The basic idea is to express a small failure probability as a product of larger conditional probabilities by introducing intermediate failure events.
In the actual implementation, samples conditional on intermediate failure events are adaptively generated to gradually populate from the frequent to rare event region.
These 'conditional samples' provide information for estimating the complementary cumulative distribution function (CCDF) of the quantity of interest (that governs failure), covering the high as well as the low probability regions.
The generation of conditional samples is not trivial but can be performed efficiently using Markov chain Monte Carlo (MCMC).
Subset simulation takes the relationship between the (input) random variables and the (output) response quantity of interest as a 'black box'.
For problems where it is possible to incorporate prior information into the reliability algorithm, it is often more efficient to use other variance reduction techniques such as importance sampling.
It has been shown that subset simulation is more efficient than traditional Monte Carlo simulation, but less efficient than line sampling, when applied to a fracture mechanics test problem.
[2] Let X be a vector of random variables and Y = h(X) be a scalar (output) response quantity of interest for which the failure probability
However this is not efficient when P(F) is small because most samples will not fail (i.e., with Y ≤ b) and in many cases an estimate of 0 results.
Subset simulation attempts to convert a rare event problem into more frequent ones.
[3] More generic and flexible version of the simulation algorithms not based on Markov chain Monte Carlo have been recently developed.
As a result, subset simulation in fact produces a set of estimates for b that corresponds to different fixed values of p = P(Y > b), rather than estimates of probabilities for fixed threshold values.
There are a number of variations of subset simulation used in different contexts in applied probability and stochastic operations research[5] [6] For example, in some variations the simulation effort to estimate each conditional probability P(Y > bi | Y > bi−1) (i = 2, ..., m) may not be fixed prior to the simulation, but may be random, similar to the splitting method in rare-event probability estimation.
[7] These versions of subset simulation can also be used to approximately sample from the distribution of X given the failure of the system (that is, conditional on the event
In that case, the relative variance of the (random) number of particles in the final level