{"title":"A random walk algorithm to estimate a lower bound of the star discrepancy","authors":"Maryam Alsolami, M. Mascagni","doi":"10.1515/mcma-2022-2125","DOIUrl":null,"url":null,"abstract":"Abstract In many Monte Carlo applications, one can substitute the use of pseudorandom numbers with quasirandom numbers and achieve improved convergence. This is because quasirandom numbers are more uniform that pseudorandom numbers. The most common measure of that uniformity is the star discrepancy. Moreover, the main error bound in quasi-Monte Carlo methods, called the Koksma–Hlawka inequality, has the star discrepancy in the formulation. A difficulty with this bound is that computing the star discrepancy is very costly. The star discrepancy can be computed by evaluating a function called the local discrepancy at a number of points. The supremum of these local discrepancy values is the star discrepancy. If we have a point set in [ 0 , 1 ] s {[0,1]^{s}} with N members, we need to compute the local discrepancy at N s {N^{s}} points. In fact, computing star discrepancy is NP-hard. In this paper, we will consider an approximate algorithm for a lower bound on the star discrepancy based on using a random walk through some of the N s {N^{s}} points. This approximation is much less expensive that computing the star discrepancy, but still accurate enough to provide information on convergence. Our numerical results show that the random walk algorithm has the same convergence rate as the Monte Carlo method, which is O ( N - 1 2 {O(N^{-\\frac{1}{2}}} ).","PeriodicalId":46576,"journal":{"name":"Monte Carlo Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2022-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monte Carlo Methods and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/mcma-2022-2125","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 1
Abstract
Abstract In many Monte Carlo applications, one can substitute the use of pseudorandom numbers with quasirandom numbers and achieve improved convergence. This is because quasirandom numbers are more uniform that pseudorandom numbers. The most common measure of that uniformity is the star discrepancy. Moreover, the main error bound in quasi-Monte Carlo methods, called the Koksma–Hlawka inequality, has the star discrepancy in the formulation. A difficulty with this bound is that computing the star discrepancy is very costly. The star discrepancy can be computed by evaluating a function called the local discrepancy at a number of points. The supremum of these local discrepancy values is the star discrepancy. If we have a point set in [ 0 , 1 ] s {[0,1]^{s}} with N members, we need to compute the local discrepancy at N s {N^{s}} points. In fact, computing star discrepancy is NP-hard. In this paper, we will consider an approximate algorithm for a lower bound on the star discrepancy based on using a random walk through some of the N s {N^{s}} points. This approximation is much less expensive that computing the star discrepancy, but still accurate enough to provide information on convergence. Our numerical results show that the random walk algorithm has the same convergence rate as the Monte Carlo method, which is O ( N - 1 2 {O(N^{-\frac{1}{2}}} ).
摘要在许多蒙特卡罗应用中,可以用拟随机数代替伪随机数的使用,从而达到提高收敛性的目的。这是因为准随机数比伪随机数更均匀。这种均匀性最常见的测量方法是恒星差异。此外,拟蒙特卡罗方法的主要误差界,称为Koksma-Hlawka不等式,在公式中具有星形差异。这个界限的一个困难是,计算恒星差异的成本非常高。星形差异可以通过在若干点上计算一个称为局部差异的函数来计算。这些局部差值的最大值是星形差值。如果我们有一个在[0,1]s {[0,1]^{s}}中有N个成员的点集,我们需要计算N s {N^{s}}点上的局部差异。事实上,计算恒星差异是np困难的。在本文中,我们将考虑一种基于随机遍历一些N s {N^{s}}点的星差下界的近似算法。这种近似方法比计算恒星差异要便宜得多,但仍然足够精确,可以提供关于收敛的信息。我们的数值结果表明,随机漫步算法具有与蒙特卡罗方法相同的收敛速度,即O(N - 1 2 {O(N^{-\frac{1}{2}}})。