Lattice Rules: How Well Do They Measure Up?

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A simple, but often effective, way to approximate an integral over the s-dimensional unit cube is to take the average of the integrand over some set P of N points. Monte Carlo methods choose P randomly and typically obtain an error of 0(N-1/2). Quasi-Monte Carlo methods attempt to decrease the error by choosing P in a deterministic (or quasi-random) way so that the points are more uniformly spread over the integration domain.