Sobol, Ilya M.; Shukhman, Boris V. QMC integration errors and quasi-asymptotics. (English) Zbl 1454.65005 Monte Carlo Methods Appl. 26, No. 3, 171-176 (2020). Summary: A crude Monte Carlo (MC) method allows to calculate integrals over a \(d\)-dimensional cube. As the number \(N\) of integration nodes becomes large, the rate of probable error of the MC method decreases as \(O(1/\sqrt{N})\). The use of quasi-random points instead of random points in the MC algorithm converts it to the quasi-Monte Carlo (QMC) method. The asymptotic error estimate of QMC integration of \(d\)-dimensional functions contains a multiplier \(1/N\). However, the multiplier \((\ln N)^d\) is also a part of the error estimate, which makes it virtually useless. We have proved that, in the general case, the QMC error estimate is not limited to the factor \(1/N\). However, our numerical experiments show that using quasi-random points of Sobol sequences with \(N=2^m\) with natural \(m\) makes the integration error approximately proportional to \(1/N\). In our numerical experiments, \(d\leq 15\), and we used \(N\leq 2^{40}\) points generated by the SOBOLSEQ16384 code published in [I. M. Sobol et al., “Construction and comparison of high-dimensional Sobol’ generators”, Wilmott Mag. 2011, No. 56, 64–79 (2011; doi:10.1002/wilm.10056)]. In this code, \(d\leq 2^{14}\) and \(N\leq 2^{63}\). MSC: 65C05 Monte Carlo methods 11K45 Pseudo-random numbers; Monte Carlo methods 65D30 Numerical integration Keywords:Monte Carlo methods; numerical integration; quasi-asymptotics; uniform distribution PDFBibTeX XMLCite \textit{I. M. Sobol} and \textit{B. V. Shukhman}, Monte Carlo Methods Appl. 26, No. 3, 171--176 (2020; Zbl 1454.65005) Full Text: DOI References: [1] D. Asotsky, E. Myshetskaya and I. M. Sobol, The average dimension of a multidimensional function for quasi-Monte Carlo estimates of an integral, Comp. Math. Math. Phys. 46 (2006), no. 12, 2061-2067. [2] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, John Wiley & Sons, New York, 1974. · Zbl 0281.10001 [3] R. Liu and A. B. Owen, Estimating mean dimensionality of analysis of variance decompositions, J. Amer. Statist. Assoc. 101 (2006), no. 474, 712-721. · Zbl 1119.62343 [4] A. B. Owen, Variance components and generalized Sobol’ indices, SIAM/ASA J. Uncertain. Quantif. 1 (2013), no. 1, 19-41. · Zbl 1459.62144 [5] B. V. Shukhman and I. M. Sobol, A limit theorem for average dimensions, Monte Carlo Methods Appl. 21 (2015), no. 2, 175-178. · Zbl 1358.11091 [6] I. M. Sobol, Asymmetric convergence of approximations of the Monte Carlo method, Comput. Math. Math. Phys. 33 (1993), no. 10, 1391-1396. [7] I. M. Sobol, Global sensitivity indices for the investigation of nonlinear mathematical models, Mat. Model. 17 (2005), no. 9, 43-52. · Zbl 1089.26500 [8] I. M. Sobol, D. Asotsky, A. Kreinin and S. Kucherenko, Construction and comparison of high-dimensional Sobol’ generators, Wilmott Mag. 2011 (2011), no. 56, 64-79. [9] I. M. Sobol and B. V. Shukhman, Quasi-Monte Carlo: A high-dimensional experiment, Monte Carlo Methods Appl. 20 (2014), no. 3, 167-171. · Zbl 1305.11066 [10] I. M. Sobol and B. V. Shukhman, On average dimensions of particle transport estimators, Monte Carlo Methods Appl. 24 (2018), no. 2, 147-151. · Zbl 1499.65007 [11] I. M. Sobol and B. V. Shukhman, Quasi-Monte Carlo method for solving Fredholm equations, Monte Carlo Methods Appl. 25 (2019), no. 3, 253-257. · Zbl 1495.65246 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.