QMC integration errors and quasi-asymptotics

Abstract

Abstract 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 / N ) {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 {1/N} . However, the multiplier ( ln ⁡ N ) d {(\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 {1/N} . However, our numerical experiments show that using quasi-random points of Sobol sequences with N = 2 m {N=2^{m}} with natural m makes the integration error approximately proportional to 1 / N {1/N} . In our numerical experiments, d ≤ 15 {d\leq 15} , and we used N ≤ 2 40 {N\leq 2^{40}} points generated by the SOBOLSEQ16384 code published in 2011. In this code, d ≤ 2 14 {d\leq 2^{14}} and N ≤ 2 63 {N\leq 2^{63}} .