# Optimal Jittered Sampling for two Points in the Unit Square

## TL;DR

This paper investigates optimal jittered sampling partitions of the unit square into two regions to minimize expected squared discrepancy, revealing a nonlinear integral equation and asymmetric solutions.

## Contribution

It introduces a novel nonlinear integral equation characterizing optimal partitions for jittered sampling in the unit square, including asymmetric solutions.

## Key findings

- Optimal partition is not symmetric.
- Derived an approximate solution to the integral equation.
- Partition minimizes expected squared discrepancy.

## Abstract

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.