Importance Sampling of Nonlinear Structures Using Adapted Process

Abstract

The reliability analysis of nonlinear-hysteretic structures subjected to uncertain earthquake excitations is an important but challenging problem in stochastic structural dynamics and performance-based earthquake engineering. The ‘first passage probability’, i.e., the probability that some response quantities of interest (e.g., interstory drifts) exceed specified threshold levels, is often of concern. Direct Monte Carlo simulation is a versatile method for estimation, but it is not efficient when the failure probability is small, which is often encountered in engineering applications. Importance sampling is a popular variance reduction technique where a change of distribution is applied in order to achieve a higher failure rate in the samples and hence variance reduction. Its success hinges on a proper choice of a user-defined ‘importance sampling distribution’. A popular choice makes use of ‘design point excitations’ that are local most probable points within the failure region in the stochastic load space. Design point excitations participate in the excitation to create large response, leading to a larger failure rate in the samples. Their use has lead to tremendous variance reduction for linear systems. For nonlinear-hysteretic systems, however, recent research shows that the gain in efficiency is much less than their linear counterpart, essentially because plastic excursions cause random phase shifts in the response that de-synchronize it from the design point excitations, undermining their effectiveness in creating large response. This work presents a new approach for constructing the importance sampling density for estimating the first passage probability of nonlinear-hysteretic systems subjected to stochastic earthquake excitations. Instead of using fixed design point excitations, the importance sampling distribution is constructed using an ‘adapted process’ whose future action can be updated based on information up to the present. Choosing the adapted process involves designing an adaptive control force algorithm in a stochastic environment that targets to drive the response to first passage failure based on information up to the present. Theoretical and computational issues related to the proposed importance sampling method shall be investigated. A stochastic control algorithm for generating the adapted process is presented and its variance reduction capability is appraised.