variance, and probability of failure can then be calculated from the results of simulations.
This method is also called direct MCS or MCS with simple random sampling (SRS).
Direct MCS is simple to implement; is robust; and is accurate with sufficiently large
samples. But the usefulness of direct MCS in reliability analysis is quite limited because
of its relatively low efficiency. For example, the probability of failure in engineering
applications is usually very small, thus the number of limit state function evaluations
required to obtain acceptable accuracy using direct MCS is very large (Chapter 5), which
makes direct MCS very time-consuming. Direct MCS is usually used as a benchmark to
verify the accuracy and compare the efficiency of other methods using approximation
concepts.
To improve the accuracy and efficiency of simple random sampling, various
simulation methods using Variance Reduction Techniques (VRT) have been developed to
reduce the variance of the output random variables.
Monte Carlo Simulation Using Variance Reduction Techniques
Rubinstein (1981) and in Melchers (1999) gave good overviews of VRT for general
Monte Carlo sampling. The VRT can be classified into different categories, such as
sampling method, correlation method, conditional expectation method, and specific
method. Sampling methods reduce the variance of the output by constraining samples to
be representative of (or distorting the samples to emphasize the important region of) the
performance function. Commonly used sampling methods include importance sampling
(Harbitz 1986), adaptive sampling, stratified sampling, Latin Hypercube sampling, and
spherical sampling. Correlation methods use techniques to achieve correlation among
random observations, functions, or different simulations to improve the accuracy of the
estimators. Some commonly used techniques are antithetic variate, common random