Scrambling
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Also named here as sobol sequence — the same set of essays touches all of them, so they are one junction rather than several.
An error bar for points that are not random
Evenly spread points integrate far better than random ones and give no error bar; random points give an error bar and integrate badly. Randomise the even points themselves — shift a lattice by a random vector, or scramble the digits of a Sobol' sequence — and both are kept: an unbiased estimate, a confidence interval from a handful of repeats, and an error that falls faster than any deterministic set's.
What a jump costs evenly spread points
Scrambled Sobol' points integrate a smooth function with an error that falls like N to the power −3/2. Put a jump in the integrand along a tilted line and the exponent halves, to −3/4, because about √N of the N boxes the points fill are cut by the jump and each of those is left to chance. A jump along an axis costs almost nothing, the cost grows with the length of the jump and the dimension, and integrating across the jump before sampling gives the whole rate back.
Named alongside it
The objects these essays reach for when they reach for this one.
Numerical integrationQuasi-monte carloSobol sequenceConfidence intervalConvergence rateCurse of dimensionalityDiscrepancyRandomnessVariance