Difficulties of Monte Carlo Eigenvalue Perturbation

In a straightforward Monte Carlo approach, a perturbation effect would be evaluated by performing two independent simulations for the perturbed and unperturbed systems. The perturbation result is then calculated by taking the difference between these two independent simulations. In this approach, the variance of the result increases as the perturbation becomes smaller, and hence the differential effect becomes difficult to observe due to a very large uncertainty [Rie86]. This can be explained by deriving the expression of the variance of a differential effect.

Let x be a random variable with an associated distribution and an unknown mean. We denote the function X(x₁, x₂, x₃,...) as an estimator of the unknown mean. The expected value of X is given by,

E(X) ,

and the variance by,

V(X) = E(X - E(X))² .

X may depend on several variables and parameters. Let us assume a small variation in a parameter (or parameters) resulting in a different unknown mean, which is estimated by the function X^*, with expectation value,

E(X^*),

and variance,

V(X^*) = E(X^* - E(X^*))² .

Therefore the expectation of the differential effect due to the small variation in the parameter(s) is given by,

E(X - X^*) = E(X) - E(X^*) ,

and the variance of the differential effect by,

V(X-X^*) = V(X) + V(X^*) - 2 cov(X,X^*) .

If the two estimates E(X) and E(X^*) are obtained by two independent Monte Carlo simulations (i.e., in the absence of any correlations between the two simulations), the covariance term in equation (3.6) vanishes and the variance is

V(X-X^*) = V(X) + V(X^*).

This is larger than either of the variances separately and may be larger than the expression given in equation (3.6) [Spa69]. Hence, it is possible to reduce the variance of the differential effect, specially for small perturbations, if one has a strong positive correlation between the unperturbed and perturbed Monte Carlo simulations.