Optimal Jittered Sampling for two points in the unit square

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 L-2-discrepancy. The optimal partitions are given by a 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. (C) 2017 Elsevier B.V. All rights reserved.