Adaptive kernel method of importance sampling

The Monte Carlo simulation is one of the most powerful methods for calculating the failure probability of structures. However, for some problems, i.e., the value of failure probability is very small, the required computational cost to reach an accurate result may be expensive. Therefore, the importance sampling method has been developed with an aim to reduce the statistical error inherent in Monte Carlo methods. The key issue involved is the construction of the importance sampling density function. In this regard, a simple Kernel method was proposed. The main drawback of the simple Kernel method is that direct Monte Carlo simulation is required to obtain the importance sampling density function. On the other hand, an adaptive sampling concept in which a starting point which is the mean of each variate is chosen to increase the efficiency of importance sampling was suggested. However, when the failure probability of a system is very small, it is unlikely to obtain the design point. The present study proposes a new algorithm called the adaptive Kernel method which combines and modifies some of the ideas from adaptive sampling and simple Kernel method to evaluate the structural reliability.; The essence of this algorithm is to select an appropriate "design point" from which the importance sampling density can be generated efficiently. The idea of the adaptive sampling is modified here to obtain the design point. The Kernel sampling density function can then be constructed using failure points generated from the design point. Finally, the failure probability can be calculated by sampling from the Kernel density function.; A number of examples were examined to illustrate the application and effectiveness of the proposed adaptive Kernel method for time invariant and time variant problems.