A product quadrature rule built from a one-dimensional rule of order has total error…
PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Computational Physics · Monte Carlo and lattice simulation of physical systems
Computational Physics, Ch Drawing numbers: random walks and Monte Carlo
Problem
A product quadrature rule built from a one-dimensional rule of order p has total error that falls as N−p/d when its N nodes are spread evenly over d dimensions (so N1/d nodes lie along each axis). A Monte Carlo estimate of the same integral has error that falls as N−1/2, whatever d is. Take p=6 throughout, corresponding to a sixth-order one-dimensional rule.
(a) By comparing the exponents of N in the two error laws, show that they fall at the same rate only at one particular dimension d∗, and give d∗ in terms of p.
(b) Evaluate d∗ for p=6.
(c) For d=10 dimensions, estimate the number of quadrature nodes NQ needed to reach a total error of 1%, using NQ−p/d=0.01, and compare it with the NMC=104 points a Monte Carlo estimate needs for the same 1% accuracy, from NMC−1/2=0.01. State which method costs fewer evaluations at d=10.
(d) Repeat part (c) at d=16 dimensions and state which method costs fewer evaluations there.