Low-discrepancy Sequence

In mathematics, a low-discrepancy sequence is a sequence with the property that for all values of N, its subsequence x₁, ..., x_N has a low discrepancy.

Roughly speaking, the discrepancy of a sequence is low if the proportion of points in the sequence falling into an arbitrary set B is close to proportional to the measure of B, as would happen on average (but not for particular samples) in the case of a uniform distribution. Specific definitions of discrepancy differ regarding the choice of B (hyperspheres, hypercubes, etc.) and how the discrepancy for every B is computed (usually normalized) and combined (usually by taking the worst value).

Low-discrepancy sequences are also called quasi-random or sub-random sequences, due to their common use as a replacement of uniformly distributed random numbers. The "quasi" modifier is used to denote more clearly that the values of a low-discrepancy sequence are neither random nor pseudorandom, but such sequences share some properties of random variables and in certain applications such as the quasi-Monte Carlo method their lower discrepancy is an important advantage.

At least three methods of numerical integration can be phrased as follows. Given a set {x₁, ..., x_N} in the interval, approximate the integral of a function f as the average of the function evaluated at those points:

If the points are chosen as x_i = i/N, this is the rectangle rule. If the points are chosen to be randomly (or pseudorandomly) distributed, this is the Monte Carlo method. If the points are chosen as elements of a low-discrepancy sequence, this is the quasi-Monte Carlo method. A remarkable result, the Koksma–Hlawka inequality (stated below), shows that the error of such a method can be bounded by the product of two terms, one of which depends only on f, and the other one is the discrepancy of the set {x₁, ..., x_N}.

It is convenient to construct the set {x₁, ..., x_N} in such a way that if a set with N+1 elements is constructed, the previous N elements need not be recomputed. The rectangle rule uses points set which have low discrepancy, but in general the elements must be recomputed if N is increased. Elements need not be recomputed in the Monte Carlo method if N is increased, but the point sets do not have minimal discrepancy. By using low-discrepancy sequences, the quasi-Monte Carlo method has the desirable features of the other two methods.