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A Metropolis random walk algorithm to estimate a lower bound of the star discrepancy

  • Maryam Alsolami ORCID logo and Michael Mascagni ORCID logo EMAIL logo
Published/Copyright: May 10, 2023

Abstract

In this paper, we introduce a new algorithm for estimating the lower bounds for the star discrepancy of any arbitrary point sets in [ 0 , 1 ] s . Computing the exact star discrepancy is known to be an NP-hard problem, so we have been looking for effective approximation algorithms. The star discrepancy can be thought of as the maximum of a function called the local discrepancy, and we will develop approximation algorithms to maximize this function. Our algorithm is analogous to the random walk algorithm described in one of our previous papers [M. Alsolami and M. Mascagni, A random walk algorithm to estimate a lower bound of the star discrepancy, Monte Carlo Methods Appl. 28 (2022), 4, 341–348.]. We add a statistical technique to the random walk algorithm by implementing the Metropolis algorithm in random walks on each chosen dimension to accept or reject this movement. We call this Metropolis random walk algorithm. In comparison to all previously known techniques, our new algorithm is superior, especially in high dimensions. Also, it can quickly determine the precise value of the star discrepancy in most of our data sets of various sizes and dimensions, or at least the lower bounds of the star discrepancy.

MSC 2010: 65C05; 65C10; 11K36

Acknowledgements

The first author would like to express her thanks to the Saudi Arabian Cultural Mission (SACM) and Umm Al-Qura University (UQU) for the scholarship and support provided throughout this research. This study is part of her doctoral research in computer science at Florida State University (FSU). Also, the authors thank the Florida State University Research Computing Center, and Pittsburgh Supercomputing Center at the University of Pittsburgh and Carnegie Mellon University for their assistance with some of the time-consuming computations described here.

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Received: 2023-02-08
Revised: 2023-03-27
Accepted: 2023-03-30
Published Online: 2023-05-10
Published in Print: 2023-06-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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