Expected uniform integration approximation under general equal measure partition

Abstract

In this paper, we study bounds of expected $L_2-$discrepancy to give mean square error of uniform integration approximation for functions in Sobolev space ${\mathcal{H}}^1(K)$, where $\mathcal{H}$ is a reproducing Hilbert space with kernel $K$. Better order $O(N^{-1-\frac{1}{d}})$ of approximation error is obtained, comparing with previously known rate $O(N^{-1})$ using crude Monte Carlo method. Secondly, we use expected $L_p-$discrepancy bound($p\ge 1$) of stratified samples to give several upper bounds of $p$-moment of integral approximation error in general Sobolev space $F_{d,q}^{\ast }$.

25 pages

References in corpus (1)

• On a partition with a lower expected ${\mathcal{L}}_2$-discrepancy than classical jittered sampling