The mean response of a detector to six dimensionless manufacturing tolerances

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Probability, statistics and stochastic methods · Numerical and computational mathematics

Error, Chance and the Machine, Ch Monte Carlo integration and variance reduction

Problem

The mean response of a detector to six dimensionless manufacturing tolerances, each scaled to lie in

[0,1]

and jointly uniform on the unit cube, is

I=\int f(\mathbf{x})\,\mathrm{d}^6x

, where the response

f

is dimensionless and smooth. A pilot run shows that

f

has variance

\sigma_f^2=0.40

over the cube and that its mean square gradient is

\int|\nabla f|^2\,\mathrm{d}^6x=6.0

. The budget is

N=10^6

evaluations of

f

. ((a)) Find the standard error of plain Monte Carlo sampling. ((b)) The cube is cut into

K^6

equal cells with one uniformly random point in each cell, using all

N

points. Find

K

, and estimate the variance and standard error of the stratified estimate and the factor by which the variance is reduced. ((c)) By what factor does the estimated stratified variance fall if

K

is doubled? Compare with the factor for plain sampling at the same

N

, and state the exponent

b

N^{-b}

for the stratified variance. ((d)) A Halton sequence is used instead. The Koksma-Hlawka bound for smooth

f

is proportional to

D_N^*=O\big((\ln N)^d/N\big)

. Evaluate

(\ln N)^d/N

N=10^6

and

d=6

, compare it with the plain-sampling scale

N^{-1/2}

, and find the number of points above which it falls below

N^{-1/2}

. Say what the bound promises at

N=10^6

, and how an error bar could be attached to the quasi-Monte Carlo estimate.

Browse the problem library