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Three Importance Sampling MethodsUNIPASS™ offers 3 different Importance Sampling Methods: Monte Carlo Importance Sampling, Sphere-based Importance Sampling, and Directional Importance Sampling Simulation. In these methods, the true limit-state functions are used and sampling density is skewed toward the MPPs. The importance sampling method is used to improve the efficiency of the Monte Carlo method. Monte Carlo Importance Sampling The sampling points are generated by using the density function which uses the MPP as the mean point in the standard normal space. An example is given in the following figure. Both the point and interval estimators of failure probability are used in this method. The number of sampling points for Monte Carlo based importance sampling method is much less than the conventional Monte Carlo simulation method.
Illustration of Monte Carlo Importance Sampling
Sphere-based ISMThe simplest technique of the importance sampling method is shown in Figure 1. In this figure, the points located inside the sphere of radius R (in standard normal space) are assumed to be in the safe domain. Therefore, no limit-state function calculation is needed for these points. Only the points that are sampled outside the radius of the sphere are used to calculate the limit-state function values. The most appropriate radius of the sphere should be less than the generalized reliability index (calculated by either FORM or SORM).
Figure 1 Sphere Based Importance Sampling Method
Because the failure probability and the associated sensitivities are of interest, the sphere-based importance sampling method can be used to perform these analyses in the same format as Monte Carlo simulation. However, because the limit-state function calculations are omitted for the sampling points inside the sphere, one cannot obtain the entire CDF/PDF of the limit-state function available in the Monte Carlo simulation. Theoretically, the CDF/PDF of limit-state function can be obtained first by identifying the MPPs for the various levels of limit state function (same as in FORM/SORM), then second by using the sphere based importance sampling method to calculate the failure probability for each level of limit-state function. This approach of CDF/PDF analysis, however, is very time consuming and is not recommended. The efficiency of this method reduces as the number of random variables increases.
Directional ISMTo improve the efficiency of the directional simulation method, the directional importance sampling method is developed by generating the unit directions according to a sampling density function which has higher probability density toward the MPPs (see Figure 2. Consequently, the sampled unit directions are concentrated toward the direction of MPP and then have a higher chance of hitting the failure surface at the area with higher probability density. This method can significantly increase the efficiency of failure probability calculation over the conventional direction simulation. Either the true or the approximated limit-state functions can be applied in this method.
Figure 2. Example of Directional Importance Sampling Method
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Last Updated 02/08/10
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