A New Adaptive Rejection Sampling Method Using Kernel Density Approximations and Its Application to Subset Simulation

Abstract

In stochastic analysis of engineering systems, the task of generating samples according to a target probability distribution involving some performance function of the system response often arises. This paper introduces an adaptive method for rejection sampling that uses adaptive kernel sampling densities (AKSD) as proposal densities for the rejection sampling algorithm in an iterative approach. The AKSD formulation relies on having available (1) a small number of samples from the target density, as well as (2) evaluations of the system performance function over some other sample set. This information is then used to establish the adaptive features of the stochastic sampling involving (1) an explicit optimization of the kernel characteristics for reduction of the computational burden, and so maximizing sampling efficiency, and (2) selection of the exact model parameters to target so that potential problems when forming proposal densities for high-dimensional vectors are avoided. Beyond this theoretical formulation of the adaptive stochastic sampling, its implementation within the context of Subset Simulation (SS) is also demonstrated, with the AKSD method utilized for generating independent conditional failure samples. Additionally, a modified rejection sampling algorithm is proposed for using AKSD in SS that can significantly reduce the required number of simulations of the system model response.