Adaptive hybrid conditional expectation approaches for reliability estimation

Abstract Conditional expectation is a simulation procedure that can be used to estimate the reliability of structures. It involves selecting an important random variable, using Monte Carlo simulation to generate sets of sample outcomes of the remaining “unimportant” random variables, estimating the conditional failure probability for each set of sample outcomes and using the average conditional failure probability as the estimate of the failure probability. This direct conditional expectation approach can be extremely effective, but only if the selected important random variable is much more important than the combined effect of the remaining random variables. In this paper, the concept of using importance sampling to generate the unimportant random variables is introduced. With a good sampling density, this greatly improves the efficiency for cases in which the selected important random variable does not dominate. To select such a density, however, one needs informations about the important regions of the failure domain. Typically, such information is not available a priori , and the direct application of this concept is limited. This concept can, however, be used in an adaptive format and two such adaptive hybrid conditional expectation approaches are developed in this paper. They are based on the idea that, as one generates sample outcomes and calculates the conditional failure probabilities at these outcomes, the knowledge of the failure domain increases. If one keeps modifying the importance sampling density to reflect this increasing state of knowledge, one can develop a good sampling density and simultaneously estimate the failure probability efficiently. Examples are used to demonstrate the efficacy of these approches and study the influence of factors like failure probability level and relative importance of random variables. Also discussed are issues such as the efficiency of the new methods (compared to FORM) and the applications to problems involving unions and intersections.