Walk on spheres and Array-RQMC

TL;DR

This paper demonstrates that Array-RQMC sampling significantly reduces variance in a walk on spheres algorithm for solving Dirichlet problems, outperforming previous methods and providing new insights into error structure via mean dimension analysis.

Contribution

The paper introduces the application of Array-RQMC to walk on spheres, achieving substantial variance reduction and developing a novel mean dimension measure for RQMC error analysis.

Findings

Array-RQMC-WOS reduces variance by 57-fold to 2290-fold at 2^17 trajectories.

Empirical convergence rates between n^{-1.4} and n^{-1.8} observed, surpassing theoretical expectations.

Mean dimension of Array-RQMC-WOS errors is higher than that of Array-MC-WOS in a gasket example.

Abstract

We use Array-RQMC sampling in a walk on spheres (WOS) algorithm for Dirichlet boundary value problems. On a collection of problems, we find that Array-RQMC-WOS reduces the Monte Carlo variance by factors ranging from $57$-fold to $2290$-fold at $n=2^{17}$ trajectories. The variance is known to be $o(1/n)$ but attains empirical rates between $n^{-1.4}$ and $n^{-1.8}$ in our examples. A simpler RQMC-WOS algorithm studied in Ho and Owen (2026) has more theoretical support but only reduced variance by 1.8 to 10.7-fold on the same set of examples. In order to explain this improvement, we introduce a column-wise mean dimension of the RQMC error based on Sobol' indices. It matches the usual mean dimension for Monte Carlo and the mean dimension of a dual lattice error for randomized lattices. We find for a gasket example from Crane et al.\ (2025) that the mean dimension of Array-RQMC-WOS errors is much higher than an analogous Array-MC-WOS algorithm has.