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

Abstract

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 [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.