Optimal Jittered Sampling for two Points in the Unit Square
This provides a theoretical advance for a special case of jittered sampling, but the result is incremental as it only covers two points and does not generalize.
The paper addresses the problem of optimal partitioning for jittered sampling in the unit square with two points, aiming to minimize expected squared L2-discrepancy. They derive an approximate solution to a nonlinear integral equation, finding that the optimal partition is asymmetric and not of equal measure.
Jittered Sampling is a refinement of the classical Monte Carlo sampling method. Instead of picking $n$ points randomly from $[0,1]^2$, one partitions the unit square into $n$ regions of equal measure and then chooses a point randomly from each partition. Currently, no good rules for how to partition the space are available. In this paper, we present a solution for the special case of subdividing the unit square by a decreasing function into two regions so as to minimize the expected squared $\mathcal{L}_2-$discrepancy. The optimal partitions are given by a \textit{highly} nonlinear integral equation for which we determine an approximate solution. In particular, there is a break of symmetry and the optimal partition is not into two sets of equal measure. We hope this stimulates further interest in the construction of good partitions.