Computational Physics

298 The Monte Carlo method

Several methods have been devised to reduce the error of the Monte Carlo integ-
rationmethod;foradiscussionseeRef.[ 9 ]. We give a brief overview here. In
order to distribute the points more homogeneously over the integration hypercube,
it is possible to subdivide the latter into smaller, equally sized subvolumes and
to choose an equal number of random points in each subvolume. This is called
‘stratified Monte Carlo’.
In many practical cases, the contributions to the integral from different regions
in the integration volume vary strongly. The MC method samples the function
homogeneously, so if the significant contributions to the integral come primarily
from a small region within the integration volume, there will be only a few MC
points for sampling the function there, leading to large statistical errors. This effect
shows up in (10.4) as a large variance offover the volume. In order to reduce this
contribution to the error, points are concentrated in the regions wherefhappens to
be large (in absolute value). More precisely, letρ(x)be a function on[a,b]which
has more or less the shape offin the sense thatf/ρis approximately constant over
the interval. We furthermore requireρto be normalised:
∫b

a

dxρ(x)=1. (10.5)

We write the integral overfas follows:
∫b

a

dxf(x)=

∫b

a

dxρ(x)

[

f(x)

ρ(x)

]

. (10.6)

The function in square brackets is reasonably flat (asρis chosen to have more or
less the shape off) and the weightρ(x)in front of this function can be included
in the integral by choosing the random pointsxiwith distributionρ(x). The Monte
Carlo result for the integral is then still given by (10.2). This reduces the error in
the result considerably as we can see by evaluating the variance (10.4) for this case:

σ^2 =

1

N







∫b

a

[

f(x)

ρ(x)

] 2

ρ(x)dx−

(∫

b

a

[

f(x)

ρ(x)

]

ρ(x)dx

) 2 



. (10.7)

If we are indeed able to chooseρsuch thatf/ρis approximately constant, then as a
result of (10.5) the expression in the braces will indeed be much smaller than in the
‘crude sampling’ case. This method is calledimportance sampling Monte Carlo.
Adaptive Monte Carlomethods also aim at concentrating the sampling points in
those regions wherefcontributes significantly to the integral, but these methods
locate these regions by probing the function at random points and require no a priori
knowledge on the functionfas in the case of importance sampling.
Note that MC integration is not susceptible to correlations in the random gener-
ator. Correlations influence the order in which the pointsxiare generated but this