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 $Θ$ in the torus $\mathbb{T}^k$. We give conditions for $\partialΘ$ 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.

翻译：我们研究了在球面行走算法中使用随机化拟蒙特卡洛方法（RQMC）求解$\mathbb{R}^d$中具有狄利克雷边界条件的函数的边值问题。对于$d=2$的调和函数，其被积函数是环面$\mathbb{T}^k$上区域$Θ$的周期指示函数。我们给出了$\partialΘ$具有$k-1$维闵可夫斯基内容的条件，从而能够利用He和Wang（2015）的结果。RQMC估计涉及多个$k$值。我们观察到，随着样本点数$n$的增加，采样方差以略优于蒙特卡洛方法的速率减小。在4种RQMC方法对5个工作示例（包括一些$d=3$和具有非零源函数的示例）的中位方差率略优于$O(n^{-1.1})$。在$n=2^{17}$时，方差缩减因子范围为1.8至10.7。四种RQMC方法中没有任何一种在所有情况下均占优。