Optimal Importance‐Sampling Density Estimator

Importance‐sampling technique has been used in recent years in conjunction with Monte Carlo simulation method to evaluate the reliability of structural systems. Since the efficiency of the importance‐sampling method depends primarily on the choice of the importance‐sampling density, the use of the kernel method to estimate the optimal importance‐sampling density is proposed. This method deviates from the current practice of prescribing the importance‐sampling density from a given parametric family of density functions. Instead, the data obtained from an initial Monte Carlo run are utilized to determine the required importance‐sampling density. The kernel method yields unbiased estimates of the probability of failure. Two measures are developed to quantify the efficiency of the kernel method relative to the basic Monte Carlo method. The first measure, called the marginal efficiency, is used as an indicator of the effectiveness of the kernel method, whereas the second measure, the overall efficiency, defines the advantage of the kernel method over the basic Monte Carlo method. Finally, a variety of example problems are used to examine the characteristics of the proposed kernel method and its efficiency over the basic Monte Carlo method.