Monte Carlo methods

Monte Carlo methods

Monte Carlo methods

are the systematic use of samples of

random numbers

in order to

estimate

parameters

of an unknown

distribution

by statistical

simulation

. Methods based on this principle of random sampling are indicated in cases where the dimensionality and/or

complexity

of a problem make straightforward numerical solutions impossible or impractical. The method is ideally

adapted

to computers, its applications are varied and many, its main drawbacks are potentially slow convergence (large

variances

of the results), and often the difficulty of estimating the statistical error (variance) of the result.

Monte Carlo

problems can be formulated as integration of a

function

f

=

(

x

→

)

over a (multi-dimensional) volume

V

, with the result

∫

V

f

⁢

𝑑

V

=

V

⁢

f

¯

where

f

¯

, the average of

f

, is obtained by exploring randomly the volume

V

.

Most easily one conceives a simple (and inefficient) hit-and-miss Monte Carlo: assume, for example, a three-dimensional volume

V

to be bounded by surfaces difficult to intersect and describe analytically; on the other hand, given a point

(

x

,

y

,

z

)

∈

V

, it is easy to decide whether it is inside or outside the boundary. In this case, a simply bounded volume which fully includes

V

can be sampled uniformly (the components

x

,

y

,

z

are generated as random numbers with uniform

probability density function

), and for each point a weight is computed, which is zero if the point is outside

V

, 1 otherwise. After

N

random numbers,

n

(

≤

N

) will have been found inside

V

, and the ratio

n

/

N

is the fraction of the sampled volume which corresponds to

V

.

Another method, crude Monte Carlo, may be used for integration: assume now the volume

V

is bounded by two functions

z

⁢

(

x

,

y

)

and

z

′

⁢

(

x

,

y

)

, both not integrable, but known for any

x

,

y

, over an interval

Δ

⁢

x

and

Δ

⁢

y

. Taking random pairs

(

x

,

y

)

, evaluating

Δ

z

=

|

z

(

x

,

y

)

=

z

′

(

x

,

y

)

|

at each point, averaging to

⟨

Δ

⁢

z

⟩

and forming

Δ

⁢

x

⁢

Δ

⁢

y

⁢

⟨

Δ

⁢

z

⟩

, gives an approximation of the volume (in this example, sampling the area with

quasirandom numbers

or, better, using standard

numerical integration

methods will lead to more precise results).

Often, the function to be sampled is, in fact, a probability density function , e.g. a matrix element in phase space. In the frequent case that regions of small values of the probability density function dominate, unacceptably many points will have to be generated by crude Monte Carlo, in other words, the convergence of the result to small statistical errors will be slow. Variance reducing techniques will then be indicated, like importance sampling or

stratified sampling

. For more reading, see [Press95], [Hammersley64], [Kalos86].

References

•

Originally from The Data Analysis Briefbook
(

http://rkb.home.cern.ch/rkb/titleA.html

http://rkb.home.cern.ch/rkb/titleA.html)

Press95

W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in C, Second edition, Cambridge University Press, 1995. (The same book exists for the

Fortran

language

). There is also an Internet version which you can work from.

Hammersley64

J.M. Hammersley and D.C. Handscomb, Monte Carlo Methods, Methuen, London, 1964.

Kalos86

M.H. Kalos and P.A. Whitlock, Monte Carlo Methods, Wiley, New York, 1986.

Title

Monte Carlo methods

Canonical name

MonteCarloMethods

Date of creation

2013-03-22 12:07:10

Last modified on

2013-03-22 12:07:10

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

7

Author

akrowne (2)

Entry type

Topic

Classification

msc 65C05

Classification

msc 11K45

Synonym

Monte Carlo

Synonym

hit-and-miss Monte Carlo

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