Low-discrepancy sequence

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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, as defined below.

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.

Roughly speaking, the discrepancy of a sequence is low if the number 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 an 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).

At least three methods of numerical integration can be phrased as follows. Given a set x₁, ..., x_N in the interval [0,1], approximate the integral of a function f as the average of the function evaluated at those points:

$\int_0^1 f(u)\,du \approx \frac{1}{N}\,\sum_{i=1}^N f(x_i).$

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, 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 another which is the discrepancy of the set x₁, ..., x_N. The Koksma-Hlawka inequality is stated below.

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.

1 Definition of discrepancy
2 Graphical examples
3 The Koksma-Hlawka inequality
4 The formula of Hlawka-Zaremba
5 The L² version of the Koksma-Hlawka inequality
6 The Erdős-Turan-Koksma inequality
7 The main conjectures
8 The best-known sequences
9 Lower bounds
10 Applications
11 References
12 External links

The Star-Discrepancy is defined as follows, using Niederreiter's notation.

$D^*_N(P) = \sup_{B\in J^*} \left| \frac{A(B;P)}{N} - \lambda_s(B) \right|$

where P is the set x₁, ..., x_N, λ_s is the s-dimensional Lebesgue measure, A(B;P) is the number of points in P that fall into B, and J^* is the set of intervals of the form

$\prod_{i=1}^s [0, u_i)$

where u_i is in the half-open interval [0, 1). Therefore

$D^*_N(P) =\|{\rm disc}\|_\infty$

where the discrepancy function is defined by

${\rm disc}(y)=\frac{A([0,y);P)}{N}-\lambda_s([0,y)).$

The points plotted below are the first 100, 1000, and 10000 elements in a sequence of the Sobol type. For comparison, 10000 elements of a sequence of pseudorandom points are also shown.

The low-discrepancy sequence was generated by TOMS algorithm 659, described by P. Bratley and B.L. Fox in ACM Transactions on Mathematical Software, vol. 14, no. 1, pp 88--100. An implementation of the algorithm in Fortran may be downloaded from Netlib, URL: http://www.netlib.org/toms/659

The first 100 points in a low-discrepancy sequence of the Sobol type.

The first 1000 points in the same sequence. These 1000 comprise the first 100, with 900 more points.

The first 10000 points in the same sequence. These 10000 comprise the first 1000, with 9000 more points.

For comparison, here are the first 10000 points in a sequence of uniformly distributed pseudorandom numbers. Regions of higher and lower density are evident.

Let Ī^s be the s-dimensional unit cube, Ī^s = [0, 1] × ... × [0, 1]. Let f have bounded variation V(f) on Ī^s in the sense of Hardy and Krause. Then for any x₁, ..., x_N in I^s = [0, 1) × ... × [0, 1),

$\left| \frac{1}{N} \sum_{i=1}^N f(x_i) - \int_{\bar I^s} f(u)\,du \right| \le V(f)\, D_N^* (x_1,\ldots,x_N).$

The Koksma-Hlawka inequality is sharp in the following sense:

For any point set x₁,...,x_N in I^s and any

ε > 0

there is a function f with bounded variation and V(f)=1 such that

$\left| \frac{1}{N} \sum_{i=1}^N f(x_i) - \int_{\bar I^s} f(u)\,du \right|>D_{N}^{*}(x_1,\ldots,x_N)-\epsilon.$

Therefore, the quality of a numerical integration rule depends only on the discrepancy D^*_N(x₁,...,x_N).

Let $D=\{1,2,\ldots,d\}$ . For $\emptyset\neq u\subseteq D$ we write

$dx_u:=\prod_{j\in u} dx_j$

and denote by $(x u,1)$ the point obtained from $x$ by replacing the coordinates not in $u$ by $1$ . Then

$\frac{1}{N} \sum_{i=1}^N f(x_i) - \int_{\bar I^s} f(u)\,du= \sum_{\emptyset\neq u\subseteq D}(-1)^{|u|} \int_{[0,1]^{|u|}}{\rm disc}(x_u,1)\frac{\partial^{|u|}}{\partial x_u}f(x_u,1) dx_u.$

