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.

翻译：我们在Dirichlet边值问题的球面游走算法中采用Array-RQMC采样。针对一组测试问题，我们发现当轨迹数$n=2^{17}$时，Array-RQMC-WOS方法可将蒙特卡罗方差降低57倍至2290倍。已知方差阶数为$o(1/n)$，但在我们的算例中实际达到$n^{-1.4}$至$n^{-1.8}$的经验收敛速率。Ho与Owen（2026）研究的更简化的RQMC-WOS算法虽具有更强理论支撑，但在相同算例上仅将方差降低1.8至10.7倍。为解释这一改进，我们基于Sobol'指数引入RQMC误差的列均值维度概念。该指标与蒙特卡罗的常规均值维度及随机格点方法的对偶格点误差均值维度相吻合。针对Crane等人（2025）的垫圈算例，我们发现Array-RQMC-WOS误差的均值维度远高于对应的Array-MC-WOS算法。