Constructions of low-discrepancy sequences

From Free net encyclopedia

There are some standard constructions of low-discrepancy sequences.

The van der Corput sequence

See main article van der Corput sequence

Let

<math>

n=\sum_{k=0}^{L-1}d_k(n)b^k </math>

be the b-ary representation of the positive integer n ≥ 1, i.e. 0 ≤ d_k(n) < b. Set

<math>

g_b(n)=\sum_{k=0}^{L-1}d_k(n)b^{-k-1}. </math>

Then there is a constant C depending only on b such that (g_b(n))_{n ≥ 1} satisfies

<math>

D^*_N(g_b(1),\dots,g_b(N))\leq C\frac{\log N}{N}. </math>

The Halton sequence

See main article Halton sequences

The Halton sequence is a natural generalization of the van der Corput sequence to higher dimensions. Let s be an arbitrary dimension and b₁, ..., b_s be arbitrary coprime integers greater than 1. Define

<math>

x(n)=(g_{b_1}(n),\dots,g_{b_s}(n)). </math>

Then there is a constant C depending only on b₁, ..., b_s, such that (x(n))_n≥1 is a s-dimensional sequence with

<math>

D^*_N(x(1),\dots,x(N))\leq C'\frac{(\log N)^s}{N}. </math>

The Hammersley set

Let b₁,...,b_s-1 be coprime positive integers greater that 1. For given s and N, the s-dimensional Hammersley set of size N is defined by

<math>

x(n)=(g_{b_1}(n),\dots,g_{b_{s-1}}(n),\frac{n}{N}) </math>

for n = 1, ..., N. Then

<math>

D^*_N(x(1),\dots,x(N))\leq C\frac{(\log N)^{s-1}}{N} </math>

where C is a constant depending only on b₁, ..., b_s−1.