The deterministic sparse grid method, also known as Smolyak’s method, is a well-established and widely used tool to tackle multivariate approximation problems, and there is a vast literature on it. Much less is known about randomized versions of the sparse grid method. In this paper we analyze randomized sparse grid algorithms, namely randomized sparse grid quadratures for multivariate integration on the $D$-dimensional unit cube $[0,1)^D$. Let $d,s \in {\mathbb {N}}$ be such that $D=d\cdot s$. The $s$-dimensional building blocks of the sparse grid quadratures are based on stratified sampling for $s=1$ and on scrambled $(0,m,s)$-nets for $s\ge 2$. The spaces of integrands and the error criterion we consider are Haar wavelet spaces with parameter $\alpha $ and the randomized error (i.e., the worst case root mean square error), respectively. We prove sharp (i.e., matching) upper and lower bounds for the convergence rates of the $N$th minimal errors for all possible combinations of the parameters $d$ and $s$. Our upper error bounds still hold if we consider as spaces of integrands Sobolev spaces of mixed dominated smoothness with smoothness parameters $1/2< \alpha < 1$ instead of Haar wavelet spaces.

中文翻译：

---

Haar小波空间多元积分的随机稀疏网格算法

确定性稀疏网格方法，也称为 Smolyak 方法，是解决多元逼近问题的成熟且广泛使用的工具，并且有大量文献。对稀疏网格方法的随机版本知之甚少。在本文中，我们分析了随机稀疏网格算法，即在$D$维单位立方体$[0,1)^D$上进行多元积分的随机稀疏网格求积。令$d,s \in {\mathbb {N}}$ 使得$D=d\cdot s$。稀疏网格正交的 $s$ 维构建块基于 $s=1$ 的分层抽样和 $s\ge 2$ 的加扰 $(0,m,s)$-nets。我们考虑的被积函数空间和误差准则分别是带有参数$\alpha$ 和随机误差（即最坏情况均方根误差）的Haar 小波空间。我们证明了参数$d$ 和$s$ 的所有可能组合的第$N$ 个最小误差的收敛速度的锐利（即匹配）上限和下限。如果我们将混合主导平滑度的被积函数 Sobolev 空间视为具有平滑度参数 $1/2< α < 1$ 代替 Haar 小波空间。