Mathematical Methods for Physics and Engineering : A Comprehensive Guide

NUMERICAL METHODS

sampling, in both value and precision. Since we knew already thatf(x)andg(x)

diverge monotonically by about 6% asxvaries over the range (0,1), we could

have made a small improvement to our control variate by scaling it by 1.03 before

using it in equation (27.51).

Antithetic variates

As a final example of a method that improves on crude Monte Carlo, and one that

is particularly useful when monotonic functions are to be integrated, we mention

the use of antithetic variates. This method relies on finding two estimatestand

t′ofθthat are strongly anticorrelated (i.e. Cov[t, t′] is large and negative) and

using the result

V[^12 (t+t′)] =^14 V[t]+^14 V[t′]+^12 Cov[t, t′].

For example, the use of^12 [f(ξ)+f(1−ξ)] instead off(ξ) involves only twice

as many evaluations off, and no more random variables, but generally gives

an improvement in precision significantly greater than this. For the integral of

f(x)=[tan−^1 (x)]^1 /^2 , using as previously a batch of ten random variables, an

estimate of 0. 623 ± 0 .018 was found. This is to be compared with the crude

Monte Carlo result, 0. 634 ± 0 .065, obtained using the same number of random

variables.

For a fuller discussion of these methods, and of theoretical estimates of their

efficiencies, the reader is referred to more specialist treatments. For practical imple-

mentation schemes, a book dedicated to scientific computing should be consulted.§

Hit or miss method

We now come to the approach that, in spirit, is closest to the activities that gave

Monte Carlo methods their name. In this approach, one or more straightforward

yes/no decisions are made on the basis of numbers drawn at random – the end

result of each trial is either a hit or a miss! In this section we are concerned

with numerical integration, but the general Monte Carlo approach, in which

one estimates a physical quantity that is hard or impossible to calculate directly

by simulating the physical processes that determine it, is widespread in modern

science. For example, the calculation of the efficiencies of detector arrays in

experiments to study elementary particle interactions are nearly always carried

out in this way. Indeed, in a normal experiment, far more simulated interactions

are generated in computers than ever actually occur when the experiment is

taking real data.

As was noted in chapter 2, the process of evaluating a one-dimensional integral
∫b
af(x)dxcan be regarded as that of finding the area between the curvey=f(x)

§e.g. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery,Numerical Recipes in C: The

ArtofScientificComputing, 2nd edn (Cambridge: Cambridge University Press, 1992).