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The principle is also illustrated in Figure 11.4. Fxj (xi)

z ‘ci Zji

Simulated sample

Figure 11.4 Principle for simulation of a random variable.

This process is the continued until all components of the vector fi have been generated.

Importance Sampling Simulation Method As already mentioned the problem in using Equation (11.45) is that the sampling function fa(x) typically is located in a region far away from the region where the indicator function /[g(x) 5 0] attains contributions. The success rate in the performed simulations is thus low. In practical reliability assessment problems where typical failure probabilities are in the order of 10 3 — le this in turn leads to the effect that the variance of the estimate of failure probability will be rather large unless a substantial amount of simulations are performed. To overcome this problem different variance reduction techniques have been proposed aiming at, with the same number of simulations to reduce the variance of the probability estimate. In the following we shall briefly consider one of the most commonly applied techniques for variance reduction in structural reliability applications, namely the importance sampling method.

The importance sampling method takes basis in the utilisation of prior information about the domain contribution to the probability integral, i.e. the region that contributes to the indicator function. Let us first assume that we know which point in the sample space x. contributes the most to the failure probability. Then by centring the simulations on this point, the important point, we would obtain a higher success rate in the simulations and the variance of the estimated failure probability would be reduced. Sampling centred on an important point may be accomplished by rewriting Equation (11.42) in the following way: () Pf = lig(x) Olfx(x)clx = l[g(x) 5 0 jfxx f,(x)dx (11.48)fs (x)

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