Mathematical Methods for Physics and Engineering : A Comprehensive Guide

27.4 NUMERICAL INTEGRATION

will have a very small variance. Further, any error in inverting the relationship

betweenηandξwill not be important sincef(η)/g(η) will be largely independent

of the value ofη.

As an example, consider the functionf(x)=[tan−^1 (x)]^1 /^2 , which is not analyt-

ically integrable over the range (0,1) but is well mimicked by the easily integrated

functiong(x)=x^1 /^2 (1−x^2 /6). The ratio of the two varies from 1.00 to 1.06 asx

varies from 0 to 1. The integral ofgover this range is 0.619 048, and so it has to

be renormalised by the factor 1.615 38. The value of the integral off(x)from0

to 1 can then be estimated by averaging the value of

[tan−^1 (η)]^1 /^2

(1.615 38)η^1 /^2 (1−^16 η^2 )

for random variablesηwhich are such thatG(η) is uniformly distributed on

(0,1). Using batches of as few as ten random numbers gave a value 0.630 forθ,

with standard deviation 0.003. The corresponding result for crude Monte Carlo,

using the same random numbers, was 0. 634 ± 0 .065. The increase in precision is

obvious, though the additional labour involved would not be justified for a single

application.

Control variates

The control-variate method is similar to, but not the same as, importance sam-

pling. Again, an analytically integrable function that mimicsf(x) in shape has

to be found. The function, known as the control variate, is first scaled so as to

matchfas closely as possible in magnitude and then its integral is found in

closed form. If we denote the scaled control variate byh(x), then the estimate of

θis computed as

t=

∫ 1

0

[f(x)−h(x)]dx+

∫ 1

0

h(x)dx. (27.51)

The first integral in (27.51) is evaluated using (crude) Monte Carlo, whilst the

second is known analytically. Although the first integral should have been ren-

dered small by the choice ofh(x), it is its variance that matters. The method relies

on the following result (see equation (30.136)):

V[t−t′]=V[t]+V[t′]−2Cov[t, t′],

and on the fact that iftestimatesθwhilstt′estimatesθ′using the same random

numbers, then the covariance oftandt′can be larger than the variance oft′,

and indeed will be so if the integrands producingθandθ′are highly correlated.

To evaluate the same integral as was estimated previously using importance

sampling, we take ash(x) the functiong(x) used there, before it was renormalised.

Again using batches of ten random numbers, the estimated value forθwas found

to be 0. 629 ± 0 .004, a result almost identical to that obtained using importance