Invariant subspace

There are some standard constructions of low-discrepancy sequences.

The van der Corput sequence

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Let

$n=\sum _{k=0}^{L-1}{d}_k(n)b^k$

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

$g_b(n)=\sum _{k=0}^{L-1}{d}_k(n)b^{-k-1}.$

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

$D_N^{*}(g_b(1),\dots ,g_b(N))\le C\frac{\log N}{N},$

where D^*_N is the star discrepancy.

The Halton sequence

File:Halton sequence 2D.svg

First 256 points of the (2,3) Halton sequence

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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

$x(n)=(g_{b_1}(n),\dots ,g_{b_s}(n)).$

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

$D_N^{*}(x(1),\dots ,x(N))\le C^{\prime }\frac{(\log N{)}^s}{N}.$

The Hammersley set

File:Hammersley set 2D.svg

2D Hammersley set of size 256

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

$x(n)=(g_{b_1}(n),\dots ,g_{b_{s-1}}(n),\frac{n}{N})$

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

$D_N^{*}(x(1),\dots ,x(N))\le C\frac{(\log N{)}^{s-1}}{N}$

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

Poisson disk sampling

Poisson disk sampling is popular in video games to rapidly placing objects in a way that appears random-looking but guarantees that every two points are separated by at least the specified minimum distance.

References

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• Quasi-Monte Carlo Simulations, http://www.puc-rio.br/marco.ind/quasi_mc.html