Crate lowdisc 

Quasi-Monte Carlo sequences.

Low-discrepancy sequences fill the unit hypercube more uniformly than pseudorandom points. For numerical integration, QMC achieves $O((\log n)^d / n)$ error vs $O(1/\sqrt{n})$ for plain MC.

Sequences

• Halton: radical-inverse construction using successive primes as bases. Simple, no precomputation, but quality degrades above ~20 dimensions due to correlation between coordinates.
• Sobol: direction-number construction with Gray-code enumeration. Better high-dimensional uniformity than Halton. Direction numbers from Joe & Kuo (2010) support up to dimension 1111.
• Owen-scrambled Sobol: random digit scrambling that preserves the low-discrepancy property while breaking systematic patterns. Enables unbiased variance estimation from a single randomized QMC sequence.

References

• Halton (1960): “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals.”
• Sobol (1967): “Distribution of points in a cube and approximate evaluation of integrals.”
• Joe & Kuo (2010): “Constructing Sobol sequences with better two-dimensional projections.”
• Owen (1995): “Randomly permuted (t,m,s)-nets and (t,s)-sequences.”
• Martinez & Williams (2026): “QMC Methods Enable Extremely Low-Dimensional Deep Generative Models.”

Structs

SobolGenerator

Incremental Sobol sequence generator using Gray-code enumeration.

Functions

halton_point

Compute a single Halton point in [0, 1)^d.

halton_sequence

Generate n Halton points in [0, 1)^d, starting from index = 1.

sobol_scrambled

Generate n Owen-scrambled Sobol points in [0, 1)^d.

sobol_sequence

Generate n Sobol points in [0, 1)^d, starting from index 1 (the origin at index 0 is skipped).