Improved expected L2-discrepancy formulas on jittered sampling

We consider the uniform integration approximation problem in a class of Sobolev spaces. This type of problem can be transformed into the study of discrepancy function for certain sampling sets. The main tools are [Formula: see text]-discrepancy methods and stratified sampling. We first extend the notion of jittered sampling (a special case of stratified sampling) to two general classes of partitions, which are based on jittered grids. Second, we compute the expected [Formula: see text]-discrepancy under these two classes of partitions, and the explicit and exact formulas are derived. These results yield better expected [Formula: see text]-discrepancy formulas than jittered sampling. Our method is to calculate the size relationship between the expected [Formula: see text]-discrepancy under the new partition models and jittered sampling. In the end, this idea can also be applied to general measure partition models, such as a partition that is no longer dependent on jittered grids. This may allow us to obtain the exact expected [Formula: see text]-discrepancy formula under partition models with appropriate sampling number conditions.