Prime Numbers

8.3 Quasi-Monte Carlo (qMC) methods 409
of successful qMC sequences: The points tend to drop into regions where
“they have not yet been.” Contrast this to direct Monte Carlo methods,
whereby—due to unbiased randomness—points will not only sometimes
“clump” together, but sometimes leave “gaps” as the points accumulate in
the domain of interest.
The Halton sequences are just one family of qMC sequences, as we discuss
in the next section. For the moment, we exhibit a typical theorem that reveals
information about how discrepancy grows as a function of the dimension:
Theorem 8.3.5 (Halton discrepancy). Denote by S ¯ B a Halton sequence for
bases ¯ B. Then the star discrepancy of the sequence satisfies
D ∗ N(S ¯ B) < D
N
+ 1
N
D
Bi − 1
i=1
2lnBi
ln N + Bi
+1
.
2
A rather intricate proof can be found in [Niederreiter 1992]. We observe that
the theorem provides an explicit upper bound for the implied big-O constant
in
D ∗
ln
N(SB)=O ¯ D
N
,
N
an error behavior foreshadowed in the introductory remarks of this section.
What is more, we can see the (unfortunate) effect of larger bases supposedly
contributing more to the discrepancy (we say supposedly because this is just
an upper bound); indeed, this effect for larger bases is seen in practice. We
note that there is a so-called N-point Hammersley point set, for which the
leading component of xn is x0 = n/N, while the rest of xn is a (D − 1)dimensional
Halton vector. This set is now N-dependent, so that it cannot be
turned into an infinite sequence. However, the Hammersley set’s discrepancy
takes the slightly superior form
D ∗
ln
N = O
D−1
N
,
N
showing how N-dependent sets can offer a slight complexity reduction.
8.3.3 Primes on Wall Street?
Testing a good qMC sequence, say estimating the volume of the unit D-ball,
is an interesting exercise. The Halton qMC sequence gives good results for
moderate dimensions, say for D up to about 10. One advantage of the Halton
sequence is that it is easy to jump ahead, so as to have several or many
computers simultaneously sampling from disjoint segments of the sequence.
The following algorithm shows how one can jump in at the n-th term, and
how to continue sequentially from there. To make the procedure especially
efficient, the digits of the index in the various bases under consideration are
constantly updated as we proceed from one index to the next.