Development of a two-phase adaptive MCMC method for efficient Bayesian model updating of complex dynamic systems

Abstract

The fundamental problem of Bayesian model updating is identifying the posterior probability density function (PDF) of uncertain model parameters. Markov chain Monte Carlo (MCMC) has been recognized to be potential for this problem. However, the main difficulty is that many samples need to be generated first to locate the regions of high probability. This could be time consuming for complex dynamic systems. A novel two-phase adaptive MCMC method is developed in this work to tackle this difficulty. A parameter-space search algorithm is used to do a systematic search to find output-equivalent points, which are in the neighborhood of the regions of high probability. To generate samples of uncertain model parameters in these regions for describing the shape of the posterior PDF, a Markov chain is constructed with these output-equivalent points as seeds. This is done by the proposed weighted MCMC algorithm based on the modified Metropolis-Hasting (MH) algorithm. To enhance the efficiency of the algorithm, a new proposal PDF is developed using the seeds. The covariance matrix of the proposal PDF is adaptively updated with the latest set of generated samples. Furthermore, the optimal prediction-error variance as a function of model parameters is analytical derived and its values at the seeds are adaptively used to estimate the acceptance ratio in the modified MH algorithm. The proposed weighted MCMC algorithm generates samples in the region of high probability adaptively without going through multiple levels, which are computationally demanding. The posterior uncertainties in model updating can be rigorously considered by the generated samples with their weights proportional to their posterior PDF values. To verify the proposed method, numerical simulation of a two-story shear building with the challenging locally identifiable and unidentifiable cases was conducted. Further verification was also conducted by updating a finite element model of a full-scaled coupled building system utilizing measured vibration data from field tests.