Anyone who does Monte Carlo simulations would at least be interested in the notion of Quasi Monte Carlo, in which points are sampled “more evenly than random.” - The Unreasonable Effectiveness of Quasirandom Sequences - (SO) / HN

As noted in OP, this can lead to O(1/n) convergence of averages to underlying target values, rather than the expected O(1/sqrt(n)) convergence (“n” is the number of probes of the underlying system). This is shown, strikingly, in Fig. 2 of the OP.