论文标题
在HAAR小波上进行多元集成的随机稀疏网格算法
Randomized sparse grid algorithms for multivariate integration on Haar-Wavelet spaces
论文作者
论文摘要
\ emph {确定性}稀疏网格方法，也称为Smolyak的方法，是一种已建立的且广泛使用的工具，可解决多元近似问题，并且有大量的文献。关于稀疏网格方法的\ emph {随机}版本知之甚少。在本文中，我们分析了随机的稀疏网格算法，即在$ d $二维单位Cube $ [0,1）^d $上进行多元集成的随机稀疏网格四倍。令$ d，s \ in \ mathbb {n} $使$ d = d \ cdot s $。稀疏网格四边形的$ s $维构件基于$ s = 1 $的分层采样，并且在炒$（0，m，s）$ - $ s \ ge 2 $上。我们认为的积分空间和误差标准分别是haar小波的空间，具有参数$α$和随机误差（即最坏情况下均方根误差）。我们证明，对于$ n $ th mininimal误差的收敛速率，参数的所有可能组合$ d $ and $ d $和$ s $。 如果我们将上的误差边界视为混合平滑度和平滑度参数的混合空间的sobolev空间$ 1/2 <α<1 $而不是HAAR小波的空间，则仍然存在。
The \emph{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 \emph{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 $α$ 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 mininimal 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 $1/2< α< 1$ instead of Haar wavelet spaces.
