BAYESIAN MODEL UPDATING USING SEQUENTIAL GAUSS-NEWTON MCMC ALGORITHM

Abstract eng:
Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods face two central difficulties when solving largescale Bayesian model updating problems; First, the large size of forward models makes the evaluation of the posterior PDF at any point in the parameter space computationally involved. Different means are used in the literature to reduce the computational cost of the forward simulations or in general the MCMC sampling. Second, the support of the posterior PDF often has a complex geometry which renders its sampling very wanting. Transitional Markov Chain Monte Carlo (TMCMC) algorithm [1] is originally developed to alleviate sampling from difficult PDFs (i.e., PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). The key idea behind this algorithm is to replace sampling from difficult distributions by sampling from a sequence of intermediate distributions converging to the posterior distribution that each can be readily sampled based on the samples from the previous intermediate distribution. In each level, an importance resampling technique is applied to the samples from the previous intermediate distribution in order to provide initial samples for the next level where a Markov chain is started from each of these initial samples using the Metropolis-Hastings (M-H) algorithm with a local Gaussian proposal PDF centered at the current sample and a covariance matrix that is a weighted covariance of the samples from the previous intermediate distribution. Although this approach facilitates sampling from very peaked, multimodal and flat PDFs, it becomes inefficient as the dimension of the parameter space grows. This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. Using the multinomial importance resampling within the TMCMC algorithm leads to a problem called sample impoverishment meaning that the diversity of the samples decreases as the simulation level increases. In this paper, we first propose to use the systematic resampling scheme to enhance the statistical efficiency of the TMCMC algorithm. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated simulated data from a three story building with Masing hysteretic model subject to a seismic excitation.

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