Prime Numbers

8.3 Quasi-Monte Carlo (qMC) methods 405
behave no better than
|I ′
1
− I| = O √N ,
where of course, the implied big-O constant depends on the dimension D, the
integrand f, and the domain R. It is interesting that the power law N −1/2 ,
though, is independent of D. By contrast, a so-called “grid” method, in which
we split the domain R into grid points, can be expected to behave no better
than
|I ′
1
− I| = O
N 1/D
,
which growth can be quite unsatisfactory, especially for large D. In fact, a grid
scheme—with few exceptions—makes practical sense only for 1- or perhaps 2dimensional
numerical integration, unless there is some special consideration
like well-behaved integrand, extra reasons to use a grid, and so on. It is easy
to see why Monte Carlo methods using random point sets have been used for
decades on numerical integration problems in D ≥ 3 dimensions.
But there is a remarkable way to improve upon direct Monte Carlo, and
in fact obtain errors such as
|I ′ − I| = O
ln D
N
,
N
or sometimes with ln D−1 powers appearing instead, depending on the
implementation (we discuss this technicality in a moment). The idea is to
use low-discrepancy sequences, a class of quasi-Monte Carlo (qMC) sequences
(some authors define a low-discrepancy sequence as one for which the behavior
of |I ′ − I| is bounded as above; see Exercise 8.32). We stress again, an
important observation is that qMC sequences are not random in the classical
sense. In fact, the points belonging to qMC sequences tend to avoid each other
(see Exercise 8.12).
We start our tour of qMC methods with a definition of discrepancy, where
it is understood that vectors drawn out of regions R consist of real-valued
components.
Definition 8.3.1. Let P be a set of at least N points in the (unit D-cube)
region R = [0, 1] D . The discrepancy of P with respect to a family F of
Lebesgue-measurable subregions of R is defined (neither DN nor D∗ be confused with dimension D) by
DN(F ; P )=sup
χ(φ; P )
− µ(φ)
φ∈F N
N is to
,
where χ(φ; P ) is the number of points of P lying in φ, andµ denotes Lebesgue
measure. Furthermore, the extreme discrepancy of P is defined by
DN(P )=DN(G; P ),