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a b s t r a c tThis paper proposes the application of sequential importance sampling (SIS) to the estimation of the probability of failure in structural reliability. SIS was developed originally in the statistical community for exploring posterior distributions and estimating normalizing constants in the context of Bayesian analysis. The basic idea of SIS is to gradually translate samples from the prior distribution to samples from the posterior distribution through a sequential reweighting operation. In the context of structural reliability, SIS can be applied to produce samples of an approximately optimal importance sampling density, which can then be used for estimating the sought probability. The transition of the samples is defined through the construction of a sequence of intermediate distributions. We present a particular choice of the intermediate distributions and discuss the properties of the derived algorithm. Moreover, we introduce two MCMC algorithms for application within the SIS procedure; one that is applicable to general problems with small to moderate number of random variables and one that is especially efficient for tackling high-dimensional problems.

Bayesian updating is a powerful method to learn and calibrate models with data and observations. Because of the difficulties involved in computing the high-dimensional integrals necessary for Bayesian updating, Markov Chain Monte Carlo (MCMC) sampling methods have been developed and successfully applied for this task. The disadvantage of MCMC methods is the difficulty of ensuring the stationarity of the Markov chain. We present an alternative to MCMC that is particularly effective for updating mechanical and other computational models, termed BUS: Bayesian Updating with Structural reliability methods. With BUS, structural reliability methods are applied to compute the posterior distribution of uncertain model parameters and model outputs in general. We propose an algorithm for the implementation of BUS, which can be interpreted as an enhancement of the classical rejection sampling algorithm for Bayesian updating. This algorithm is based on the subset simulation and its efficiency is not dependent on the number of random variables in the model. The method is demonstrated by application to parameter identification in a dynamic system, Bayesian updating of the material parameters of a structural system, and Bayesian updating of a random-field-based FE model of a geotechnical site.

The Transitional Markov Chain Monte Carlo (TMCMC) method is a widely used method for Bayesian updating and Bayesian model class selection. The method is based on successively sampling from a sequence of distributions that gradually approach the posterior target distribution. The samples of the intermediate distributions are used to obtain an estimate of the evidence, which is needed in the context of Bayesian model class selection. We discuss the properties of the TMCMC method and propose the following three modifications to the TMCMC method: (1) The sample weights should be adjusted after each MCMC step. (2) A burn-in period in the MCMC sampling step can improve the posterior approximation. (3) The scale of the proposal distribution of the MCMC algorithm can be selected adaptively to achieve a near-optimal acceptance rate. We compare the performance of the proposed modifications with the original TMCMC method by means of three example problems. The proposed modifications reduce the bias in the estimate of the evidence, and improve the convergence behavior of posterior estimates.