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However, when one wants to estimate an integral for a ‘non-well-behaved’ it does not need each time a biased roulette. To do this the integral of equation above can be written as b f (x) I (x) (x)dx (3.14) a where ρ(x) is a probability density function that is large where it is thought that the function will be large. The integral can now be approximated by choosing values of the integration variable randomly from ρ(x) in the range [a, b], instead of from the uniform distribution, and averaging over the values of ƒ (xi)/ρ (xi) that are obtained 1n f(x) I i (3.15) ni1 (xi) this formula is the same as equation (3.13) in the case of a uniform distribution follows because the probability distribution function for the uniform distribution is 1/(b-a). The use of a function ρ in this way to enhance sampling in certain regions is known as importance sampling. Choosing perfect weight function is same as solving integral The stochastic method outlined above cannot usually compete with numerical methods of the type given in equation (3.12) if there is a small number of integration variables. However, the number of function evaluations required by simple discretization schemes for the estimation of an integral becomes prohibitively large as the number of dimensions, Ndim, increases. To see this, suppose that n points are chosen for the discretization in each direction, and then the number of function evaluations required is Ndim. It is in these cases that stochastic methods are often the only realistic approaches for tackling the problem. 41PDF Image | MOLECULAR SIMULATION STUDIES IN THE SUPERCRITICAL REGION
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