Prime Numbers

8.3 Quasi-Monte Carlo (qMC) methods 407
all that discrepancy is of profound practical importance. Moreover, there are
some surprising new results that go some distance, as we shall see, to explain
why actual qMC experiments are sometimes fare much better—provide far
more accuracy—than the discrepancy bounds imply.
A qMC sequence S should generally be one of low D ∗ , and it is in the
construction of such S that number theory becomes involved. The first thing
we need to observe is that there is a subtle distinction between a point-set
discrepancy and the discrepancy of a sequence. Take D = 1 dimension for
example, in which case the point set
P =
has D∗ N
sequence S that enjoys the property D∗ N
1 3
1
, ,...,1 −
2N 2N 2N
(P )=1/(2N). On the other hand, there exists no countably infinite
(S) =O(1/N ). In fact, it was shown
by [Schmidt 1972] that if S is countably infinite, then for infinitely many N,
D ∗ ln N
N(S) ≥ c
N ,
where c is an absolute constant (i.e., independent of N and S). Actually, the
constant can be taken to be c =3/50 [Niederreiter 1992], but the main point
is that the requirement of an infinite qMC sequence, from which a researcher
may draw arbitrarily large numbers of contiguous samples, gives rise to special
considerations of error. The point set P above with its discrepancy 1/(2N) is
allowed because, of course, the members of the sequence themselves depend
on N.
8.3.2 Specific qMC sequences
We are now prepared to construct some low-star-discrepancy sequences.
A primary goal will be to define a practical low-discrepancy sequence
for any given prime p, by counting in a certain clever fashion through
base-p representations of integers. We shall start with a somewhat more
general description for arbitrary base-B representations. For more than one
dimension, a set of pairwise coprime bases will be used.
Definition 8.3.3. For an integer base B ≥ 2, the van der Corput sequence
for base B is the sequence
SB =(ρB(n)) , n =0, 1, 2,...,
where ρB is the radical-inverse function, defined on nonnegative integers n,
with presumed base-B representation n =
i niB i ,by:
ρB(n) =
niB −i−1
i