac{\alpha}{2}}(x^2+x+1)$, with $\alpha=0.4$ (cf. Example \ref{['Ex:1']}). It is observed that SMC rapidly achieves machine-level precision, whereas MC struggles to reach high accuracy.
• Figure 1: The path of walk on $2$-D irregular domain
• Figure 1: The discrete $E_N^\infty$-errors for $u_1(x)$ in \ref{['solu1']} against the iteration number $K$. Left: $M = 50, N_x =2$; Right: $\alpha =0.4, N_x = 2$.
• Figure 2: The discrete $E_N^\infty$-errors for $u_2(x)$ in \ref{['solu1']}. Left: with fixed $M = 10$; Right: take $\alpha = 0.4, N_x =6$.
• Figure 3: The discrete $E_N^\infty$-errors for example 2. Left: $M = 10$; Right: $\alpha=0.4, N_x=6$.

Theorems & Definitions (11)

• Lemma 3.1
• Proof 1
• Theorem 3.1
• Proof 2
• Remark 3.2
• Lemma 4.1
• Lemma 4.1
• Proof 3
• Theorem 4.1
• Proof 4