Reference S.P. Mudur and S.N. Pattanik, "Monte Carlo Methods for Computer Graphics", State of the Art Reports, Eurographics'93, Barcelona, Sept. 6-10, 1993.:
The Monte Carlo method is a numerical method for solving mathematical problems using stochastic sampling. Look at a simple Monte Carlo technique known as the "hit or miss" method.
Assume we have a function f(x) and we want to find the integral of f(x) from x = 0.0 to x = 1.0. We can draw a unit square, choose N random points within the square. For each point we determine if it lies below the f(x) curve, if it does then we increment a counter Hit. Then the area under the curve is given by Hit/N. N = Hit + Miss.
Note two features of this method. First, an algorithm is written to perfom one random test and then repeated N times. Second, the statistical measure of error is proportional to (D/N)^1/2 where D depends on the particular Monte Carlo technique used.
Another method for finding an integral is called Monte Carlo quadrature. This consists of the following steps:
1. Let G = òg(x) dx be the integral to be evaluated. 2.
Rewrite g(x) as f(x)g'(x) where f(x) is a probability distribtion
function and g'(x) = g(x)/f(x). Note that for a uniform random
distribution in the interval [a,b] the pdf = 1/(b-a). 3. Sample f
(x) for a xi, i.e. obtain some xi from f(x). 4. For each such
sample xi evaluate g'(x). 5. Perform steps 3 and 4 N times. Then
One value of the variance that is frequently used is the
sample variance S2 where: S2 = 1/(N-1)SNi=1 (g'(x) -
So we can improve the accuracy by increasing the number of samples N or by decreasing the variance. We can decrease the variance by eliminating potential samples with unusually high or low values.
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