Expected uniform integration approximation under general equal measure partition

TL;DR

This paper derives improved bounds on the expected discrepancy for uniform integration in Sobolev spaces, leading to more accurate mean square error estimates for stratified sampling methods.

Contribution

It establishes a better order of approximation error for uniform integration using stratified samples, surpassing the crude Monte Carlo rate.

Findings

Achieved an approximation error order of O(N^{-1-1/d})

Provided upper bounds for p-moment errors in Sobolev spaces

Enhanced understanding of discrepancy bounds for stratified sampling

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 $\mathcal{H}^{\mathbf{1}}(K)$, 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 $F_{d,q}^{*}$.