Applying the Cauchy-Schwarz inequality for integrals and sums to the Hlawka-Zaremba identity, we obtain an $L 2$ version of the Koksma-Hlawka inequality:

$\left|\frac{1}{N} \sum_{i=1}^N f(x_i) - \int_{\bar I^s} f(u)\,du\right|\le \|f\|_{d}\,{\rm disc}_{d}(\{t_i\}),$

where

${\rm disc}_{d}(\{t_i\})=\left(\sum_{\emptyset\neq u\subseteq D} \int_{[0,1]^{|u|}}{\rm disc}(x_u,1)^2 dx_u\right)^{1/2}$

and

$\|f\|_{d}=\left(\sum_{u\subseteq D} \int_{[0,1]^{|u|}} \left|\frac{\partial^{|u|}}{\partial x_u}f(x_u,1)\right|^2 dx_u\right)^{1/2}.$

It is computationally hard to find the exact value of the discrepancy of large point sets. The Erdős-Turán-Koksma inequality provides an upper bound.

Let x₁,...,x_N be points in I^s and H be an arbitrary positive integer. Then

$D_{N}^{*}(x_1,\ldots,x_N)\leq \left(\frac{3}{2}\right)^s \left( \frac{2}{H+1}+ \sum_{0<\|h\|_{\infty}\leq H}\frac{1}{r(h)} \left| \frac{1}{N} \sum_{n=1}^{N} e^{2\pi i\langle h,x_n\rangle} \right| \right)$

where

$r(h)=\prod_{i=1}^s\max\{1,|h_i|\}\quad\mbox{for}\quad h=(h_1,\ldots,h_s)\in\Z^s.$

Conjecture 1. There is a constant c_s depending only on s, such that

$D_{N}^{*}(x_1,\ldots,x_N)\geq c_s\frac{(\ln N)^{s-1}}{N}$

for any finite point set x₁,...,x_N.

Conjecture 2. There is a constant c^'_s depending only on s, such that

$D_{N}^{*}(x_1,\ldots,x_N)\geq c'_s\frac{(\ln N)^{s}}{N}$

for any infinite sequence x₁,x₂,x₃,....

These conjectures are equivalent. They have been proved for s ≤ 2 by W. M. Schmidt. In higher dimensions, the corresponding problem is still open. The best-known lower bounds are due to K. F. Roth.

Constructions of sequences are known (due to Faure, Halton, Hammersley, Sobol, Niederreiter and van der Corput) such that

$D_{N}^{*}(x_1,\ldots,x_N)\leq C\frac{(\ln N)^{s}}{N}.$

where C is a certain constant, depending on the sequence. After Conjecture 2, these sequences are believed to have the best possible order of convergence. See also: Halton sequences.

Let s = 1. Then

$D_N^*(x_1,\ldots,x_N)\geq\frac{1}{2N}$

for any finite point set x₁, ..., x_N.

Let s = 2. W. M. Schmidt proved that for any finite point set x₁, ..., x_N,

$D_N^*(x_1,\ldots,x_N)\geq C\frac{\log N}{N}$

where

$C=\max_{a\geq3}\frac{1}{16}\frac{a-2}{a\log a}=0.02333...$

For arbitrary dimensions s > 1, K.F. Roth proved that

$D_N^*(x_1,\ldots,x_N)\geq\frac{1}{2^{4s}}\frac{1}{((s-1)\log2)^\frac{s-1}{2}}\frac{\log^{\frac{s-1}{2}}N}{N}$

for any finite point set x₁, ..., x_N. This bound is the best known for s > 3.

Harald Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. Society for Industrial and Applied Mathematics, 1992. ISBN 0-89871-295-5
Michael Drmota and Robert F. Tichy, Sequences, discrepancies and applications, Lecture Notes in Math., 1651, Springer, Berlin, 1997, ISBN 3-540-62606-9
William H. Press, Brian P. Flannery, Saul A. Teukolsky, William T. Vetterling. Numerical Recipes in C. Cambridge, UK: Cambridge University Press, second edition 1992. ISBN 0-521-43108-5 (see Section 7.7 for a less technical discussion of low-discrepancy sequences)

Collected Algorithms of the ACM (See algorithms 647, 659, and 738.)

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