Randomized quasi-Monte Carlo for walk on spheres

TL;DR

This paper explores the application of randomized quasi-Monte Carlo methods to walk on spheres algorithms for solving boundary value problems, demonstrating improved variance reduction over traditional Monte Carlo in various dimensions.

Contribution

It provides new conditions for applying RQMC to harmonic functions with boundary regions, and compares multiple RQMC methods showing consistent variance reduction improvements.

Findings

Sampling variances decrease slightly faster than Monte Carlo rates.

Variance reduction factors ranged from 1.8 to 10.7 at 2^17 samples.

No single RQMC method consistently outperformed others.

Abstract

We investigate the use of randomized quasi-Monte Carlo (RQMC) in walk on spheres algorithms to solve boundary value problems for functions with Dirichlet boundary conditions in $\mathbb{R}^d$. For harmonic functions with $d=2$, the integrands of interest are periodic indicator functions over regions $\Theta$ in the torus $\mathbb{T}^k$. We give conditions for $\partial\Theta$ to have $k-1$ dimensional Minkowski content which allows us to use results of He and Wang (2015). The RQMC estimates involve multiple values of $k$. We see sampling variances decreasing with the number $n$ of sample points at slightly better than Monte Carlo rates. The median variance rate in $4$ RQMC methods over $5$ worked examples, including some with $d=3$ and some with nonzero source functions, was slightly better than $O(n^{-1.1})$. The variance reduction factors ranged from $1.8$ to $10.7$ at $n=2^{17}$. None of the four RQMC methods dominated the others.