Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Partition Principle Revisited: Non-Equal Volume Designs Achieve Minimal Expected Star Discrepancy

Xiaoda Xu

Abstract

We study the expected star discrepancy under a newly designed class of non-equal volume partitions. The main contributions are twofold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed non-equal volume partitions yield stratified sampling point sets with lower expected star discrepancy than classical jittered sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z)) < \mathbb{E}(D^{*}_{N}(Y))$, where $Y$ and $Z$ represent jittered sampling and our non-equal volume partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our non-equal volume partition models, which improve upon existing bounds for jittered sampling. Our results provide a theoretical foundation for using non-equal volume partitions in high-dimensional numerical integration.

The Partition Principle Revisited: Non-Equal Volume Designs Achieve Minimal Expected Star Discrepancy

Abstract

We study the expected star discrepancy under a newly designed class of non-equal volume partitions. The main contributions are twofold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed non-equal volume partitions yield stratified sampling point sets with lower expected star discrepancy than classical jittered sampling. Specifically, we prove that , where and represent jittered sampling and our non-equal volume partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our non-equal volume partition models, which improve upon existing bounds for jittered sampling. Our results provide a theoretical foundation for using non-equal volume partitions in high-dimensional numerical integration.
Paper Structure (33 sections, 4 theorems, 52 equations)

This paper contains 33 sections, 4 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Our Contributions
  4. Technical Overview
  5. Preliminaries
  6. Star Discrepancy and Sampling Methods
  7. Non-Equal Volume Partition Model
  8. $\delta$-Covers and Discretization
  9. Main Results
  10. Comparison of Expected Star Discrepancy
  11. Explicit Upper Bounds
  12. Proofs of Main Results
  13. Preliminary Lemmas
  14. Proof of Theorem \ref{['thm:main1']}
  15. Step 1: Integral representation of expectation.
  16. ...and 18 more sections

Key Result

Theorem 3.1

Let $m, d \in \mathbb{N}$ with $m \geq d \geq 2$, $b \in [\frac{3}{2m}, \frac{2}{m}]$, and $N = m^d$. Let: Then:

Theorems & Definitions (12)

  • Definition 2.1: Star Discrepancy
  • Definition 2.2: Jittered Sampling
  • Definition 2.3: $\delta$-Cover
  • Theorem 3.1: Strong Partition Principle for Star Discrepancy
  • Remark 3.2
  • Theorem 3.3: Upper Bounds for Non-Equal Volume Partitions
  • Remark 3.4
  • Lemma 4.1: Bernstein's Inequality
  • Lemma 4.2: $L_2$-Discrepancy Difference for Non-Equal Volume Partition
  • proof : Proof Sketch
  • ...and 2 more