English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The $L_{\infty}$ star discrepancy is a measure for the regularity of a finiteset of points taken from $[0,1)^d$. Low discrepancy point sets are highlyrelevant for Quasi-Monte Carlo methods in numerical integration and severalother applications. Unfortunately, computing the $L_{\infty}$ star discrepancyof a given point set is known to be a hard problem, with the best exactalgorithms falling short for even moderate dimensions around 8. However,despite the difficulty of finding the global maximum that defines the$L_{\infty}$ star discrepancy of the set, local evaluations at selected pointsare inexpensive. This makes the problem tractable by black-box optimizationapproaches.  In this work we compare 8 popular numerical black-box optimization algorithmson the $L_{\infty}$ star discrepancy computation problem, using a wide set ofinstances in dimensions 2 to 15. We show that all used optimizers perform verybadly on a large majority of the instances and that in many cases random searchoutperforms even the more sophisticated solvers. We suspect thatstate-of-the-art numerical black-box optimization techniques fail to capturethe global structure of the problem, an important shortcoming that may guidetheir future development.  We also provide a parallel implementation of the best-known algorithm tocompute the discrepancy.

Computing Star Discrepancies with Numerical Black-Box Optimization Algorithms