Numerical integration
Named by 2 essays across one field — each of them below, with the objects they name alongside it.
Also named here as quasi-monte carlo — the same set of essays touches all of them, so they are one junction rather than several.
Points on a lattice that see almost nothing
Average a function over the points of a carefully tilted lattice and the error can fall like one over the square of the number of points — far faster than random sampling, and faster than the most evenly spread sequences. The reason is that a lattice rule is blind to only a thin set of frequencies, its dual lattice, and a smooth periodic function has almost nothing there.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Quasi-monte carloConfidence intervalDiscrepancyDual latticeFourier seriesLatticePeriodicityRandomnessScramblingSobol sequenceVariance