Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

averaged:

t=

1

n

∑n

i=1

f(ξi). (27.49)

Stratified sampling

Here the range ofxis broken up intoksubranges,

0=α 0 <α 1 <···<αk=1,

and crude Monte Carlo evaluation is carried out in each subrange. The estimate

E[t] is then calculated as

E[t]=

∑k

j=1

∑nj

i=1

αj−αj− 1

nj

f

(

αj− 1 +ξij(αj−αj− 1 )

)

. (27.50)

This is an unbiased estimator ofθwith variance

σt^2 =

∑k

j=1

αj−αj− 1

nj

∫αj

αj− 1

[f(x)]^2 dx−

∑k

j=1

1

nj

[∫

αj

αj− 1

f(x)dx

] 2

.

This variance can be made less than that for crude Monte Carlo, whilst using

the same total number of random numbers,n=

∑

nj, if the differences between

the average values off(x) in the various subranges are significantly greater than

the variations infwithin each subrange. It is easier administratively to make all

subranges equal in length, but better, if it can be managed, to make them such

that the variations infare approximately equal in all the individual subranges.

Importance sampling

Although we cannot integratef(x) analytically – we would not be using Monte

Carlo methods if we could – if we can find another functiong(x)thatcanbe

integrated analytically and mimics the shape offthen the variance in the estimate

ofθcan be reduced significantly compared with that resulting from the use of

crude Monte Carlo evaluation.

∫Firstly, if necessary the functiongmust be renormalised, so thatG(x)=
x
0 g(y)dyhas the propertyG(1) = 1. Clearly, it also has the propertyG(0) = 0.
Then, since

θ=

∫ 1

0

f(x)

g(x)

dG(x),

it follows that finding the expectation value off(η)/g(η) using a random number

η, distributed in such a way thatξ=G(η) is uniformly distributed on (0,1), is

equivalent to estimatingθ. This involves being able to find the inverse function

ofG; a discussion of how to do this is given towards the end of this subsection.

Ifg(η) mimicsf(η) well,f(η)/g(η) will be nearly constant and the estimation