Quasi-Monte Carlo: Low-Discrepancy vs Pseudorandom Integration

Sampling Method (3D view)

Simulation

Points per frame 10

Show cube edges

Live Estimate

Samples N

π (true)

3.14159

π̂ pseudorandom

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error

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π̂ Halton

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error

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How it works

A Monte Carlo integrator estimates a volume by sampling points uniformly in a bounding box and counting the fraction that land inside the target shape. Here the box is the cube [-1,1]³ (volume 8) and the target is the inscribed unit sphere (volume 4π/3), so:

P(inside) = (4π/3) / 8 = π/6
π̂ = 6 · (points inside sphere) / (total points)

Plain pseudorandom sampling (each coordinate drawn independently from Math.random()) has estimator error that shrinks only as O(1/√N) — points clump and leave gaps by chance, so quadrupling N only halves the error.

A low-discrepancy sequence avoids that randomness. This simulator uses the Halton sequence: each coordinate is generated by the van der Corput construction in a different prime base (2, 3, 5), which deterministically fills the cube more evenly than random draws:

vdC(i, b) = Σ dₖ(i) · b^(-k-1)   (digits of i in base b, reversed after the point)
Halton point i = ( vdC(i,2), vdC(i,3), vdC(i,5) )

Because it avoids clustering, Halton's discrepancy (a measure of how unevenly points cover the space) shrinks close to O(1/N) for smooth integrands like a sphere's indicator — visibly outrunning pseudorandom sampling's error as N grows, which is exactly why quasi-Monte Carlo methods are used in high-dimensional finance and rendering integrals where every extra sample is expensive.

Pseudorandom / Halton buttons — choose which sequence's points are drawn in the 3D view (both estimators keep running underneath either way, so the error comparison stays live).
Points per frame — how many new samples are drawn each animation frame.
Play/Pause, Reset — control and restart the sampling run.

Inside sphere Outside sphere

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